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\section*{Introduction to Cobordism and Complex Oriented Cohomology}


\vspace{.5em} \hrule \vspace{.5em}
$\,$

This page collects introductory seminar notes to the concepts of [[generalized (Eilenberg-Steenrod) cohomology theory]], basics of [[cobordism theory]] and [[complex oriented cohomology]].

$\,$

\emph{The [[category]] of those [[generalized cohomology theories]] that are equipped with a universal ``[[complex oriented cohomology theory|complex]] [[orientation in generalized cohomology|orientation]]'' happens to unify within it the abstract structure theory of [[stable homotopy theory]] with the concrete richness of the [[differential topology]] of [[cobordism theory]] and of the [[arithmetic geometry]] of [[formal group laws]], such as [[elliptic curves]]. In the seminar we work through classical results in [[algebraic topology]], organized such as to give in the end a first glimpse of the modern picture of [[chromatic homotopy theory]].}

$\,$

For background on [[stable homotopy theory]] see \emph{[[Introduction to Stable homotopy theory]]}.

For application to/of the [[Adams spectral sequence]] see \emph{[[Introduction to the Adams Spectral Sequence]]}

$\,$


\vspace{.5em} \hrule \vspace{.5em}
$\,$

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{S1GeneralizedCohomology}{Generalized cohomology}\dotfill \pageref*{S1GeneralizedCohomology} \linebreak
\noindent\hyperlink{GeneralizedHomologyAndCohomologyFunctors}{Generalized cohomology functors}\dotfill \pageref*{GeneralizedHomologyAndCohomologyFunctors} \linebreak
\noindent\hyperlink{reduced_cohomology}{Reduced cohomology}\dotfill \pageref*{reduced_cohomology} \linebreak
\noindent\hyperlink{unreduced_cohomology}{Unreduced cohomology}\dotfill \pageref*{unreduced_cohomology} \linebreak
\noindent\hyperlink{RelationBetweenUnreducedAndReducedCohomology}{Relation between unreduced and reduced cohomology}\dotfill \pageref*{RelationBetweenUnreducedAndReducedCohomology} \linebreak
\noindent\hyperlink{generalized_homology_functors}{Generalized homology functors}\dotfill \pageref*{generalized_homology_functors} \linebreak
\noindent\hyperlink{multiplicative_cohomology_theories}{Multiplicative cohomology theories}\dotfill \pageref*{multiplicative_cohomology_theories} \linebreak
\noindent\hyperlink{BrownRepresentabilityTheorem}{Brown representability theorem}\dotfill \pageref*{BrownRepresentabilityTheorem} \linebreak
\noindent\hyperlink{BrownRepresentabilityTraditional}{Traditional discussion}\dotfill \pageref*{BrownRepresentabilityTraditional} \linebreak
\noindent\hyperlink{BrownRepresentabilityAppliedToOrdinaryCohomology}{Application to ordinary cohomology}\dotfill \pageref*{BrownRepresentabilityAppliedToOrdinaryCohomology} \linebreak
\noindent\hyperlink{homotopytheoretic_discussion}{Homotopy-theoretic discussion}\dotfill \pageref*{homotopytheoretic_discussion} \linebreak
\noindent\hyperlink{S2MilnorExactSequence}{Milnor exact sequence}\dotfill \pageref*{S2MilnorExactSequence} \linebreak
\noindent\hyperlink{Lim1}{$Lim^1$}\dotfill \pageref*{Lim1} \linebreak
\noindent\hyperlink{ThomTheorem}{Bordism and Thom's theorem}\dotfill \pageref*{ThomTheorem} \linebreak
\noindent\hyperlink{bordism}{Bordism}\dotfill \pageref*{bordism} \linebreak
\noindent\hyperlink{SectionThomTheorem}{Thom's theorem}\dotfill \pageref*{SectionThomTheorem} \linebreak
\noindent\hyperlink{ThomIsomorphism}{Thom isomorphism}\dotfill \pageref*{ThomIsomorphism} \linebreak
\noindent\hyperlink{thomgysin_sequence}{Thom-Gysin sequence}\dotfill \pageref*{thomgysin_sequence} \linebreak
\noindent\hyperlink{OrientationAndFiberIntegration}{Orientation in generalized cohomology}\dotfill \pageref*{OrientationAndFiberIntegration} \linebreak
\noindent\hyperlink{universal_orientation}{Universal $E$-orientation}\dotfill \pageref*{universal_orientation} \linebreak
\noindent\hyperlink{complex_projective_space}{Complex projective space}\dotfill \pageref*{complex_projective_space} \linebreak
\noindent\hyperlink{ComplexOrientatioon}{Complex orientation}\dotfill \pageref*{ComplexOrientatioon} \linebreak
\noindent\hyperlink{ComplexOrientedCohomologyTheory}{Complex oriented cohomology}\dotfill \pageref*{ComplexOrientedCohomologyTheory} \linebreak
\noindent\hyperlink{ChernClasses}{Chern classes}\dotfill \pageref*{ChernClasses} \linebreak
\noindent\hyperlink{existence}{Existence}\dotfill \pageref*{existence} \linebreak
\noindent\hyperlink{splitting_principle}{Splitting principle}\dotfill \pageref*{splitting_principle} \linebreak
\noindent\hyperlink{ConnerFloydChernClasses}{Conner-Floyd Chern classes}\dotfill \pageref*{ConnerFloydChernClasses} \linebreak
\noindent\hyperlink{FormalGroupLaws}{Formal group laws of first CF-Chern classes}\dotfill \pageref*{FormalGroupLaws} \linebreak
\noindent\hyperlink{formal_group_laws}{Formal group laws}\dotfill \pageref*{formal_group_laws} \linebreak
\noindent\hyperlink{formal_group_laws_from_complex_orientation}{Formal group laws from complex orientation}\dotfill \pageref*{formal_group_laws_from_complex_orientation} \linebreak
\noindent\hyperlink{the_universal_1d_commutative_formal_group_law_and_lazards_theorem}{The universal 1d commutative formal group law and Lazard's theorem}\dotfill \pageref*{the_universal_1d_commutative_formal_group_law_and_lazards_theorem} \linebreak
\noindent\hyperlink{ComplexCobordismCohomology}{Complex cobordism}\dotfill \pageref*{ComplexCobordismCohomology} \linebreak
\noindent\hyperlink{connerfloydchern_classes_are_thom_classes}{Conner-Floyd-Chern classes are Thom classes}\dotfill \pageref*{connerfloydchern_classes_are_thom_classes} \linebreak
\noindent\hyperlink{ComplexOrientationAsRingSpectrumMaps}{Complex orientation as ring spectrum maps}\dotfill \pageref*{ComplexOrientationAsRingSpectrumMaps} \linebreak
\noindent\hyperlink{HomologyOfMU}{Homology of $M U$}\dotfill \pageref*{HomologyOfMU} \linebreak
\noindent\hyperlink{QuillenTheoremOnMU}{Milnor-Quillen theorem on $M U$}\dotfill \pageref*{QuillenTheoremOnMU} \linebreak
\noindent\hyperlink{landweber_exact_functor_theorem}{Landweber exact functor theorem}\dotfill \pageref*{landweber_exact_functor_theorem} \linebreak
\noindent\hyperlink{outlook_geometry_of_}{Outlook: Geometry of $Spec(MU)$}\dotfill \pageref*{outlook_geometry_of_} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


$\,$

\textbf{Outline.} We start with two classical topics of [[algebraic topology]] that first run independently in parallel:

\begin{itemize}%
\item \hyperlink{S1GeneralizedCohomology}{1) Generalized cohomology}


\item \hyperlink{S2CobordismTheory}{2) Cobordism theory}



\end{itemize}
The development of either of these happens to give rise to the concept of \emph{[[spectra]]} and via this concept it turns out that both topics are intimately related. The unification of both is our third topic

\begin{itemize}%
\item \hyperlink{ComplexOrientedCohomologyTheory}{3) Complex oriented cohomology}

\end{itemize}
$\,$

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96}).

\hypertarget{S1GeneralizedCohomology}{}\subsection*{{Generalized cohomology}}\label{S1GeneralizedCohomology}

\textbf{Idea.} The concept that makes [[algebraic topology]] be about methods of [[homological algebra]] applied to [[topology]] is that of [[generalized homology]] and [[generalized cohomology]]: these are [[covariant functors]] or [[contravariant functors]], respectively,

\begin{displaymath}
Spaces \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}
from (sufficiently nice) [[topological spaces]] to $\mathbb{Z}$-[[graded abelian groups]], such that a few key properties of the [[homotopy types]] of topological spaces is preserved as one passes them from [[Ho(Top)]] to the much more tractable [[abelian category]] [[Ab]].

\textbf{Literature.} (\hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, chapters 7,8 and 12}, \hyperlink{Kochman96}{Kochman 96, 3.4, 4.2}, \hyperlink{Schwede12}{Schwede 12, II.6})

\hypertarget{GeneralizedHomologyAndCohomologyFunctors}{}\subsubsection*{{Generalized cohomology functors}}\label{GeneralizedHomologyAndCohomologyFunctors}

\textbf{Idea.} A [[generalized (Eilenberg-Steenrod) cohomology]] theory is such a contravariant functor which satisfies the key properties exhibited by [[ordinary cohomology]] (as computed for instance by [[singular cohomology]]), notably [[homotopy invariance]] and [[excision]], \emph{except} that its value on the point is not required to be concentrated in degree 0. Dually for [[generalized homology]]. There are two versions of the axioms, one for [[reduced cohomology]], and they are equivalent if properly set up.

An important example of a generalised cohomology theory other than [[ordinary cohomology]] is [[topological K-theory]]. The other two examples of key relevance below are [[cobordism cohomology]] and [[stable cohomotopy]].

\textbf{Literature.} (\hyperlink{Switzer75}{Switzer 75, section 7}, \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, section 12 and section 9}, \hyperlink{Kochman96}{Kochman 96, 3.4}).

$\,$

\hypertarget{reduced_cohomology}{}\paragraph*{{Reduced cohomology}}\label{reduced_cohomology}

The traditional formulation of reduced generalized cohomology in terms of point-set topology is this:

\begin{defn}
\label{ReducedGeneralizedCohomology}\hypertarget{ReducedGeneralizedCohomology}{}
A \textbf{[[reduced cohomology theory]]} is

\begin{enumerate}%
\item a [[functor]]

\begin{displaymath}
\tilde E^\bullet
   \;\colon\;
  (Top^{\ast/}_{CW})^{op} \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}
from the [[opposite category|opposite]] of [[pointed topological spaces]] ([[CW-complexes]]) to $\mathbb{Z}$-[[graded abelian groups]] (``[[cohomology groups]]''), in components

\begin{displaymath}
\tilde E
    \;\colon\;
  (X \stackrel{f}{\longrightarrow} Y)
    \mapsto
  (\tilde E^\bullet(Y)
    \stackrel{f^\ast}{\longrightarrow}
  \tilde E^\bullet(X))
  \,,
\end{displaymath}

\item equipped with a [[natural isomorphism]] of degree +1, to be called the \textbf{[[suspension isomorphism]]}, of the form

\begin{displaymath}
\sigma_E
\;\colon\;
\tilde E^\bullet(-)
\overset{\simeq}{\longrightarrow}
\tilde E^{\bullet +1}(\Sigma -)
\end{displaymath}


\end{enumerate}
such that:

\begin{enumerate}%
\item \textbf{([[homotopy invariance]])} If $f_1,f_2 \colon X \longrightarrow Y$ are two morphisms of pointed topological spaces such that there is a (base point preserving) [[homotopy]] $f_1 \simeq f_2$ between them, then the induced [[homomorphisms]] of abelian groups are [[equality|equal]]

\begin{displaymath}
f_1^\ast = f_2^\ast
  \,.
\end{displaymath}

\item \textbf{(exactness)} For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced [[mapping cone]] (\href{Introduction+to+Stable+homotopy+theory+--+P#ConeAndMappingCylinder}{def.}), then this gives an [[exact sequence]] of graded abelian groups

\begin{displaymath}
\tilde E^\bullet(Cone(i))
\overset{j^\ast}{\longrightarrow}
\tilde E^\bullet(X)
\overset{i^\ast}{\longrightarrow}
\tilde E^\bullet(A)
\,.
\end{displaymath}


\end{enumerate}
\end{defn}
(e.g. \hyperlink{AguilarGitlerPrieto02}{AGP 02, def. 12.1.4})

This is equivalent (prop. \ref{HomotopyTheoreticVersionOfCohomologyFunctorDefIsEquivalent} below) to the following more succinct homotopy-theoretic definition:

\begin{defn}
\label{ReducedGeneralizedCohomologyHomotopyTheoretically}\hypertarget{ReducedGeneralizedCohomologyHomotopyTheoretically}{}
A \textbf{reduced [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]]} is a [[functor]]

\begin{displaymath}
\tilde E^\bullet
   \;\colon\;
  Ho(Top^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}
from the [[opposite category|opposite]] of the pointed [[classical homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#ClassicalHomotopyCategory}{def.}, \href{Introduction+to+Stable+homotopy+theory+--+P#ClassicalPointedHomotopyCategory}{def.}), to $\mathbb{Z}$-[[graded abelian groups]], and equipped with [[natural isomorphisms]], to be called the \textbf{[[suspension isomorphism]]} of the form

\begin{displaymath}
\sigma
    \;\colon\;
  \tilde E^{\bullet +1}(\Sigma -)
    \overset{\simeq}{\longrightarrow}
  \tilde E^\bullet(-)
\end{displaymath}
such that:

\begin{itemize}%
\item \textbf{(exactness)} it takes [[homotopy cofiber]] sequences in $Ho(Top^{\ast/})$ (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyFiber}{def.}) to [[exact sequence|exact sequences]].

\end{itemize}
\end{defn}
As a consequence (prop. \ref{HomotopyTheoreticVersionOfCohomologyFunctorDefIsEquivalent} below), we find yet another equivalent definition:

\begin{defn}
\label{ReducedGeneralizedCohomologyHomotopyHomotopicalFunctor}\hypertarget{ReducedGeneralizedCohomologyHomotopyHomotopicalFunctor}{}
A \textbf{reduced [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]]} is a [[functor]]

\begin{displaymath}
\tilde E^\bullet
   \;\colon\;
  (Top^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}
from the [[opposite category|opposite]] of the category of [[pointed topological spaces]] to $\mathbb{Z}$-[[graded abelian groups]], such that

\begin{itemize}%
\item \textbf{(WHE)} it takes [[weak homotopy equivalences]] to isomorphisms

\end{itemize}
and equipped with [[natural isomorphism]], to be called the \textbf{[[suspension isomorphism]]} of the form

\begin{displaymath}
\sigma
    \;\colon\;
  \tilde E^{\bullet +1}(\Sigma -)
    \overset{\simeq}{\longrightarrow}
  \tilde E^\bullet(-)
\end{displaymath}
such that

\begin{itemize}%
\item \textbf{(exactness)} it takes [[homotopy cofiber]] sequences in $Ho(Top^{\ast/})$ (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyFiber}{def.}), to [[exact sequence|exact sequences]].

\end{itemize}
\end{defn}
\begin{prop}
\label{HomotopyTheoreticVersionOfCohomologyFunctorDefIsEquivalent}\hypertarget{HomotopyTheoreticVersionOfCohomologyFunctorDefIsEquivalent}{}
The three definitions

\begin{itemize}%
\item def. \ref{ReducedGeneralizedCohomology}


\item def. \ref{ReducedGeneralizedCohomologyHomotopyTheoretically}


\item def. \ref{ReducedGeneralizedCohomologyHomotopyHomotopicalFunctor}



\end{itemize}
are indeed equivalent.

\end{prop}
\begin{proof}
Regarding the equivalence of def. \ref{ReducedGeneralizedCohomology} with def. \ref{ReducedGeneralizedCohomologyHomotopyTheoretically}:

By the existence of the [[classical model structure on topological spaces]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopQuillenModelStructure}{thm.}), the characterization of its [[homotopy category of a model category|homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects}{cor.}) and the existence of [[CW-approximations]], the homotopy invariance axiom in def. \ref{ReducedGeneralizedCohomology} is equivalent to the functor passing to the classical pointed homotopy category. In view of this and since on CW-complexes the standard topological mapping cone construction is a model for the [[homotopy cofiber]] (\href{Introduction+to+Stable+homotopy+theory+--+P#StandardTopologicalMappingConeIsHomotopyCofiber}{prop.}), this gives the equivalence of the two versions of the exactness axiom.

Regarding the equivalence of def. \ref{ReducedGeneralizedCohomologyHomotopyTheoretically} with def. \ref{ReducedGeneralizedCohomologyHomotopyHomotopicalFunctor}:

This is the [[universal property]] of the [[classical homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#UniversalPropertyOfHomotopyCategoryOfAModelCategory}{thm.}) which identifies it with the [[localization]] (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyCategoryOfACategoryWithWeakEquivalences}{def.}) of $Top^{\ast/}$ at the weak homotopy equivalences (\href{Introduction+to+Stable+homotopy+theory+--+P#TopQuillenModelStructure}{thm.}), together with the existence of [[CW approximations]] (\href{Introduction+to+Stable+homotopy+theory+--+P#EveryTopologicalSpaceWeaklyEquivalentToACWComplex}{rmk.}): jointly this says that, up to [[natural isomorphism]], there is a bijection between functors $F$ and $\tilde F$ in the following diagram (which is filled by a natural isomorphism itself):

\begin{displaymath}
\itexarray{
    Top^{op} &\overset{F}{\longrightarrow}& Ab^{\mathbb{Z}}
    \\
    {}^{\mathllap{\gamma_{Top}}}\downarrow & \nearrow_{\mathrlap{\tilde F}}
    \\
    Ho(Top)^{op}\simeq (Top_{CW})/_\sim
  }
\end{displaymath}
where $F$ sends weak homotopy equivalences to isomorphisms and where $(-)_\sim$ means identifying homotopic maps.

\end{proof}
Prop. \ref{HomotopyTheoreticVersionOfCohomologyFunctorDefIsEquivalent} naturally suggests (e.g. \hyperlink{LurieHigherAlgebra}{Lurie 10, section 1.4}) that the concept of generalized cohomology be formulated in the generality of any abstract homotopy theory ([[model category]]), not necessarily that of (pointed) topological spaces:

\begin{defn}
\label{GeneralizedCohomologyOnGeneralInfinityCategory}\hypertarget{GeneralizedCohomologyOnGeneralInfinityCategory}{}
Let $\mathcal{C}$ be a [[model category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#ModelCategory}{def.}) with $\mathcal{C}^{\ast/}$ its [[slice model structure|pointed model category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#ModelStructureOnSliceCategory}{prop.}).

A \textbf{reduced additive [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]]} on $\mathcal{C}$ is

\begin{enumerate}%
\item a [[functor]]

\begin{displaymath}
\tilde E^\bullet \;\colon \; Ho(\mathcal{C}^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}

\item a [[natural isomorphism]] (``[[suspension isomorphisms]]'') of degree +1

\begin{displaymath}
\sigma \; \colon \; \tilde E^\bullet \longrightarrow \tilde E^{\bullet+1} \circ \Sigma
\end{displaymath}


\end{enumerate}
such that

\begin{itemize}%
\item \textbf{(exactness)} $\tilde E^\bullet$ takes [[homotopy cofiber sequences]] to [[exact sequences]].

\end{itemize}
\end{defn}
Finally we need the following terminology:

\begin{defn}
\label{AdditiveOrdinary}\hypertarget{AdditiveOrdinary}{}
Let $\tilde E^\bullet$ be a [[reduced cohomology theory]] according to either of def. \ref{ReducedGeneralizedCohomology}, def. \ref{ReducedGeneralizedCohomologyHomotopyTheoretically}, def. \ref{ReducedGeneralizedCohomologyHomotopyHomotopicalFunctor} or def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}.

We say $\tilde E^\bullet$ is \textbf{additive} if in addition

\begin{itemize}%
\item \textbf{([[wedge axiom]])} For $\{X_i\}_{i \in I}$ any set of pointed CW-complexes, then the canonical morphism

\begin{displaymath}
\tilde E^\bullet(\vee_{i \in I} X_i)
\longrightarrow
\prod_{i \in I} \tilde E^\bullet(X_i)
\end{displaymath}
from the functor applied to their [[wedge sum]] (\href{Introduction+to+Stable+homotopy+theory+--+P#WedgeSumAsCoproduct}{def.}), to the [[product]] of its values on the wedge summands, is an [[isomorphism]].



\end{itemize}
We say $\tilde E^\bullet$ is \textbf{ordinary} if its value on the [[0-sphere]] $S^0$ is concentrated in degree 0:

\begin{itemize}%
\item \textbf{(Dimension)} $\tilde E^{\bullet\neq 0}(\mathbb{S}^0) \simeq 0$.

\end{itemize}
If $\tilde E^\bullet$ is not ordinary, one also says that it is \textbf{generalized} or \textbf{extraordinary}.

A \textbf{[[homomorphism]] of reduced cohomology theories}

\begin{displaymath}
\eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet
\end{displaymath}
is a [[natural transformation]] between the underlying functors which is compatible with the suspension isomorphisms in that all the following [[commuting square|squares commute]]

\begin{displaymath}
\itexarray{
    \tilde E^\bullet(X) &\overset{\eta_X}{\longrightarrow}&  \tilde F^\bullet(X)
    \\
    {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}}
    \\
    \tilde E^{\bullet + 1}(\Sigma X)
    &\overset{\eta_{\Sigma X}}{\longrightarrow}&
    \tilde F^{\bullet + 1}(\Sigma X)
  }
  \,.
\end{displaymath}
\end{defn}
We now discuss some constructions and consequences implied by the concept of reduced cohomology theories:

\begin{defn}
\label{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory}\hypertarget{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory}{}
Given a generalized cohomology theory $(E^\bullet,\delta)$ on some $\mathcal{C}$ as in def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, and given a [[homotopy cofiber sequence]] in $\mathcal{C}$ (\href{Introduction+to+Stable+homotopy+theory+--+P#LongFiberSequence}{prop.}),

\begin{displaymath}
X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z
  \stackrel{coker(g)}{\longrightarrow}
  \Sigma X
  \,,
\end{displaymath}
then the corresponding \textbf{[[connecting homomorphism]]} is the composite

\begin{displaymath}
\partial
    \;\colon\;
  E^\bullet(X)
   \stackrel{\sigma}{\longrightarrow}
  E^{\bullet+1}(\Sigma X)
   \stackrel{coker(g)^\ast}{\longrightarrow}
  E^{\bullet+1}(Z)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory}\hypertarget{LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory}{}
The [[connecting homomorphisms]] of def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory} are parts of [[long exact sequences]]

\begin{displaymath}
\cdots
   \stackrel{\partial}{\longrightarrow}
  E^{\bullet}(Z)
    \longrightarrow
  E^\bullet(Y)
    \longrightarrow
  E^\bullet(X)
    \stackrel{\partial}{\longrightarrow}
  E^{\bullet+1}(Z)
   \to \cdots
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the defining exactness of $E^\bullet$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, and the way this appears in def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory}, using that $\sigma$ is by definition an isomorphism.

\end{proof}
\hypertarget{unreduced_cohomology}{}\paragraph*{{Unreduced cohomology}}\label{unreduced_cohomology}

Given a reduced [[generalized cohomology theory]] as in def. \ref{ReducedGeneralizedCohomology}, we may ``un-reduce'' it and evaluate it on unpointed topological spaces $X$ simply by evaluating it on $X_+$ (\href{Introduction+to+Stable+homotopy+theory+--+P#BasePointAdjoined}{def.}). It is conventional to further generalize to [[relative cohomology]] and evaluate on unpointed subspace inclusions $i \colon A \hookrightarrow X$, taken as placeholders for their [[mapping cones]] $Cone(i_+)$ (\href{Introduction+to+Stable+homotopy+theory+--+P#UnreducedMappingConeAsReducedConeOfBasedPointAdjoined}{prop.}).

In the following a \emph{pair} $(X,U)$ refers to a [[subspace]] inclusion of [[topological spaces]] $U \hookrightarrow X$. Whenever only one space is mentioned, the subspace is assumed to be the [[empty set]] $(X, \emptyset)$. Write $Top_{CW}^{\hookrightarrow}$ for the category of such pairs (the [[full subcategory]] of the [[arrow category]] of $Top_{CW}$ on the inclusions). We identify $Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow}$ by $X \mapsto (X,\emptyset)$.

\begin{defn}
\label{GeneralizedCohomologyTheory}\hypertarget{GeneralizedCohomologyTheory}{}
A \emph{[[cohomology theory]]} (unreduced, [[relative cohomology|relative]]) is

\begin{enumerate}%
\item a [[functor]]

\begin{displaymath}
E^\bullet : (Top_{CW}^{\hookrightarrow})^{op} \to Ab^{\mathbb{Z}}
\end{displaymath}
to the category of $\mathbb{Z}$-[[graded abelian groups]],


\item a [[natural transformation]] of degree +1, to be called the \textbf{[[connecting homomorphism]]}, of the form

\begin{displaymath}
\delta_{(X,A)}
\;\colon\;
E^\bullet(A, \emptyset) \to E^{\bullet + 1}(X, A)
\,.
\end{displaymath}


\end{enumerate}
such that:

\begin{enumerate}%
\item \textbf{(homotopy invariance)} For $f \colon (X_1,A_1) \to (X_2,A_2)$ a [[homotopy equivalence]] of pairs, then

\begin{displaymath}
E^\bullet(f)
   \;\colon\;
  E^\bullet(X_2,A_2) \stackrel{\simeq}{\longrightarrow} E^\bullet(X_1,A_1)
\end{displaymath}
is an [[isomorphism]];


\item \textbf{(exactness)} For $A \hookrightarrow X$ the induced sequence

\begin{displaymath}
\cdots
\to
E^n(X, A)
\longrightarrow
E^n(X)
\longrightarrow
E^n(A)
\stackrel{\delta}{\longrightarrow}
E^{n+1}(X, A)
\to
\cdots
\end{displaymath}
is a [[long exact sequence]] of [[abelian groups]].


\item \textbf{([[excision]])} For $U \hookrightarrow A \hookrightarrow X$ such that $\overline{U} \subset Int(A)$, then the natural inclusion of the pair $i \colon (X-U, A-U) \hookrightarrow (X, A)$ induces an isomorphism

\begin{displaymath}
E^\bullet(i)
   \;\colon\;
  E^n(X, A)
   \overset{\simeq}{\longrightarrow}
  E^n(X-U, A-U)
\end{displaymath}


\end{enumerate}
We say $E^\bullet$ is \textbf{additive} if it takes [[coproducts]] to [[products]]:

\begin{itemize}%
\item \textbf{(additivity)} If $(X, A) = \coprod_i (X_i, A_i)$ is a [[coproduct]], then the canonical comparison morphism

\begin{displaymath}
E^n(X, A) \overset{\simeq}{\longrightarrow} \prod_i E^n(X_i, A_i)
\end{displaymath}
is an [[isomorphism]] from the value on $(X,A)$ to the [[product]] of values on the summands.



\end{itemize}
We say $E^\bullet$ is \textbf{ordinary} if its value on the point is concentrated in degree 0

\begin{itemize}%
\item \textbf{(Dimension)}: $E^{\bullet \neq 0}(\ast,\emptyset) = 0$.

\end{itemize}
A [[homomorphism]] of unreduced cohomology theories

\begin{displaymath}
\eta \;\colon\; E^\bullet \longrightarrow F^\bullet
\end{displaymath}
is a [[natural transformation]] of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these [[commuting square|squares commute]]:

\begin{displaymath}
\itexarray{
     E^\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F^\bullet(A,\emptyset)
     \\
     {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}}
     \\
     E^{\bullet +1}(X,A)
       &\overset{\eta_{(X,A)}}{\longrightarrow}&
     F^{\bullet +1}(X,A)
  }
  \,.
\end{displaymath}
\end{defn}
e.g. (\hyperlink{AguilarGitlerPrieto02}{AGP 02, def. 12.1.1}).

\begin{defn}
\label{AlternativeFormulationOfExcisionAxiom}\hypertarget{AlternativeFormulationOfExcisionAxiom}{}
The excision axiom in def. \ref{GeneralizedCohomologyTheory} is equivalent to the following statement:

For all $A,B \hookrightarrow X$ with $X = Int(A) \cup Int(B)$, then the inclusion

\begin{displaymath}
i \colon (A, A \cap B) \longrightarrow (X,B)
\end{displaymath}
induces an isomorphism,

\begin{displaymath}
i^\ast
    \;\colon\;
  E^\bullet(X, B)
    \overset{\simeq}{\longrightarrow}
  E^\bullet(A, A \cap B)
\end{displaymath}
\end{defn}
(e.g \hyperlink{Switzer75}{Switzer 75, 7.2})

\begin{proof}
In one direction, suppose that $E^\bullet$ satisfies the original excision axiom. Given $A,B$ with $X = \Int(A) \cup Int(B)$, set $U \coloneqq X-A$ and observe that

\begin{displaymath}
\begin{aligned}
     \overline{U}
        & = \overline{X-A}
     \\ & = X- Int(A)
     \\ & \subset Int(B)
  \end{aligned}
\end{displaymath}
and that

\begin{displaymath}
(X-U, B-U) = (A, A \cap B)
  \,.
\end{displaymath}
Hence the excision axiom implies $E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B)$.

Conversely, suppose $E^\bullet$ satisfies the alternative condition. Given $U \hookrightarrow A \hookrightarrow X$ with $\overline{U} \subset Int(A)$, observe that we have a cover

\begin{displaymath}
\begin{aligned}
          Int(X-U) \cup Int(A)
      & = (X - \overline{U}) \cap \Int(A)
   \\ & \supset (X - Int(A)) \cap Int(A)
   \\ & = X
  \end{aligned}
\end{displaymath}
and that

\begin{displaymath}
(X-U, (X-U) \cap A) = (X-U, A - U)
  \,.
\end{displaymath}
Hence

\begin{displaymath}
E^\bullet(X-U,A-U)
  \simeq
  E^\bullet(X-U, (X-U)\cap A)
  \simeq
  E^\bullet(X,A)
  \,.
\end{displaymath}
\end{proof}
The following lemma shows that the dependence in pairs of spaces in a generalized cohomology theory is really a stand-in for evaluation on [[homotopy cofibers]] of inclusions.

\begin{lemma}
\label{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient}\hypertarget{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient}{}
Let $E^\bullet$ be an cohomology theory, def. \ref{GeneralizedCohomologyTheory}, and let $A \hookrightarrow X$. Then there is an isomorphism

\begin{displaymath}
E^\bullet(X,A)
    \stackrel{\simeq}{\longrightarrow}
  E^\bullet(X \cup Cone(A), \ast)
\end{displaymath}
between the value of $E^\bullet$ on the pair $(X,A)$ and its value on the unreduced [[mapping cone]] of the inclusion (\href{Introduction+to+Stable+homotopy+theory+--+P#UnreducedCone}{rmk.}), relative to a basepoint.

If moreover $A \hookrightarrow X$ is (the [[retract]] of) a [[relative cell complex]] inclusion, then also the morphism in cohomology induced from the [[quotient]] map $p \;\colon\; (X,A)\longrightarrow (X/A, \ast)$ is an [[isomorphism]]:

\begin{displaymath}
E^\bullet(p)
    \;\colon\;
  E^\bullet(X/A,\ast)
    \longrightarrow
  E^\bullet(X,A)
  \,.
\end{displaymath}
\end{lemma}
(e.g \hyperlink{AguilarGitlerPrieto02}{AGP 02, corollary 12.1.10})

\begin{proof}
Consider $U \coloneqq (Cone(A)-A \times \{0\}) \hookrightarrow Cone(A)$, the cone on $A$ minus the base $A$. We have

\begin{displaymath}
( X\cup Cone(A)-U, Cone(A)-U) \simeq (X,A)
\end{displaymath}
and hence the first isomorphism in the statement is given by the excision axiom followed by homotopy invariance (along the contraction of the cone to the point).

Next consider the quotient of the [[mapping cone]] of the inclusion:

\begin{displaymath}
( X\cup Cone(A), Cone(A) ) \longrightarrow (X/A,\ast)
  \,.
\end{displaymath}
If $A \hookrightarrow X$ is a cofibration, then this is a [[homotopy equivalence]] since $Cone(A)$ is contractible and since by the dual [[factorization lemma]] (\href{Introduction+to+Stable+homotopy+theory+--+P#FactorizationLemma}{lem.}) and by the invariance of homotopy fibers under weak equivalences (\href{Introduction+to+Stable+homotopy+theory+--+P#FiberOfFibrationIsCompatibleWithWeakEquivalences}{lem.}), $X \cup Cone(A)\to X/A$ is a weak homotopy equivalence, hence, by the universal property of the [[classical homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#UniversalPropertyOfHomotopyCategoryOfAModelCategory}{thm.}) a homotopy equivalence on CW-complexes.

Hence now we get a composite isomorphism

\begin{displaymath}
E^\bullet(X/A,\ast)
    \overset{\simeq}{\longrightarrow}
  E^\bullet( X\cup Cone(A), Cone(A) )
    \overset{\simeq}{\longrightarrow}
  E^\bullet(X,A)
  \,.
\end{displaymath}
\end{proof}
\begin{example}
\label{GeneralizedCohomologyOnHomotopyQuotientMaps}\hypertarget{GeneralizedCohomologyOnHomotopyQuotientMaps}{}
As an important special case of : Let $(X,x)$ be a [[pointed topological space|pointed]] [[CW-complex]]. For $p\colon (Cone(X), X) \to (\Sigma X,\{x\})$ the quotient map from the reduced cone on $X$ to the [[reduced suspension]], then

\begin{displaymath}
E^\bullet(p)
  \;\colon\;
  E^\bullet(Cone(X),X)
   \overset{\simeq}{\longrightarrow}
  E^\bullet(\Sigma X, \{x\})
\end{displaymath}
is an isomorphism.

\end{example}
\begin{prop}
\label{ExactSequenceOfATriple}\hypertarget{ExactSequenceOfATriple}{}
\textbf{(exact sequence of a triple)}

For $E^\bullet$ an unreduced generalized cohomology theory, def. \ref{GeneralizedCohomologyTheory}, then every inclusion of two consecutive subspaces

\begin{displaymath}
Z \hookrightarrow Y \hookrightarrow X
\end{displaymath}
induces a [[long exact sequence]] of cohomology groups of the form

\begin{displaymath}
\cdots
   \to
  E^{q-1}(Y,Z)
    \stackrel{\bar \delta}{\longrightarrow}
  E^q(X,Y)
    \stackrel{}{\longrightarrow}
  E^q(X,Z)
    \stackrel{}{\longrightarrow}
  E^q(Y,Z)
    \to
  \cdots
\end{displaymath}
where

\begin{displaymath}
\bar \delta
    \;\colon \;
  E^{q-1}(Y,Z)
   \longrightarrow
  E^{q-1}(Y)
   \stackrel{\delta}{\longrightarrow}
  E^{q}(X,Y)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Apply the [[braid lemma]] to the interlocking long exact sequences of the three pairs $(X,Y)$, $(X,Z)$, $(Y,Z)$:



(graphics from \href{http://math.stackexchange.com/a/1180681/58526}{this Maths.SE comment}, showing the dual situation for homology)

See \href{braid+lemma#ExactSequenceForTripleInGeneralizedHomology}{here} for details.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The exact sequence of a triple in prop. \ref{ExactSequenceOfATriple} is what gives rise to the [[Cartan-Eilenberg spectral sequence]] for $E$-cohomology of a [[CW-complex]] $X$.

\end{remark}
\begin{example}
\label{ExtractingSuspensionIsomorphismFromUnreducedCohomology}\hypertarget{ExtractingSuspensionIsomorphismFromUnreducedCohomology}{}
For $(X,x)$ a [[pointed topological space]] and $Cone(X) = (X \wedge (I_+))/X$ its reduced [[cone]], the long exact sequence of the triple $(\{x\}, X, Cone(X))$, prop. \ref{ExactSequenceOfATriple},

\begin{displaymath}
0
   \simeq
   E^q(Cone(X), \{x\})
     \longrightarrow
   E^q(X,\{x\})
     \overset{\bar \delta}{\longrightarrow}
   E^{q+1}(Cone(X),X)
     \longrightarrow
   E^{q+1}(Cone(X), \{x\})
   \simeq 0
\end{displaymath}
exhibits the [[connecting homomorphism]] $\bar \delta$ here as an [[isomorphism]]

\begin{displaymath}
\bar \delta
     \;\colon\;
   E^q(X,\{x\})
     \overset{\simeq}{\longrightarrow}
   E^{q+1}(Cone(X),X)
  \,.
\end{displaymath}
This is the \emph{[[suspension isomorphism]]} extracted from the unreduced cohomology theory, see def. \ref{FromUnreducedToReducedCohomology} below.

\end{example}
\begin{prop}
\label{MayerVietorisSequenceInGeneralizedCohomology}\hypertarget{MayerVietorisSequenceInGeneralizedCohomology}{}
\textbf{([[Mayer-Vietoris sequence]])}

Given $E^\bullet$ an unreduced cohomology theory, def. \ref{GeneralizedCohomologyTheory}. Given a topological space covered by the [[interior]] of two spaces as $X = Int(A) \cup Int(B)$, then for each $C \subset A \cap B$ there is a [[long exact sequence]] of cohomology groups of the form

\begin{displaymath}
\cdots
  \to
  E^{n-1}(A \cap B , C)
  \overset{\bar \delta}{\longrightarrow}
  E^n(X,C)
  \longrightarrow
  E^n(A,C) \oplus E^n(B,C)
  \longrightarrow
  E^n(A \cap B, C)
  \to
  \cdots
  \,.
\end{displaymath}
\end{prop}
e.g. (\hyperlink{Switzer75}{Switzer 75, theorem 7.19}, \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, theorem 12.1.22})

\hypertarget{RelationBetweenUnreducedAndReducedCohomology}{}\paragraph*{{Relation between unreduced and reduced cohomology}}\label{RelationBetweenUnreducedAndReducedCohomology}

\begin{defn}
\label{FromUnreducedToReducedCohomology}\hypertarget{FromUnreducedToReducedCohomology}{}
\textbf{(unreduced to reduced cohomology)}

Let $E^\bullet$ be an [[generalized (Eilenberg-Steenrod) cohomology|unreduced cohomology theory]], def. \ref{GeneralizedCohomologyTheory}. Define a reduced cohomology theory, def. \ref{ReducedGeneralizedCohomology} $(\tilde E^\bullet, \sigma)$ as follows.

For $x \colon \ast \to X$ a [[pointed topological space]], set

\begin{displaymath}
\tilde E^\bullet(X,x) \coloneqq E^\bullet(X,\{x\})
  \,.
\end{displaymath}
This is clearly [[functor|functorial]]. Take the [[suspension isomorphism]] to be the composite

\begin{displaymath}
\sigma
   \;\colon\;
  \tilde E^{\bullet+1}(\Sigma X)
   =
  E^{\bullet+1}(\Sigma X, \{x\})
   \overset{E^\bullet(p)}{\longrightarrow}
  E^{\bullet+1}(Cone(X),X)
    \overset{\bar \delta^{-1}}{\longrightarrow}
  E^\bullet(X,\{x\})
   =
  \tilde E^{\bullet}(X)
\end{displaymath}
of the isomorphism $E^\bullet(p)$ from example \ref{GeneralizedCohomologyOnHomotopyQuotientMaps} and the [[inverse]] of the isomorphism $\bar \delta$ from example \ref{ExtractingSuspensionIsomorphismFromUnreducedCohomology}.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The construction in def. \ref{FromUnreducedToReducedCohomology} indeed gives a reduced cohomology theory.

\end{prop}
(e.g \hyperlink{Switzer75}{Switzer 75, 7.34})

\begin{proof}
We need to check the \hyperlink{ExactnessUnreduced}{exactness axiom} given any $A\hookrightarrow X$. By lemma \ref{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient} we have an isomorphism

\begin{displaymath}
\tilde E^\bullet(X \cup Cone(A))
  =
  E^\bullet(X \cup Cone(A), \{\ast\})
    \overset{\simeq}{\longrightarrow}
  E^\bullet(X,A)
  \,.
\end{displaymath}
Unwinding the constructions shows that this makes the following [[commuting diagram|diagram commute]]:

\begin{displaymath}
\itexarray{
     \tilde E^\bullet(X\cup Cone(A))
        &\overset{\simeq}{\longrightarrow}&
     E^\bullet(X,A)
     \\
     \downarrow && \downarrow
     \\
     \tilde E^\bullet(X) &=& E^\bullet(X,\{x\})
     \\
     \downarrow && \downarrow
     \\
     \tilde E^\bullet(A) &=& E^\bullet(A,\{a\})
  }
  \,,
\end{displaymath}
where the vertical sequence on the right is exact by prop. \ref{ExactSequenceOfATriple}. Hence the left vertical sequence is exact.

\end{proof}
\begin{defn}
\label{ReducedToUnreducedGeneralizedCohomology}\hypertarget{ReducedToUnreducedGeneralizedCohomology}{}
\textbf{(reduced to unreduced cohomology)}

Let $(\tilde E^\bullet, \sigma)$ be a [[reduced cohomology theory]], def. \ref{ReducedGeneralizedCohomology}. Define an unreduced cohomolog theory $E^\bullet$, def. \ref{GeneralizedCohomologyTheory}, by

\begin{displaymath}
E^\bullet(X,A)
  \coloneqq
  \tilde E^\bullet( X_+ \cup Cone(A_+))
\end{displaymath}
and let the connecting homomorphism be as in def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory}.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The construction in def. \ref{ReducedToUnreducedGeneralizedCohomology} indeed yields an unreduced cohomology theory.

\end{prop}
e.g. (\hyperlink{Switzer75}{Switzer 75, 7.35})

\begin{proof}
Exactness holds by prop. \ref{LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory}. For excision, it is sufficient to consider the alternative formulation of lemma \ref{AlternativeFormulationOfExcisionAxiom}. For CW-inclusions, this follows immediately with lemma \ref{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient}.

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
The constructions of def. \ref{ReducedToUnreducedGeneralizedCohomology} and def. \ref{FromUnreducedToReducedCohomology} constitute a pair of [[functors]] between then [[categories]] of reduced cohomology theories, def. \ref{ReducedGeneralizedCohomology} and unreduced cohomology theories, def. \ref{GeneralizedCohomologyTheory} which exhbit an [[equivalence of categories]].

\end{theorem}
\begin{proof}
(\ldots{}careful with checking the respect for suspension iso and connecting homomorphism..)

To see that there are [[natural isomorphisms]] relating the two composites of these two functors to the identity:

One composite is

\begin{displaymath}
\begin{aligned}
    E^\bullet
    & \mapsto
    (\tilde E^\bullet \colon (X,x) \mapsto E^\bullet(X,\{x\}))
    \\
    & \mapsto
    ((E')^\bullet \colon (X,A) \mapsto E^\bullet( X_+ \cup Cone(A_+) ), \ast)
  \end{aligned}
  \,,
\end{displaymath}
where on the right we have, from the construction, the reduced mapping cone of the original inclusion $A \hookrightarrow X$ with a base point adjoined. That however is isomorphic to the unreduced mapping cone of the original inclusion (prop.- P\#UnreducedMappingConeAsReducedConeOfBasedPointAdjoined)). With this the natural isomorphism is given by lemma \ref{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient}.

The other composite is

\begin{displaymath}
\begin{aligned}
    \tilde E^\bullet
     & \mapsto
    (E^\bullet \colon (X,A) \mapsto \tilde E^\bullet(X_+ \cup Cone(A_+)))
    \\
    & \mapsto
    ((\tilde E')^\bullet \colon X \mapsto \tilde E^\bullet(X_+ \cup Cone(*_+)))
  \end{aligned}
\end{displaymath}
where on the right we have the reduced mapping cone of the point inclusion with a point adoined. As before, this is isomorphic to the unreduced mapping cone of the point inclusion. That finally is clearly homotopy equivalent to $X$, and so now the natural isomorphism follows with homotopy invariance.

\end{proof}
Finally we record the following basic relation between reduced and unreduced cohomology:

\begin{prop}
\label{UnreducedCohomologyIsReducedPlusPointValue}\hypertarget{UnreducedCohomologyIsReducedPlusPointValue}{}
Let $E^\bullet$ be an unreduced cohomology theory, and $\tilde E^\bullet$ its reduced cohomology theory from def. \ref{FromUnreducedToReducedCohomology}. For $(X,\ast)$ a pointed topological space, then there is an identification

\begin{displaymath}
E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast)
\end{displaymath}
of the unreduced cohomology of $X$ with the [[direct sum]] of the reduced cohomology of $X$ and the unreduced cohomology of the base point.

\end{prop}
\begin{proof}
The pair $\ast \hookrightarrow X$ induces the sequence

\begin{displaymath}
\cdots
   \to
  E^{\bullet-1}(\ast)
    \stackrel{\delta}{\longrightarrow}
  \tilde E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(\ast)
    \stackrel{\delta}{\longrightarrow}
  \tilde E^{\bullet+1}(X)
    \to
  \cdots
\end{displaymath}
which by the exactness clause in def. \ref{GeneralizedCohomologyTheory} is [[exact sequence|exact]].

Now since the composite $\ast \to X \to \ast$ is the identity, the morphism $E^\bullet(X) \to E^\bullet(\ast)$ has a [[section]] and so is in particular an [[epimorphism]]. Therefore, by exactness, the [[connecting homomorphism]] vanishes, $\delta = 0$ and we have a [[short exact sequence]]

\begin{displaymath}
0
    \to
  \tilde E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(\ast)
    \to
  0
\end{displaymath}
with the right map an epimorphism. Hence this is a [[split exact sequence]] and the statement follows.

\end{proof}
\hypertarget{generalized_homology_functors}{}\paragraph*{{Generalized homology functors}}\label{generalized_homology_functors}

All of the above has a dual version with [[generalized cohomology]] replaced by [[generalized homology]]. For ease of reference, we record these dual definitions:

\begin{defn}
\label{ReducedGeneralizedHomology}\hypertarget{ReducedGeneralizedHomology}{}
A \textbf{reduced homology theory} is a [[functor]]

\begin{displaymath}
\tilde E_\bullet
   \;\colon\;
  (Top^{\ast/}_{CW}) \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}
from the category of [[pointed topological spaces]] ([[CW-complexes]]) to $\mathbb{Z}$-[[graded abelian groups]] (``[[homology groups]]''), in components

\begin{displaymath}
\tilde E _\bullet
    \;\colon\;
  (X \stackrel{f}{\longrightarrow} Y)
    \mapsto
  (\tilde E_\bullet(X)
    \stackrel{f_\ast}{\longrightarrow}
  \tilde E_\bullet(Y))
  \,,
\end{displaymath}
and equipped with a [[natural isomorphism]] of degree +1, to be called the \textbf{[[suspension isomorphism]]}, of the form

\begin{displaymath}
\sigma
    \;\colon\;
  \tilde E_\bullet(-)
    \overset{\simeq}{\longrightarrow}
  \tilde E_{\bullet +1}(\Sigma -)
\end{displaymath}
such that:

\begin{enumerate}%
\item \textbf{([[homotopy invariance]])} If $f_1,f_2 \colon X \longrightarrow Y$ are two morphisms of pointed topological spaces such that there is a (base point preserving) [[homotopy]] $f_1 \simeq f_2$ between them, then the induced [[homomorphisms]] of abelian groups are [[equality|equal]]

\begin{displaymath}
f_1_\ast = f_2_\ast
  \,.
\end{displaymath}

\item \textbf{(exactness)} For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced [[mapping cone]], then this gives an [[exact sequence]] of graded abelian groups

\begin{displaymath}
\tilde E_\bullet(A)
\overset{i_\ast}{\longrightarrow}
\tilde E_\bullet(X)
\overset{j_\ast}{\longrightarrow}
\tilde E_\bullet(Cone(i))
\,.
\end{displaymath}


\end{enumerate}
We say $\tilde E_\bullet$ is \textbf{additive} if in addition

\begin{itemize}%
\item \textbf{([[wedge axiom]])} For $\{X_i\}_{i \in I}$ any set of pointed CW-complexes, then the canonical morphism

\begin{displaymath}
\oplus_{i \in I} \tilde E_\bullet(X_i)
    \longrightarrow
  \tilde E^\bullet(\vee_{i \in I} X_i)
\end{displaymath}
from the [[direct sum]] of the value on the summands to the value on the [[wedge sum]] (prop.- P\#WedgeSumAsCoproduct)), is an [[isomorphism]].



\end{itemize}
We say $\tilde E_\bullet$ is \textbf{ordinary} if its value on the [[0-sphere]] $S^0$ is concentrated in degree 0:

\begin{itemize}%
\item \textbf{(Dimension)} $\tilde E_{\bullet\neq 0}(\mathbb{S}^0) \simeq 0$.

\end{itemize}
A [[homomorphism]] of reduced cohomology theories

\begin{displaymath}
\eta \;\colon\; \tilde E_\bullet \longrightarrow \tilde F_\bullet
\end{displaymath}
is a [[natural transformation]] between the underlying functors which is compatible with the suspension isomorphisms in that all the following [[commuting square|squares commute]]

\begin{displaymath}
\itexarray{
    \tilde E_\bullet(X) &\overset{\eta_X}{\longrightarrow}&  \tilde F_\bullet(X)
    \\
    {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}}
    \\
    \tilde E_{\bullet + 1}(\Sigma X)
    &\overset{\eta_{\Sigma X}}{\longrightarrow}&
    \tilde F_{\bullet + 1}(\Sigma X)
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{GeneralizedHomologyTheory}\hypertarget{GeneralizedHomologyTheory}{}
A \textbf{homology theory} (unreduced, [[relative cohomology|relative]]) is a [[functor]]

\begin{displaymath}
E_\bullet : (Top_{CW}^{\hookrightarrow}) \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}
to the category of $\mathbb{Z}$-[[graded abelian groups]], as well as a [[natural transformation]] of degree +1, to be called the \textbf{[[connecting homomorphism]]}, of the form

\begin{displaymath}
\delta_{(X,A)}
    \;\colon\;
  E_{\bullet + 1}(X, A)
    \longrightarrow
  E^\bullet(A, \emptyset)
  \,.
\end{displaymath}
such that:

\begin{enumerate}%
\item \textbf{(homotopy invariance)} For $f \colon (X_1,A_1) \to (X_2,A_2)$ a [[homotopy equivalence]] of pairs, then

\begin{displaymath}
E_\bullet(f)
   \;\colon\;
  E_\bullet(X_1,A_1) \stackrel{\simeq}{\longrightarrow} E_\bullet(X_2,A_2)
\end{displaymath}
is an [[isomorphism]];


\item \textbf{(exactness)} For $A \hookrightarrow X$ the induced sequence

\begin{displaymath}
\cdots
\to
E_{n+1}(X, A)
\stackrel{\delta}{\longrightarrow}
E_n(A)
\longrightarrow
E_n(X)
\longrightarrow
E_n(X, A)
\to
\cdots
\end{displaymath}
is a [[long exact sequence]] of [[abelian groups]].


\item \textbf{([[excision]])} For $U \hookrightarrow A \hookrightarrow X$ such that $\overline{U} \subset Int(A)$, then the natural inclusion of the pair $i \colon (X-U, A-U) \hookrightarrow (X, A)$ induces an isomorphism

\begin{displaymath}
E_\bullet(i)
   \;\colon\;
  E_n(X-U, A-U)
   \overset{\simeq}{\longrightarrow}
  E_n(X, A)
\end{displaymath}


\end{enumerate}
We say $E^\bullet$ is \textbf{additive} if it takes [[coproducts]] to [[direct sums]]:

\begin{itemize}%
\item \textbf{(additivity)} If $(X, A) = \coprod_i (X_i, A_i)$ is a [[coproduct]], then the canonical comparison morphism

\begin{displaymath}
\oplus_i E^n(X_i, A_i) \overset{\simeq}{\longrightarrow} E^n(X, A)
\end{displaymath}
is an [[isomorphism]]from the [[direct sum]] of the value on the summands, to the value on the total pair.



\end{itemize}
We say $E_\bullet$ is \textbf{ordinary} if its value on the point is concentrated in degree 0

\begin{itemize}%
\item \textbf{(Dimension)}: $E_{\bullet \neq 0}(\ast,\emptyset) = 0$.

\end{itemize}
A [[homomorphism]] of unreduced homology theories

\begin{displaymath}
\eta \;\colon\; E_\bullet \longrightarrow F_\bullet
\end{displaymath}
is a [[natural transformation]] of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these [[commuting square|squares commute]]:

\begin{displaymath}
\itexarray{
     E_{\bullet +1}(X,A)
       &\overset{\eta_{(X,A)}}{\longrightarrow}&
     F_{\bullet +1}(X,A)
     \\
     {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}}
     \\
     E_\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F^\bullet(A,\emptyset)
  }
  \,.
\end{displaymath}
\end{defn}
\hypertarget{multiplicative_cohomology_theories}{}\paragraph*{{Multiplicative cohomology theories}}\label{multiplicative_cohomology_theories}

The [[generalized cohomology theories]] considered above assign \emph{[[cohomology groups]]}. It is familiar from [[ordinary cohomology]] with [[coefficients]] not just in a group but in a [[ring]], that also the cohomology groups inherit compatible ring structure. The generalization of this phenomenon to generalized cohomology theories is captured by the concept of [[multiplicative cohomology theories]]:

\begin{defn}
\label{PairingOfUnreducedCohomologyTheories}\hypertarget{PairingOfUnreducedCohomologyTheories}{}
Let $E_1, E_2, E_3$ be three unreduced [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theories]] (\href{generalized+cohomology+theory#GeneralizedCohomologyTheory}{def.}). A \textbf{pairing of cohomology theories}

\begin{displaymath}
\mu \;\colon\; E_1 \Box E_2 \longrightarrow E_3
\end{displaymath}
is a [[natural transformation]] (of functors on $(Top_{CW}^{\hookrightarrow}\times Top_{CW}^{\hookrightarrow})^{op}$) of the form

\begin{displaymath}
\mu_{n_1,n_2}
   \;\colon\;
  E_1^{n_1}(X,A)
    \otimes
  E_2^{n_2}(Y,B)
    \longrightarrow
  E_3^{n_1 + n_2}(X\times Y \;,\; A\times Y \cup X \times B)
\end{displaymath}
such that this is compatible with the connecting homomorphisms $\delta_i$ of $E_i$, in that the following are [[commuting squares]]

\begin{displaymath}
\itexarray{
    E_1^{n_1}(A)
      \otimes
    E_2^{n_2}(Y,B)
      &\overset{\delta_1 \otimes id_2}{\longrightarrow}&
    E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1+1, n_2}}}
    \\
   \underoverset
     {E_3^{n_1 + n_2}(A \times Y  \cup  X \times B , X \times B)}
     {E_3^{n_1 + n_2}(A \times Y, A \times B)}
     {\simeq}
     &\overset{\delta_3}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X \times Y, A \times B)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    E_1^{n_1}(X,A)
      \otimes
    E_2^{n_2}(B)
      &\overset{(-1)^{n_1} id_1 \otimes \delta_2}{\longrightarrow}&
    E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{n_1, n_2 + 1}}}
    \\
   \underoverset
     {E_3^{n_1 + n_2}(A \times Y \cup X \times B , A \times Y)}
     {E_3^{n_1 + n_2}(X \times B, A \times B)}
     {\simeq}
     &\overset{\delta_3}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X \times Y, A \times B)
  }
  \,,
\end{displaymath}
where the isomorphisms in the bottom left are the \href{generalized+cohomology+theory##excision}{excision isomorphisms}.

\end{defn}
\begin{defn}
\label{MultiplicativeCohomologyTheory}\hypertarget{MultiplicativeCohomologyTheory}{}
An (unreduced) \textbf{multiplicative cohomology theory} is an unreduced [[generalized cohomology theory]] theory $E$ (def. \ref{GeneralizedCohomologyTheory}) equipped with

\begin{enumerate}%
\item (external multiplication) a pairing (def. \ref{PairingOfUnreducedCohomologyTheories}) of the form $\mu \;\colon\; E \Box E  \longrightarrow E$;


\item (unit) an element $1 \in E^0(\ast)$



\end{enumerate}
such that

\begin{enumerate}%
\item ([[associativity]]) $\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id)$;


\item ([[unitality]]) $\mu(1\otimes x) = \mu(x \otimes 1) = x$ for all $x \in E^n(X,A)$.



\end{enumerate}
The mulitplicative cohomology theory is called \textbf{commutative} (often considered by default) if in addition

\begin{itemize}%
\item \textbf{(graded commutativity)}

\begin{displaymath}
\itexarray{
    E^{n_1}(X,A) \otimes E^{n_2}(Y,B)
    &\overset{(u \otimes v) \mapsto (-1)^{n_1 n_2} (v \otimes u) }{\longrightarrow}&
    E^{n_2}(Y,B) \otimes E^{n_1}_{X,A}
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_2,n_1}}}
    \\
    E^{n_1 + n_2}( X \times Y , A \times Y \cup X \times B)
    &\underset{(switch_{(X,A), (Y,B)})^\ast}{\longrightarrow}&
    E^{n_1 + n_2}( Y \times X , B \times X \cup Y \times A)
  }
  \,.
\end{displaymath}


\end{itemize}
 Given a multiplicative cohomology theory $(E, \mu, 1)$, its \textbf{[[cup product]]} is the composite of the above external multiplication with pullback along the [[diagonal]] maps $\Delta_{(X,A)} \colon (X,A) \longrightarrow (X\times X, A \times X \cup X \times A)$;

\begin{displaymath}
(-) \cup (-)
   \;\colon\;
  E^{n_1}(X,A)
   \otimes
  E^{n_2}(X,A)
    \overset{\mu_{n_1,n_2}}{\longrightarrow}
  E^{n_1 + n_2}( X \times X, \; A \times X \cup X \times A)
    \overset{\Delta^\ast_{(X,A)}}{\longrightarrow}
  E^{n_1 + n_2}(X, \; A \cup B)
  \,.
\end{displaymath}
\end{defn}
e.g. (\href{multiplicative+cohomology+theory#TamakiKono06}{Tamaki-Kono 06, II.6})

\begin{prop}
\label{RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}\hypertarget{RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}{}
Let $(E,\mu,1)$ be a multiplicative cohomology theory, def. \ref{MultiplicativeCohomologyTheory}. Then

\begin{enumerate}%
\item For every space $X$ the \hyperlink{InternalMultiplicationOfMultiplicativeCohomologyTheory}{cup product} gives $E^\bullet(X)$ the structure of a $\mathbb{Z}$-[[graded ring]], which is graded-commutative if $(E,\mu,1)$ is commutative.


\item For every pair $(X,A)$ the external multiplication $\mu$ gives $E^\bullet(X,A)$ the structure of a left and right [[module]] over the graded ring $E^\bullet(\ast)$.


\item All pullback morphisms respect the left and right action of $E^\bullet(\ast)$ and the connecting homomorphisms respect the right action and the left action up to multiplication by $(-1)^{n_1}$



\end{enumerate}
\end{prop}
\begin{proof}
Regarding the third point:

For pullback maps this is the [[natural transformation|naturality]] of the external product: let $f \colon (X,A) \longrightarrow (Y,B)$ be a morphism in $Top_{CW}^{\hookrightarrow}$ then naturality says that the following square commutes:

\begin{displaymath}
\itexarray{
    E^{n_1}(\ast) \otimes E^{n_2}(Y,B)
    &\overset{\mu_{n_1,n_2}}{\longrightarrow}&
    E^{n_1 + n_2}(Y, B)
    \\
    {}^{\mathllap{(id,f^\ast)}}\downarrow && \downarrow^{\mathrlap{f^\ast}}
    \\
    E^{n_1}(\ast) \otimes E^{n_2}(X,A)
    &\overset{\mu_{n_1,n_2}}{\longrightarrow}&
    E^{n_1 + n_2}(Y,B)
  }
  \,.
\end{displaymath}
For connecting homomorphisms this is the (graded) commutativity of the squares in def. \ref{MultiplicativeCohomologyTheory}:

\begin{displaymath}
\itexarray{
    E^{n_1}(\ast)\otimes E^{n_2}(A)
      &\overset{(-1)^{n_1} (id, \delta)}{\longrightarrow}&
    E^{n_1}(\ast) \otimes E^{n_2 + 2}(X)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}}
    \\
    E^{n_1 + n_2}(A)
      &\overset{\delta}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X,B)
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{BrownRepresentabilityTheorem}{}\subsubsection*{{Brown representability theorem}}\label{BrownRepresentabilityTheorem}

\textbf{Idea.} Given any [[functor]] such as the generalized (co)homology functor \hyperlink{GeneralizedHomologyAndCohomologyFunctors}{above}, an important question to ask is whether it is a \emph{[[representable functor]]}. Due to the $\mathbb{Z}$-grading and the [[suspension isomorphisms]], if a generalized (co)homology functor is representable at all, it must be represented by a $\mathbb{Z}$-indexed sequence of [[pointed topological spaces]] such that the [[reduced suspension]] of one is comparable to the next one in the list. This is a \emph{[[spectrum]]} or more specifically: a \emph{[[sequential spectrum]]} .

Whitehead observed that indeed every [[spectrum]] represents a generalized (co)homology theory. The \emph{[[Brown representability theorem]]} states that, conversely, every generalized (co)homology theory is represented by a spectrum, subject to conditions of additivity.

As a first application, [[Eilenberg-MacLane spectra]] representing [[ordinary cohomology]] may be characterized via Brown representability.

\textbf{Literature.} (\hyperlink{Switzer75}{Switzer 75, section 9}, \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, section 12}, \hyperlink{Kochman96}{Kochman 96, 3.4})

\hypertarget{BrownRepresentabilityTraditional}{}\paragraph*{{Traditional discussion}}\label{BrownRepresentabilityTraditional}

Write $Top_{{\geq 1}}^{\ast/} \hookrightarrow Top^{\ast/}$ for the [[full subcategory]] of \emph{[[connected topological space|connected]]} [[pointed topological spaces]]. Write $Set^{\ast/}$ for the category of [[pointed sets]].

\begin{defn}
\label{BrownFunctorTraditional}\hypertarget{BrownFunctorTraditional}{}
A \textbf{[[Brown functor]]} is a functor

\begin{displaymath}
F\;\colon \; Ho(Top_{\geq 1}^{\ast/})^{op} \longrightarrow Set^{\ast/}
\end{displaymath}
(from the [[opposite category|opposite]] of the [[classical homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#ClassicalHomotopyCategory}{def.}, \href{Introduction+to+Stable+homotopy+theory+--+P#ClassicalPointedHomotopyCategory}{def.}) of [[connected topological space|connected]] [[pointed topological space|pointed]] [[topological spaces]]) such that

\begin{enumerate}%
\item \textbf{(additivity)} $F$ takes small coproducts ([[wedge sums]]) to [[products]];


\item \textbf{(Mayer-Vietoris)} If $X = Int(A) \cup Int(B)$ then for all $x_A \in F(A)$ and $x_B \in F(B)$ such that $(x_A)|_{A \cap B} = (x_B)|_{A \cap B}$ then there exists $x_X \in F(X)$ such that $x_A = (x_X)|_A$ and $x_B = (x_X)|_B$.



\end{enumerate}
\end{defn}
\begin{prop}
\label{EveryComponentOfAdditiveReducedCohomologyIsBrownFunctor}\hypertarget{EveryComponentOfAdditiveReducedCohomologyIsBrownFunctor}{}
For every \hyperlink{WedgeAxiom}{additive} [[reduced cohomology theory]] $\tilde E^\bullet(-) \colon Ho(Top^{\ast/})^{op}\to Set^{\ast/}$ (def. \ref{ReducedGeneralizedCohomologyHomotopyTheoretically}) and for each degree $n \in \mathbb{N}$, the restriction of $\tilde E^n(-)$ to connected spaces is a [[Brown functor]] (def. \ref{BrownFunctorTraditional}).

\end{prop}
\begin{proof}
Under the relation between reduced and unreduced cohomology \hyperlink{RelationBetweenUnreducedAndReducedCohomology}{above}, this follows from the [[exact sequence|exactness]] of the [[Mayer-Vietoris sequence]] of prop. \ref{MayerVietorisSequenceInGeneralizedCohomology}.

\end{proof}
\begin{theorem}
\label{BrownRepresentabilityForTraditionalBrownFunctors}\hypertarget{BrownRepresentabilityForTraditionalBrownFunctors}{}
\textbf{(Brown representability)}

Every [[Brown functor]] $F$ (def. \ref{BrownFunctorTraditional}) is [[representable functor|representable]], hence there exists $X \in Top_{\geq 1}^{\ast/}$ and a [[natural isomorphism]]

\begin{displaymath}
[-,X]_{\ast} \overset{\simeq}{\longrightarrow} F(-)
\end{displaymath}
(where $[-,-]_\ast$ denotes the [[hom-functor]] of $Ho(Top_{\geq 1}^{\ast/})$ (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyCategoryOfPointedModelStructureIsEnrichedInPointedSets}{exmpl.})).

\end{theorem}
(e.g. \hyperlink{AguilarGitlerPrieto02}{AGP 02, theorem 12.2.22})

\begin{remark}
\label{ConnectivityInTraditionalBrownRepresentability}\hypertarget{ConnectivityInTraditionalBrownRepresentability}{}
A key subtlety in theorem \ref{BrownRepresentabilityForTraditionalBrownFunctors} is the restriction to \emph{connected} pointed topological spaces in def. \ref{BrownFunctorTraditional}. This comes about since the proof of the theorem requires that continuous functions $f \colon X \longrightarrow Y$ that induce isomorphisms on pointed homotopy classes

\begin{displaymath}
[S^n,X]_\ast \longrightarrow [S^n,Y]_\ast
\end{displaymath}
for all $n$ are [[weak homotopy equivalences]] (For instance in \hyperlink{AguilarGitlerPrieto02}{AGP 02} this is used in the proof of theorem 12.2.19 there). But $[S^n,X]_\ast = \pi_n(X,x)$ gives the $n$th [[homotopy group]] of $X$ \emph{only} for the canonical basepoint, while for a weak homotopy equivalence in general one needs to consider the homotopy groups at all possible basepoints, at least one for each connected component. But so if one does assume that all spaces involved are connected, hence only have one connected component, then indeed weak homotopy equivalences are equivalently those maps $X\to Y$ making all the $[S^n,X]_\ast \longrightarrow [S^n,Y]_\ast$ into isomorphisms.

See also example \ref{TheClassicalPointedConnectedHomotopyCategoryAsDomainForTheAbstractBrownRepresentabilityTheorem} below.

\end{remark}
The representability result applied degreewise to an additive reduced cohomology theory will yield (prop. \ref{AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum} below) the following concept.

\begin{defn}
\label{OmegaSpectrum}\hypertarget{OmegaSpectrum}{}
An \textbf{[[Omega-spectrum]]} $X$ (\href{Introduction+to+Stable+homotopy+theory+--+1-1#OmegaSpectrum}{def.}) is

\begin{enumerate}%
\item a sequence $\{X_n\}_{n \in \mathbb{N}}$ of [[pointed topological spaces]] $X_n \in Top^{\ast/}$


\item [[weak homotopy equivalences]]

\begin{displaymath}
\tilde \sigma_n
  \;\colon\;
  X_n \underoverset{\in W_{cl}}{\tilde \sigma_n}{\longrightarrow} \Omega X_{n+1}
\end{displaymath}
for each $n \in \mathbb{N}$, form each space to the [[loop space]] of the following space.



\end{enumerate}
\end{defn}
\begin{prop}
\label{AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum}\hypertarget{AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum}{}
Every \hyperlink{WedgeAxiom}{additive} [[reduced cohomology theory]] $\tilde E^\bullet(-) \colon (Top_{CW}^\ast)^{op} \longrightarrow Ab^{\mathbb{Z}}$ according to def. \ref{ReducedGeneralizedCohomologyHomotopyTheoretically}, is [[representable functor|represented]] by an [[Omega-spectrum]] $E$ (def. \ref{OmegaSpectrum}) in that in each degree $n \in \mathbb{N}$

\begin{enumerate}%
\item $\tilde E^n(-)$ is represented by some $E_n \in Ho(Top^{\ast/})$;


\item the \hyperlink{SuspensionIsomorphismForReducedGeneralizedCohomology}{suspension isomorphism} $\sigma_n$ of $\tilde E^\bullet$ is represented by the structure map $\tilde \sigma_n$ of the Omega-spectrum in that for all $X \in Top^{\ast/}$ the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    \tilde E^{n}(X)
      &\overset{\sigma_n(X)}{\longrightarrow}&
      &\longrightarrow&
    \tilde E^{n+1}(\Sigma X)
    \\
    {}^{\mathllap{\simeq}}\downarrow
      && &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    [X,E_n]_\ast
      &\overset{[X,\tilde \sigma_n]_\ast}{\longrightarrow}&
    [X, \Omega E_{n+1}]_\ast
      &\simeq&
    [\Sigma X, E_{n+1}]_\ast
  }
  \,,
\end{displaymath}
where $[-,-]_\ast \coloneqq Hom_{Ho(Top_{\geq 1}^{\ast/})}$ denotes the [[hom-sets]] in the [[classical pointed homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+P#ClassicalModelStructureOnPointedTopologicalSpaces}{def.}) and where in the bottom right we have the $(\Sigma\dashv \Omega)$-[[adjunction]] isomorphism (\href{Introduction+to+Stable+homotopy+theory+--+P#SuspensionAndLoopAreAdjointOnHomotopyCategory}{prop.}).



\end{enumerate}
\end{prop}
\begin{proof}
If it were not for the connectedness clause in def. \ref{BrownFunctorTraditional} (remark \ref{ConnectivityInTraditionalBrownRepresentability}), then theorem \ref{BrownRepresentabilityForTraditionalBrownFunctors} with prop. \ref{EveryComponentOfAdditiveReducedCohomologyIsBrownFunctor} would immediately give the existence of the $\{E_n\}_{n \in \mathbb{N}}$ and the remaining statement would follow immediately with the [[Yoneda lemma]], which says in particular that morphisms between [[representable functors]] are in [[natural bijection]] with the morphisms of objects that represent them.

The argument with the connectivity condition in Brown representability taken into account is essentially the same, just with a little bit more care:

For $X$ a [[pointed topological space]], write $X^{(0)}$ for the connected component of its basepoint. Observe that the [[loop space]] of a pointed topological space only depends on this connected component:

\begin{displaymath}
\Omega X \simeq \Omega (X^{(0)})
  \,.
\end{displaymath}
Now for $n \in \mathbb{N}$, to show that $\tilde E^n(-)$ is representable by some $E_n \in Ho(Top^{\ast/})$, use first that the restriction of $\tilde E^{n+1}$ to connected spaces is represented by some $E_{n+1}^{(0)}$. Observe that the [[reduced suspension]] of any $X \in Top^{\ast/}$ lands in $Top_{\geq 1}^{\ast/}$. Therefore the $(\Sigma\dashv \Omega)$-[[adjunction]] isomorphism (\href{Introduction+to+Stable+homotopy+theory+--+P#SuspensionAndLoopAreAdjointOnHomotopyCategory}{prop.}) implies that $\tilde E^{n+1}(\Sigma(-))$ is represented on \emph{all} of $Top^{\ast/}$ by $\Omega E_{n+1}^{(0)}$:

\begin{displaymath}
\tilde E^{n+1}(\Sigma X)
    \simeq
  [\Sigma X, E_{n+1}^{(0)}]_\ast
    \simeq
  [X, \Omega E_{n+1}^{(0)}]_\ast
    \simeq
  [X, \Omega E_{n+1}]_\ast
  \,,
\end{displaymath}
where $E_{n+1}$ is any pointed topological space with the given connected component $E_{n+1}^{(0)}$.

Now the \hyperlink{SuspensionIsomorphismForReducedGeneralizedCohomology}{suspension isomorphism} of $\tilde E$ says that $E_n \in Ho(Top^{\ast/})$ representing $\tilde E^n$ exists and is given by $\Omega E_{n+1}^{(0)}$:

\begin{displaymath}
\tilde E^n(X) \simeq \tilde E^{n+1}(\Sigma, X)\simeq [X,\Omega E_{n+1}]
\end{displaymath}
for any $E_{n+1}$ with connected component $E_{n+1}^{(0)}$.

This completes the proof. Notice that running the same argument next for $(n+1)$ gives a representing space $E_{n+1}$ such that its connected component of the base point is $E_{n+1}^{(0)}$ found before. And so on.

\end{proof}
Conversely:

\begin{prop}
\label{SpectrumRepresentsCohomologyTheoryDegreewise}\hypertarget{SpectrumRepresentsCohomologyTheoryDegreewise}{}
Every [[Omega-spectrum]] $E$, def. \ref{OmegaSpectrum}, represents an \hyperlink{WedgeAxiom}{additive} [[reduced cohomology theory]] def. \ref{ReducedGeneralizedCohomology} $\tilde E^\bullet$ by

\begin{displaymath}
\tilde E^n(X)
    \coloneqq
  [X,E_n]_\ast
\end{displaymath}
with \hyperlink{SuspensionIsomorphismForReducedGeneralizedCohomology}{suspension isomorphism} given by

\begin{displaymath}
\sigma_n
    \;\colon\;
  \tilde E^n(X)
    =
  [X,E_n]_\ast
    \overset{[X,\tilde \sigma_n]}{\longrightarrow}
  [X, \Omega E_{n+1}]_\ast
    \overset{\simeq}{\to}
  [\Sigma X, E_{n+1}]
    =
  \tilde E^{n+1}(\Sigma X)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
The \hyperlink{WedgeAxiom}{additivity} is immediate from the construction. The \hyperlink{ReducedExactnessAxiom}{exactnes} follows from the [[long exact sequences]] of [[homotopy cofiber sequences]] given by \href{Introduction+to+Stable+homotopy+theory+--+P#LongFiberSequence}{this prop.}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
If we consider the [[stable homotopy category]] $Ho(Spectra)$ of [[spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#TheStableHomotopyCategory}{def.}) and consider any [[topological space]] $X$ in terms of its [[suspension spectrum]] $\Sigma^\infty X \in Ho(Spectra)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-1#SuspensionSpectrum}{exmpl.}), then the statement of prop. \ref{SpectrumRepresentsCohomologyTheoryDegreewise} is more succinctly summarized by saying that the [[graded abelian group|graded]] reduced cohomology groups of a topological space $X$ represented by an [[Omega-spectrum]] $E$ are the hom-groups

\begin{displaymath}
\tilde E^\bullet(X)
   \;\simeq\;
  [\Sigma^\infty X, \Sigma^\bullet E]
\end{displaymath}
in the [[stable homotopy category]], into all the [[suspensions]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#StableModelStructureOnSequentiaSpectraIsStableModelCategory}{thm.}) of $E$.

This means that more generally, for $X \in Ho(Spectra)$ any spectrum, it makes sense to consider

\begin{displaymath}
\tilde E^\bullet(X)
    \;\coloneqq\;
  [X,\Sigma^\bullet E]
\end{displaymath}
to be the graded reduced generalized $E$-cohomology groups of the spectrum $X$.

See also in \emph{[[Introduction to Stable homotopy theory -- 1|part 1]]} \href{Introduction+to+Stable+homotopy+theory+--+1-1#ForASpectrumXGeneralizedECohomology}{this example}.

\end{remark}
\hypertarget{BrownRepresentabilityAppliedToOrdinaryCohomology}{}\paragraph*{{Application to ordinary cohomology}}\label{BrownRepresentabilityAppliedToOrdinaryCohomology}

\begin{example}
\label{EilenbergMacLaneSpectrum}\hypertarget{EilenbergMacLaneSpectrum}{}
Let $A$ be an [[abelian group]]. Consider [[singular cohomology]] $H^n(-,A)$ with [[coefficients]] in $A$. The corresponding [[reduced cohomology]] evaluated on [[n-spheres]] satisfies

\begin{displaymath}
\tilde H^n(S^q,A)
  \simeq
  \left\{
    \itexarray{
      A & if \; q = n
      \\
      0 & otherwise
    }
  \right.
\end{displaymath}
Hence singular cohomology is a \hyperlink{S1GeneralizedCohomology}{generalized cohomology theory} which is ``[[ordinary cohomology]]'' in the sense of def. \ref{AdditiveOrdinary}.

Applying the [[Brown representability theorem]] as in prop. \ref{AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum} hence produces an [[Omega-spectrum]] (def. \ref{OmegaSpectrum}) whose $n$th component space is characterized as having [[homotopy groups]] concentrated in degree $n$ on $A$. These are called \emph{[[Eilenberg-MacLane spaces]]} $K(A,n)$

\begin{displaymath}
\pi_q(K(A,n))
    \simeq
  \left\{
    \itexarray{
      A & if \; q = n
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Here for $n \gt 0$ then $K(A,n)$ is connected, therefore with an essentially unique basepoint, while $K(A,0)$ is (homotopy equivalent to) the underlying set of the group $A$.

Such spectra are called \textbf{[[Eilenberg-MacLane spectra]]} $H A$:

\begin{displaymath}
(H A)_n \simeq K(A,n)
  \,.
\end{displaymath}
\end{example}
As a consequence of example \ref{EilenbergMacLaneSpectrum} one obtains the uniqueness result of Eilenberg-Steenrod:

\begin{prop}
\label{}\hypertarget{}{}
Let $\tilde E_1$ and $\tilde E_2$ be ordinary (def. \ref{AdditiveOrdinary}) [[generalized (Eilenberg-Steenrod) cohomology theories]]. If there is an isomorphism

\begin{displaymath}
\tilde E_1(S^0) \simeq \tilde E_2(S^0)
\end{displaymath}
of [[cohomology groups]] of the [[0-sphere]], then there is an [[isomorphism]] of cohomology theories

\begin{displaymath}
\tilde E_1 \overset{\simeq}{\longrightarrow} \tilde E_2
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, theorem 12.3.6})

\hypertarget{homotopytheoretic_discussion}{}\paragraph*{{Homotopy-theoretic discussion}}\label{homotopytheoretic_discussion}

Using abstract [[homotopy theory]] in the guise of [[model category]] theory (see the \emph{[[Introduction to Stable homotopy theory -- P|lecture notes on classical homotopy theory]]), the traditional proof and further discussion of the [[Brown representability theorem]] \hyperlink{BrownRepresentabilityTraditional}{above} becomes more transparent (\hyperlink{LurieHigherAlgebra}{Lurie 10, section 1.4.1}, for exposition see also \href{Brown+representability+theorem#Mathew11}{Mathew 11}).}

This abstract homotopy-theoretic proof uses the general concept of [[homotopy colimits]] in [[model categories]] as well as the concept of [[derived hom-spaces]] (``[[(∞,1)-category|∞-categories]]''). Even though in the accompanying [[Introduction to Stable homotopy theory -- P|Lecture notes on classical homotopy theory]] these concepts are only briefly indicated, the following is included for the interested reader.

\begin{defn}
\label{BrownFunctorOnInfinityCategory}\hypertarget{BrownFunctorOnInfinityCategory}{}
Let $\mathcal{C}$ be a [[model category]]. A [[functor]]

\begin{displaymath}
F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
\end{displaymath}
(from the [[opposite category|opposite]] of the [[homotopy category of a model category|homotopy category]] of $\mathcal{C}$ to [[Set]])

is called a \textbf{[[Brown functor]]} if

\begin{enumerate}%
\item it sends small [[coproducts]] to [[products]];


\item it sends [[homotopy pushouts]] in $\mathcal{C}\to Ho(\mathcal{C})$ to [[weak pullbacks]] in [[Set]] (see remark \ref{WeakPullbacks}).



\end{enumerate}
\end{defn}
\begin{remark}
\label{WeakPullbacks}\hypertarget{WeakPullbacks}{}
A \emph{[[weak pullback]]} is a diagram that satisfies the existence clause of a [[pullback]], but not necessarily the uniqueness condition. Hence the second clause in def. \ref{BrownFunctorOnInfinityCategory} says that for a [[homotopy pushout]] square

\begin{displaymath}
\itexarray{
    Z &\longrightarrow& X
    \\
    \downarrow &\swArrow& \downarrow
    \\
    Y &\longrightarrow& X \underset{Z}{\sqcup}Y
  }
\end{displaymath}
in $\mathcal{C}$, then the induced universal morphism

\begin{displaymath}
F\left(X \underset{Z}{\sqcup}Y\right)
  \stackrel{epi}{\longrightarrow}
  F(X) \underset{F(Z)}{\times} F(Y)
\end{displaymath}
into the actual [[pullback]] is an [[epimorphism]].

\end{remark}
\begin{defn}
\label{CompactGenerationByCogroupObjects}\hypertarget{CompactGenerationByCogroupObjects}{}
Say that a [[model category]] $\mathcal{C}$ is \textbf{compactly generated by cogroup objects closed under suspensions} if

\begin{enumerate}%
\item $\mathcal{C}$ is [[compactly generated (∞,1)-category|generated]] by a set

\begin{displaymath}
\{S_i \in \mathcal{C}\}_{i \in I}
\end{displaymath}
of [[compact object in an (infinity,1)-category|compact objects]] (i.e. every object of $\mathcal{C}$ is a [[homotopy colimit]] of the objects $S_i$.)


\item each $S_i$ admits the structure of a [[cogroup]] object in the [[homotopy category of a model category|homotopy category]] $Ho(\mathcal{C})$;


\item the set $\{S_i\}$ is closed under forming [[reduced suspensions]].



\end{enumerate}
\end{defn}
\begin{example}
\label{SuspensionsAreHCogroupObjects}\hypertarget{SuspensionsAreHCogroupObjects}{}
\textbf{([[suspensions are H-cogroup objects]])}

Let $\mathcal{C}$ be a [[model category]] and $\mathcal{C}^{\ast/}$ its pointed model category (\href{Introduction+to+Stable+homotopy+theory+--+P#ModelStructureOnSliceCategory}{prop.}) with [[zero object]] (\href{Introduction+to+Stable+homotopy+theory+--+P#PointedObjectsHaveZeroObject}{rmk.}). Write $\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0$ for the [[reduced suspension]] functor.

Then the [[fold map]]

\begin{displaymath}
\Sigma X \coprod \Sigma X
    \simeq
  0 \underset{X}{\sqcup} 0 \underset{X}{\sqcup} 0
    \longrightarrow
  0 \underset{X}{\sqcup} X \underset{X}{\sqcup} 0
  \simeq
  0 \underset{X}{\sqcup} 0
  \simeq
  \Sigma X
\end{displaymath}
exhibits [[cogroup]] structure on the image of any [[suspension object]] $\Sigma X$ in the [[homotopy category of an (∞,1)-category|homotopy category]].

This is equivalently the [[group]]-structure of the first ([[fundamental group|fundamental]]) [[homotopy group]] of the values of [[representable functor|functor co-represented]] by $\Sigma X$:

\begin{displaymath}
Ho(\mathcal{C})(\Sigma X, -)
   \;\colon\;
  Y
   \mapsto
  Ho(\mathcal{C})(\Sigma X, Y)
   \simeq
  Ho(\mathcal{C})(X, \Omega Y)
   \simeq
  \pi_1 Ho(\mathcal{C})(X, Y)
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{TheClassicalPointedConnectedHomotopyCategoryAsDomainForTheAbstractBrownRepresentabilityTheorem}\hypertarget{TheClassicalPointedConnectedHomotopyCategoryAsDomainForTheAbstractBrownRepresentabilityTheorem}{}
In bare [[pointed homotopy types]] $\mathcal{C} = Top^{\ast/}_{Quillen}$, the ([[homotopy types]] of) [[n-spheres]] $S^n$ are [[cogroup]] objects for $n \geq 1$, but not for $n = 0$, by example \ref{SuspensionsAreHCogroupObjects}. And of course they are [[compact object in an (∞,1)-category|compact objects]].

So while $\{S^n\}_{n \in \mathbb{N}}$ generates all of the homotopy theory of $Top^{\ast/}$, the latter is \emph{not} an example of def. \ref{CompactGenerationByCogroupObjects} due to the failure of $S^0$ to have [[cogroup]] structure.

Removing that generator, the homotopy theory generated by $\{S^n\}_{{n \in \mathbb{N}} \atop {n \geq 1}}$ is $Top^{\ast/}_{\geq 1}$, that of \emph{[[connected object|connected]]} [[pointed homotopy types]]. This is one way to see how the connectedness condition in the classical version of Brown representability theorem arises. See also remark \ref{ConnectivityInTraditionalBrownRepresentability} above.

\end{example}
See also (\hyperlink{LurieHigherAlgebra}{Lurie 10, example 1.4.1.4})

In homotopy theories compactly generated by cogroup objects closed under forming suspensions, the following strenghtening of the [[Whitehead theorem]] holds.

\begin{prop}
\label{WhiteheadTheoremForCompactGenerationByCogroupObjects}\hypertarget{WhiteheadTheoremForCompactGenerationByCogroupObjects}{}
In a homotopy theory compactly generated by cogroup objects $\{S_i\}_{i \in I}$ closed under forming suspensions, according to def. \ref{CompactGenerationByCogroupObjects}, a morphism $f\colon X \longrightarrow Y$ is an [[equivalence in an (infinity,1)-category|equivalence]] precisely if for each $i \in I$ the induced function of maps in the [[homotopy category of a model category|homotopy category]]

\begin{displaymath}
Ho(\mathcal{C})(S_i,f)
   \;\colon\;
  Ho(\mathcal{C})(S_i,X)
   \longrightarrow
  Ho(\mathcal{C})(S_i,Y)
\end{displaymath}
is an [[isomorphism]] (a [[bijection]]).

\end{prop}
(\hyperlink{LurieHigherAlgebra}{Lurie 10, p. 114, Lemma star})

\begin{proof}
By the [[(∞,1)-Yoneda lemma|∞-Yoneda lemma]], the morphism $f$ is a weak equivalence precisely if for all objects $A \in \mathcal{C}$ the induced morphism of [[derived hom-spaces]]

\begin{displaymath}
\mathcal{C}(A,f)
  \;\colon\;
  \mathcal{C}(A,X)
   \longrightarrow
  \mathcal{C}(A,Y)
\end{displaymath}
is an equivalence in $Top_{Quillen}$. By assumption of compact generation and since the hom-functor $\mathcal{C}(-,-)$ sends [[homotopy colimits]] in the first argument to [[homotopy limits]], this is the case precisely already if it is the case for $A \in \{S_i\}_{i \in I}$.

Now the maps

\begin{displaymath}
\mathcal{C}(S_i,f)
  \;\colon\;
  \mathcal{C}(S_i,X)
   \longrightarrow
  \mathcal{C}(S_i,Y)
\end{displaymath}
are weak equivalences in $Top_{Quillen}$ if they are [[weak homotopy equivalences]], hence if they induce [[isomorphisms]] on all [[homotopy groups]] $\pi_n$ for \textbf{all basepoints}.

It is this last condition of testing on all basepoints that the assumed [[cogroup]] structure on the $S_i$ allows to do away with: this cogroup structure implies that $\mathcal{C}(S_i,-)$ has the structure of an $H$-group, and this implies (by group multiplication), that all [[connected components]] have the same homotopy groups, hence that all homotopy groups are independent of the choice of basepoint, up to isomorphism.

Therefore the above morphisms are equivalences precisely if they are so under applying $\pi_n$ based on the connected component of the [[zero morphism]]

\begin{displaymath}
\pi_n\mathcal{C}(S_i,f)
    \;\colon\;
  \pi_n \mathcal{C}(S_i,X)
     \longrightarrow
  \pi_n\mathcal{C}(S_i,Y)
  \,.
\end{displaymath}
Now in this pointed situation we may use that

\begin{displaymath}
\begin{aligned}
    \pi_n \mathcal{C}(-,-) & \simeq \pi_0 \mathcal{C}(-,\Omega^n(-))
    \\
    & \simeq \pi_0\mathcal{C}(\Sigma^n(-),-)
    \\
    & \simeq Ho(\mathcal{C})(\Sigma^n(-),-)
  \end{aligned}
\end{displaymath}
to find that $f$ is an equivalence in $\mathcal{C}$ precisely if the induced morphisms

\begin{displaymath}
Ho(\mathcal{C})(\Sigma^n S_i, f)
    \;\colon\;
  Ho(\mathcal{C})(\Sigma^n S_i,X)
    \longrightarrow
  Ho(\mathcal{C})(\Sigma^n S_i,Y)
\end{displaymath}
are isomorphisms for all $i \in I$ and $n \in \mathbb{N}$.

Finally by the assumption that each suspension $\Sigma^n S_i$ of a generator is itself among the set of generators, the claim follows.

\end{proof}
\begin{theorem}
\label{BrownRepresentabilityOnPresentableInfinityCategories}\hypertarget{BrownRepresentabilityOnPresentableInfinityCategories}{}
\textbf{(Brown representability)}

Let $\mathcal{C}$ be a [[model category]] compactly generated by cogroup objects closed under forming suspensions, according to def. \ref{CompactGenerationByCogroupObjects}. Then a [[functor]]

\begin{displaymath}
F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
\end{displaymath}
(from the [[opposite category|opposite]] of the [[homotopy category of a model category|homotopy category]] of $\mathcal{C}$ to [[Set]]) is [[representable functor|representable]] precisely if it is a [[Brown functor]], def. \ref{BrownFunctorOnInfinityCategory}.

\end{theorem}
(\hyperlink{LurieHigherAlgebra}{Lurie 10, theorem 1.4.1.2})

\begin{proof}
Due to the version of the Whitehead theorem of prop. \ref{WhiteheadTheoremForCompactGenerationByCogroupObjects} we are essentially reduced to showing that [[Brown functors]] $F$ are representable on the $S_i$. To that end consider the following lemma. (In the following we notationally identify, via the [[Yoneda lemma]], objects of $\mathcal{C}$, hence of $Ho(\mathcal{C})$, with the functors they [[representable functor|represent]].)

Lemma ($\star$): \emph{Given $X \in \mathcal{C}$ and $\eta \in F(X)$, hence $\eta \colon X \to F$, then there exists a morphism $f \colon X \to X'$ and an [[extension]] $\eta' \colon X' \to F$ of $\eta$ which induces for each $S_i$ a [[bijection]] $\eta'\circ (-) \colon PSh(Ho(\mathcal{C}))(S_i,X') \stackrel{\simeq}{\longrightarrow} Ho(\mathcal{C})(S_i,F) \simeq F(S_i)$.}

To see this, first notice that we may directly find an extension $\eta_0$ along a map $X\to X_o$ such as to make a [[surjection]]: simply take $X_0$ to be the [[coproduct]] of \textbf{all} possible elements in the codomain and take

\begin{displaymath}
\eta_0
  \;\colon\;
   X
     \sqcup
   \left(
     \underset{{i \in I,} \atop {\gamma \colon S_i \stackrel{}{\to} F}}{\coprod}
     S_i
   \right)
   \longrightarrow
   F
\end{displaymath}
to be the canonical map. (Using that $F$, by assumption, turns coproducts into products, we may indeed treat the coproduct in $\mathcal{C}$ on the left as the coproduct of the corresponding functors.)

To turn the surjection thus constructed into a bijection, we now successively form quotients of $X_0$. To that end proceed by [[induction]] and suppose that $\eta_n \colon X_n \to F$ has been constructed. Then for $i \in I$ let

\begin{displaymath}
K_i
    \coloneqq
  ker
  \left(
     Ho(\mathcal{C})(S_i, X_n)
        \stackrel{\eta_n \circ (-)}{\longrightarrow}
     F(S_i)
   \right)
\end{displaymath}
be the [[kernel]] of $\eta_n$ evaluated on $S_i$. These $K_i$ are the pieces that need to go away in order to make a bijection. Hence define $X_{n+1}$ to be their joint [[homotopy cofiber]]

\begin{displaymath}
X_{n+1}
     \coloneqq
   coker\left(
     \left(
        \underset{{i \in I,} \atop {\gamma \in K_i}}{\sqcup} S_i
     \right)
     \overset{(\gamma)_{{i \in I} \atop {\gamma\in K_i}}}{\longrightarrow}
     X_n
  \right)
  \,.
\end{displaymath}
Then by the assumption that $F$ takes this homotopy cokernel to a [[weak limit|weak]] [[fiber]] (as in remark \ref{WeakPullbacks}), there exists an extension $\eta_{n+1}$ of $\eta_n$ along $X_n \to X_{n+1}$:

Then by the assumption that $F$ takes this homotopy cokernel to a [[weak limit|weak]] [[fiber]] (as in remark \ref{WeakPullbacks}), there exists an extension $\eta_{n+1}$ of $\eta_n$ along $X_n \to X_{n+1}$:

\begin{displaymath}
\itexarray{
    \left(
      \underset{{i \in I}\atop {\gamma \in K_i}}{\sqcup} S_i
    \right)
    &\overset{(\gamma)_{{i \in I}\atop \gamma \in K_i}}{\longrightarrow}&
    X_n
    &\overset{\eta_n}{\longrightarrow}& F
    \\
    \downarrow &(po^{h})& \downarrow & \nearrow_{\mathrlap{\exists \eta_{n+1}}}
    \\
    \ast &\longrightarrow& X_{n+1}
  }
  \;\;\;\;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;\;\;\;
  \itexarray{
    && F(X_{n+1}) &\longrightarrow& \ast
    \\
    &{}^{\mathllap{\exists \eta_{n+1}}}\nearrow& \downarrow^{\mathrlap{epi}} && \downarrow
    \\
    \ast &\overset{\eta_n}{\longrightarrow}& ker\left((\gamma^\ast\right)_{{i \in I} \atop {\gamma \in K_i}}) &\longrightarrow& \ast
    \\
    &{}_{\mathllap{\eta_n}}\searrow& \downarrow &(pb)& \downarrow
    \\
    && F(X_n)
    &\underset{(\gamma^\ast)_{{i \in I} \atop {\gamma \in K_i}} }{\longrightarrow}&
    \underset{{i \in I}\atop {\gamma\in K_i}}{\prod}F(S_i)
  }
  \,.
\end{displaymath}
It is now clear that we want to take

\begin{displaymath}
X' \coloneqq \underset{\rightarrow}{\lim}_n X_n
\end{displaymath}
and extend all the $\eta_n$ to that colimit. Since we have no condition for evaluating $F$ on colimits other than pushouts, observe that this [[sequential colimit]] is equivalent to the following pushout:

\begin{displaymath}
\itexarray{
    \underset{n}{\sqcup} X_n
    &\longrightarrow& \underset{n}{\sqcup} X_{2n}
    \\
    \downarrow && \downarrow
    \\
    \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X'
  }
  \,,
\end{displaymath}
where the components of the top and left map alternate between the identity on $X_n$ and the above successor maps $X_n \to X_{n+1}$. Now the excision property of $F$ applies to this pushout, and we conclude the desired extension $\eta' \colon X' \to F$:

\begin{displaymath}
\itexarray{
    && \underset{n}{\sqcup} X_n
    \\
    & \swarrow && \searrow
    \\
    \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' &\longleftarrow& \underset{n}{\sqcup} X_{2n}
    \\
    & {}_{\mathllap{(\eta_{2n+1})_{n}}}\searrow& \downarrow^{\mathrlap{\exists \eta}} & \swarrow_{\mathrlap{(\eta_{2n})_n}}
    \\
    && F
  }
  \;\;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;\;
  \itexarray{
    && F(X')
    \\
    &{}^{\mathllap{\exists \eta}}\nearrow& \downarrow^{\mathrlap{epi}}
    \\
    &\ast \overset{(\eta_n)_n}{\longrightarrow}& \underset{\longleftarrow}{\lim}_n F(X_n)
    \\
    & \swarrow && \searrow
    \\
    \underset{n}{\prod}F(X_{2n+1})
    && &&
    \underset{n}{\prod}(X_{2n})
    \\
    & \searrow && \swarrow
    \\
    && \underset{n}{\prod}F(X_n)
  }
  \,,
\end{displaymath}
It remains to confirm that this indeed gives the desired bijection. Surjectivity is clear. For injectivity use that all the $S_i$ are, by assumption, [[compact object|compact]], hence they may be taken inside the [[sequential colimit]]:

\begin{displaymath}
\itexarray{
    && X_{n(\gamma)}
    \\
    &{}^{\mathllap{ \exists \hat \gamma}}\nearrow& \downarrow
    \\
    S_i
      &\overset{\gamma}{\longrightarrow}&
    X'
      =
    \underset{\longrightarrow}{\lim}_n X_n
  }
  \,.
\end{displaymath}
With this, injectivity follows because by construction we quotiented out the kernel at each stage. Because suppose that $\gamma$ is taken to zero in $F(S_i)$, then by the definition of $X_{n+1}$ above there is a factorization of $\gamma$ through the point:

\begin{displaymath}
\itexarray{
    0 \colon & S_i
    &\overset{\hat \gamma}{\longrightarrow}&
    X_{n(\gamma)}
    &\overset{\eta_n}{\longrightarrow}& F
    \\
    & \downarrow && \downarrow &
    \\
    & \ast &\longrightarrow& X_{n(\gamma)+1}
    \\
    & && \downarrow
    \\
    & && X'
  }
\end{displaymath}
This concludes the proof of Lemma ($\star$).

Now apply the construction given by this lemma to the case $X_0 \coloneqq 0$ and the unique $\eta_0 \colon 0 \stackrel{\exists !}{\to} F$. Lemma $(\star)$ then produces an object $X'$ which represents $F$ on all the $S_i$, and we want to show that this $X'$ actually represents $F$ generally, hence that for every $Y \in \mathcal{C}$ the function

\begin{displaymath}
\theta \coloneqq \eta'\circ (-)
  \;\colon\;
  Ho(\mathcal{C})(Y,X')
   \stackrel{}{\longrightarrow}
  F(Y)
\end{displaymath}
is a [[bijection]].

First, to see that $\theta$ is surjective, we need to find a preimage of any $\rho \colon Y \to F$. Applying Lemma $(\star)$ to $(\eta',\rho)\colon X'\sqcup Y \longrightarrow F$ we get an extension $\kappa$ of this through some $X' \sqcup Y \longrightarrow Z$ and the morphism on the right of the following commuting diagram:

\begin{displaymath}
\itexarray{
    Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z)
    \\
    & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\kappa \circ (-)}}
    \\
    && F(-)
  }
  \,.
\end{displaymath}
Moreover, Lemma $(\star)$ gives that evaluated on all $S_i$, the two diagonal morphisms here become isomorphisms. But then prop. \ref{WhiteheadTheoremForCompactGenerationByCogroupObjects} implies that $X' \longrightarrow Z$ is in fact an equivalence. Hence the component map $Y \to Z \simeq Z$ is a lift of $\kappa$ through $\theta$.

Second, to see that $\theta$ is injective, suppose $f,g \colon Y \to X'$ have the same image under $\theta$. Then consider their [[homotopy pushout]]

\begin{displaymath}
\itexarray{
    Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X'
    \\
    \downarrow && \downarrow
    \\
    Y &\longrightarrow& Z
  }
\end{displaymath}
along the [[codiagonal]] of $Y$. Using that $F$ sends this to a [[weak pullback]] by assumption, we obtain an extension $\bar \eta$ of $\eta'$ along $X' \to Z$. Applying Lemma $(\star)$ to this gives a further extension $\bar \eta' \colon Z' \to Z$ which now makes the following diagram

\begin{displaymath}
\itexarray{
    Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z)
    \\
    & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\bar \eta' \circ (-)}}
    \\
    && F(-)
  }
\end{displaymath}
such that the diagonal maps become isomorphisms when evaluated on the $S_i$. As before, it follows via prop. \ref{WhiteheadTheoremForCompactGenerationByCogroupObjects} that the morphism $h \colon X' \longrightarrow Z'$ is an equivalence.

Since by this construction $h\circ f$ and $h\circ g$ are homotopic

\begin{displaymath}
\itexarray{
    Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X'
    \\
    \downarrow && \downarrow & \searrow^{\mathrlap{\stackrel{h}{\simeq}}}
    \\
    Y &\longrightarrow& Z &\longrightarrow& Z'
  }
\end{displaymath}
it follows with $h$ being an equivalence that already $f$ and $g$ were homotopic, hence that they represented the same element.

\end{proof}
\begin{prop}
\label{CohomologyFunctorOnInfinityCategoryIsBrownFunctor}\hypertarget{CohomologyFunctorOnInfinityCategoryIsBrownFunctor}{}
Given a reduced additive cohomology functor $H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, its underlying [[Set]]-valued functors $H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set$ are [[Brown functors]], def. \ref{BrownFunctorOnInfinityCategory}.

\end{prop}
\begin{proof}
The first condition on a [[Brown functor]] holds by definition of $H^\bullet$. For the second condition, given a [[homotopy pushout]] square

\begin{displaymath}
\itexarray{
    X_1 &\stackrel{f_1}{\longrightarrow}& Y_1
    \\
    \downarrow^{} && \downarrow
    \\
    X_2 &\stackrel{f_2}{\longrightarrow}& Y_2
  }
\end{displaymath}
in $\mathcal{C}$, consider the induced morphism of the [[long exact sequences]] given by prop. \ref{LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory}

\begin{displaymath}
\itexarray{
    H^\bullet(coker(f_2))
      &\longrightarrow&
    H^\bullet(Y_2)
      &\stackrel{f^\ast_2}{\longrightarrow}&
    H^\bullet(X_2)
      &\stackrel{}{\longrightarrow}&
    H^{\bullet+1}(\Sigma coker(f_2))
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    H^\bullet(coker(f_1))
      &\longrightarrow&
    H^\bullet(Y_1)
      &\stackrel{f^\ast_1}{\longrightarrow}&
    H^\bullet(X_1)
      &\stackrel{}{\longrightarrow}&
    H^{\bullet+1}(\Sigma coker(f_1))
  }
\end{displaymath}
Here the outer vertical morphisms are [[isomorphisms]], as shown, due to the [[pasting law]] (see also at \emph{\href{homotopy+pullback#FiberwiseRecognitionInStableCase}{fiberwise recognition of stable homotopy pushouts}}). This means that the [[four lemma]] applies to this diagram. Inspection shows that this implies the claim.

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[model category]] which satisfies the conditions of theorem \ref{BrownRepresentabilityOnPresentableInfinityCategories}, and let $(H^\bullet, \delta)$ be a reduced additive [[generalized cohomology]] functor on $\mathcal{C}$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}. Then there exists a [[spectrum object]] $E \in Stab(\mathcal{C})$ such that

\begin{enumerate}%
\item $H\bullet$ is degreewise [[representable functor|represented]] by $E$:

\begin{displaymath}
H^\bullet \simeq Ho(\mathcal{C})(-,E_\bullet)
  \,,
\end{displaymath}

\item the [[suspension isomorphism]] $\delta$ is given by the structure morphisms $\tilde \sigma_n \colon E_n \to \Omega E_{n+1}$ of the spectrum, in that

\begin{displaymath}
\delta
    \colon
  H^n(-)
   \simeq
  Ho(\mathcal{C})(-,E_n)
   \stackrel{Ho(\mathcal{C})(-,\tilde\sigma_n) }{\longrightarrow}
  Ho(\mathcal{C})(-,\Omega E_{n+1})
    \simeq
  Ho(\mathcal{C})(\Sigma (-), E_{n+1})
    \simeq
  H^{n+1}(\Sigma(-))
  \,.
\end{displaymath}


\end{enumerate}
\end{cor}
\begin{proof}
Via prop. \ref{CohomologyFunctorOnInfinityCategoryIsBrownFunctor}, theorem \ref{BrownRepresentabilityOnPresentableInfinityCategories} gives the first clause. With this, the second clause follows by the [[Yoneda lemma]].

\end{proof}
\hypertarget{S2MilnorExactSequence}{}\subsubsection*{{Milnor exact sequence}}\label{S2MilnorExactSequence}

\textbf{Idea.} One tool for computing [[generalized cohomology]] [[cohomology groups|groups]] via ``[[inverse limits]]'' are \emph{[[Milnor exact sequences]]}. For instance the generalized cohomology of the [[classifying space]] $B U(1)$ plays a key role in the [[complex oriented cohomology]]-theory discussed \hyperlink{ComplexOrientedCohomologyTheory}{below}, and via the equivalence $B U(1) \simeq \mathbb{C}P^\infty$ to the [[homotopy type]] of the infinite [[complex projective space]] (def. \ref{ComplexProjectiveSpace}), which is the [[direct limit]] of finite dimensional projective spaces $\mathbb{C}P^n$, this is an [[inverse limit]] of the generalized cohomology groups of the $\mathbb{C}P^n$s. But what really matters here is the [[derived functor]] of the [[limit]]-operation -- the [[homotopy limit]] -- and the [[Milnor exact sequence]] expresses how the naive limits receive corrections from higher ``[[lim{\tt \symbol{94}}1]]-terms''. In practice one mostly proceeds by verifying conditions under which these corrections happen to disappear, these are the \emph{[[Mittag-Leffler conditions]]}.

We need this for instance for the computation of [[Conner-Floyd Chern classes]] \hyperlink{ConnerFloydChernClasses}{below}.

\textbf{Literature.} (\hyperlink{Switzer75}{Switzer 75, section 7 from def. 7.57 on}, \hyperlink{Kochman96}{Kochman 96, section 4.2}, \hyperlink{GoerssJardine99}{Goerss-Jardine 99, section VI.2}, )

\hypertarget{Lim1}{}\paragraph*{{$Lim^1$}}\label{Lim1}

\begin{defn}
\label{TheBoundaryMapDefiningLim1}\hypertarget{TheBoundaryMapDefiningLim1}{}
Given a [[tower]] $A_\bullet$ of [[abelian groups]]

\begin{displaymath}
\cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0
\end{displaymath}
write

\begin{displaymath}
\partial
    \;\colon\;
  \underset{n}{\prod} A_n
    \longrightarrow
  \underset{n}{\prod} A_n
\end{displaymath}
for the homomorphism given by

\begin{displaymath}
\partial \;\colon\;
    (a_n)_{n \in \mathbb{N}}
  \mapsto
    (a_n - f_n(a_{n+1}))_{n \in \mathbb{N}}.
\end{displaymath}
\end{defn}
\begin{remark}
\label{LimitAsKernelAnalogousToLim1}\hypertarget{LimitAsKernelAnalogousToLim1}{}
The [[limit]] of a sequence as in def. \ref{TheBoundaryMapDefiningLim1} -- hence the group $\underset{\longleftarrow}{\lim}_n A_n$ universally equipped with morphisms $\underset{\longleftarrow}{\lim}_n A_n \overset{p_n}{\to} A_n$ such that all

\begin{displaymath}
\itexarray{
     && \underset{\longleftarrow}{\lim}_n A_n
     \\
     & {}^{\mathllap{p_{n+1}}}\swarrow && \searrow^{\mathrlap{p_n}}
     \\
     A_{n+1} && \overset{f_n}{\longrightarrow} && A_n
  }
\end{displaymath}
[[commuting diagram|commute]] -- is equivalently the [[kernel]] of the morphism $\partial$ in def. \ref{TheBoundaryMapDefiningLim1}.

\end{remark}
\begin{defn}
\label{Lim1ViaCokernel}\hypertarget{Lim1ViaCokernel}{}
Given a [[tower]] $A_\bullet$ of [[abelian groups]]

\begin{displaymath}
\cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0
\end{displaymath}
then $\underset{\longleftarrow}{\lim}^1 A_\bullet$ is the [[cokernel]] of the map $\partial$ in def. \ref{TheBoundaryMapDefiningLim1}, hence the group that makes a [[long exact sequence]] of the form

\begin{displaymath}
0 \to
  \underset{\longleftarrow}{\lim}_n A_n
  \longrightarrow
  \underset{n}{\prod} A_n
   \stackrel{\partial}{\longrightarrow}
  \underset{n}{\prod} A_n
    \longrightarrow
  \underset{\longleftarrow}{\lim}^1_n A_n
  \to 0
  \,,
\end{displaymath}
\end{defn}
\begin{prop}
\label{PropertiesOfLim1}\hypertarget{PropertiesOfLim1}{}
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. \ref{Lim1ViaCokernel}) satisfies

\begin{enumerate}%
\item for every [[short exact sequence]] $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 \;\;\; \in Ab^{(\mathbb{N}, \geq)}$ then the induced sequence

\begin{displaymath}
0
 \to
 \underset{\longleftarrow}{\lim}_n A_n
 \to
 \underset{\longleftarrow}{\lim}_n B_n
 \to
 \underset{\longleftarrow}{\lim}_n C_n
 \to
 \underset{\longleftarrow}{\lim}_n^1  A_n
 \to
 \underset{\longleftarrow}{\lim}_n^1 B_n
 \to
 \underset{\longleftarrow}{\lim}_n^1 C_n
 \to 0
\end{displaymath}
is a [[long exact sequence]] of abelian groups;


\item if $A_\bullet$ is a tower such that all maps are [[surjections]], then $\underset{\longleftarrow}{\lim}^1_n A_n \simeq 0$.



\end{enumerate}
\end{prop}
(e.g. \hyperlink{Switzer75}{Switzer 75, prop. 7.63}, \hyperlink{GoerssJardine96}{Goerss-Jardine 96, section VI. lemma 2.11})

\begin{proof}
For the first property: Given $A_\bullet$ a tower of abelian groups, write

\begin{displaymath}
L^\bullet(A_\bullet)
  \coloneqq
  \left[
    0
   \to
  \underset{deg \, 0}{\underbrace{\underset{n}{\prod} A_n}}
    \overset{\partial}{\longrightarrow}
  \underset{deg\, 1}{\underbrace{\underset{n}{\prod} A_n}}
   \to
   0
  \right]
\end{displaymath}
for the homomorphism from def. \ref{TheBoundaryMapDefiningLim1} regarded as the single non-trivial differential in a [[cochain complex]] of abelian groups. Then by remark \ref{LimitAsKernelAnalogousToLim1} and def. \ref{Lim1ViaCokernel} we have $H^0(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim} A_\bullet$ and $H^1(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim}^1 A_\bullet$.

With this, then for a short exact sequence of towers $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ the long exact sequence in question is the [[long exact sequence in homology]] of the corresponding short exact sequence of complexes

\begin{displaymath}
0
    \to
  L^\bullet(A_\bullet)
    \longrightarrow
  L^\bullet(B_\bullet)
    \longrightarrow
  L^\bullet(C_\bullet)
    \to
  0
  \,.
\end{displaymath}
For the second statement: If all the $f_k$ are surjective, then inspection shows that the homomorphism $\partial$ in def. \ref{TheBoundaryMapDefiningLim1} is surjective. Hence its [[cokernel]] vanishes.

\end{proof}
\begin{lemma}
\label{TowersOfAbelianGroupsHasEnoughInjectives}\hypertarget{TowersOfAbelianGroupsHasEnoughInjectives}{}
The category $Ab^{(\mathbb{N}, \geq)}$ of [[towers]] of [[abelian groups]] has [[enough injectives]].

\end{lemma}
\begin{proof}
The functor $(-)_n \colon Ab^{(\mathbb{N}, \geq)} \to Ab$ that picks the $n$-th component of the tower has a [[right adjoint]] $r_n$, which sends an abelian group $A$ to the tower

\begin{displaymath}
r_n
  \coloneqq
  \left[
    \cdots \overset{id}{\to}
    A \overset{id}{\to}
    \underset{= (r_n)_{n+1}}{\underbrace{A}} \overset{id}{\to}
    \underset{= (r_n)_n}{\underbrace{A}}
       \overset{id}{\to}
    \underset{= (r_n)_{n-1}}{\underbrace{0}}
    \to
    0
    \to
    \cdots
    \to
    0
    \to
    0
  \right]
  \,.
\end{displaymath}
Since $(-)_n$ itself is evidently an [[exact functor]], its right adjoint preserves injective objects (\href{injective+object#RightAdjointsOfExactFunctorsPreserveInjectives}{prop.}).

So with $A_\bullet \in Ab^{(\mathbb{N}, \geq)}$, let $A_n \hookrightarrow \tilde A_n$ be an injective resolution of the abelian group $A_n$, for each $n \in \mathbb{N}$. Then

\begin{displaymath}
A_\bullet
    \overset{(\eta_n)_{n \in \mathbb{N}}}{\longrightarrow}
  \underset{n \in \mathbb{R}}{\prod}
  r_n A_n
  \hookrightarrow
  \underset{n \in \mathbb{N}}{\prod} r_n \tilde A_n
\end{displaymath}
is an injective resolution for $A_\bullet$.

\end{proof}
\begin{prop}
\label{Lim1IsDerivedLimit}\hypertarget{Lim1IsDerivedLimit}{}
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. \ref{Lim1ViaCokernel}) is the [[derived functor in homological algebra|first right derived functor]] of the [[limit]] functor $\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab$.

\end{prop}
\begin{proof}
By lemma \ref{TowersOfAbelianGroupsHasEnoughInjectives} there are [[enough injectives]] in $Ab^{(\mathbb{N}, \geq)}$. So for $A_\bullet \in Ab^{(\mathbb{N}, \geq)}$ the given tower of abelian groups, let

\begin{displaymath}
0
    \to
  A_\bullet
    \overset{j^0}{\longrightarrow}
  J^0_\bullet
    \overset{j^1}{\longrightarrow}
  J^1_\bullet
    \overset{j^2}{\longrightarrow}
  J^2_\bullet
    \overset{}{\longrightarrow}
  \cdots
\end{displaymath}
be an [[injective resolution]]. We need to show that

\begin{displaymath}
\underset{\longleftarrow}{\lim}^1 A_\bullet
    \simeq
  ker(\underset{\longleftarrow}{\lim}(j^2))/im(\underset{\longleftarrow}{\lim}(j^1))
  \,.
\end{displaymath}
Since limits preserve [[kernels]], this is equivalently

\begin{displaymath}
\underset{\longleftarrow}{\lim}^1 A_\bullet
    \simeq
  (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1))
\end{displaymath}
Now observe that each injective $J^q_\bullet$ is a tower of epimorphism. This follows by the defining [[right lifting property]] applied against the monomorphisms of towers of the following form

\begin{displaymath}
\itexarray{
      \cdots &\to & 0 &\to& 0 &\longrightarrow& 0 &\longrightarrow& \mathbb{Z} &\overset{id}{\longrightarrow}& \cdots &\overset{id}{\longrightarrow}& \mathbb{Z} &\overset{id}{\longrightarrow}& \mathbb{Z}
     \\
     \cdots
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{incl}}
     &&
     \downarrow^{\mathrlap{id}}
     &&
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{id}}
     \\
     \cdots &\to& 0 &\to& 0 &\to & \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \cdots &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z}
  }
\end{displaymath}
Therefore by the second item of prop. \ref{PropertiesOfLim1} the long exact sequence from the first item of prop. \ref{PropertiesOfLim1} applied to the [[short exact sequence]]

\begin{displaymath}
0
   \to
  A_\bullet
   \overset{j^0}{\longrightarrow}
  J^0_\bullet
   \overset{j^1}{\longrightarrow}
  ker(j^2)_\bullet
   \to
  0
\end{displaymath}
becomes

\begin{displaymath}
0
     \to
   \underset{\longleftarrow}{\lim} A_\bullet
    \overset{\underset{\longleftarrow}{\lim} j^0}{\longrightarrow}
  \underset{\longleftarrow}{\lim} J^0_\bullet
    \overset{\underset{\longleftarrow}{\lim}j^1}{\longrightarrow}
  \underset{\longleftarrow}{\lim}(ker(j^2)_\bullet)
    \longrightarrow
  \underset{\longleftarrow}{\lim}^1 A_\bullet
    \longrightarrow
  0
  \,.
\end{displaymath}
Exactness of this sequence gives the desired identification $\underset{\longleftarrow}{\lim}^1 A_\bullet
    \simeq
  (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1))
 \,.$

\end{proof}
\begin{prop}
\label{AbstractCharacterizationOfLim1}\hypertarget{AbstractCharacterizationOfLim1}{}
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. \ref{Lim1ViaCokernel}) is in fact the unique functor, up to [[natural isomorphism]], satisfying the conditions in prop. \ref{AbstractCharacterizationOfLim1}.

\end{prop}
\begin{proof}
The proof of prop. \ref{Lim1IsDerivedLimit} only used the conditions from prop. \ref{PropertiesOfLim1}, hence any functor satisfying these conditions is the first right derived functor of $\underset{\longleftarrow}{\lim}$, up to natural isomorphism.

\end{proof}
The following is a kind of double dual version of the $\lim^1$ construction which is sometimes useful:

\begin{lemma}
\label{lim1AndExt1}\hypertarget{lim1AndExt1}{}
Given a [[cotower]]

\begin{displaymath}
A_\bullet = (A_0 \overset{f_0}{\to} A _1 \overset{f_1}{\to} A_2 \to \cdots)
\end{displaymath}
of [[abelian groups]], then for every abelian group $B \in Ab$ there is a [[short exact sequence]] of the form

\begin{displaymath}
0
    \to
  \underset{\longleftarrow}{\lim}^1_n Hom(A_n, B)
    \longrightarrow
  Ext^1( \underset{\longrightarrow}{\lim}_n A_n, B )
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n Ext^1( A_n, B)
    \to
  0
  \,,
\end{displaymath}
where $Hom(-,-)$ denotes the [[hom-object|hom-group]], $Ext^1(-,-)$ denotes the first [[Ext]]-group (and so $Hom(-,-) = Ext^0(-,-)$).

\end{lemma}
\begin{proof}
Consider the homomorphism

\begin{displaymath}
\tilde \partial
  \;\colon\;
  \underset{n}{\oplus} A_n
    \longrightarrow
  \underset{n}{\oplus} A_n
\end{displaymath}
which sends $a_n \in A_n$ to $a_n - f_n(a_n)$. Its [[cokernel]] is the [[colimit]] over the cotower, but its [[kernel]] is trivial (in contrast to the otherwise [[formal dual|formally dual]] situation in remark \ref{LimitAsKernelAnalogousToLim1}). Hence (as opposed to the long exact sequence in def. \ref{Lim1ViaCokernel}) there is a [[short exact sequence]] of the form

\begin{displaymath}
0
    \to
  \underset{n}{\oplus} A_n
    \overset{\tilde \partial}{\longrightarrow}
  \underset{n}{\oplus} A_n
    \overset{}{\longrightarrow}
  \underset{\longrightarrow}{lim}_n A_n
   \to
  0
  \,.
\end{displaymath}
Every short exact sequence gives rise to a [[long exact sequence]] of [[derived functors]] (\href{derived+functor+in+homological+algebra#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence}{prop.}) which in the present case starts out as

\begin{displaymath}
0
    \to
  Hom(\underset{\longrightarrow}{\lim}_n A_n,B)
    \longrightarrow
  \underset{n}{\prod} Hom( A_n, B )
    \overset{\partial}{\longrightarrow}
  \underset{n}{\prod} Hom( A_n, B )
    \longrightarrow
  Ext^1(\underset{\longrightarrow}{\lim}_n A_n,B)
    \longrightarrow
  \underset{n}{\prod} Ext^1(  A_n, B )
    \overset{\partial}{\longrightarrow}
  \underset{n}{\prod} Ext^1(  A_n, B )
    \longrightarrow
  \cdots
\end{displaymath}
where we used that [[direct sum]] is the [[coproduct]] in abelian groups, so that homs out of it yield a [[product]], and where the morphism $\partial$ is the one from def. \ref{TheBoundaryMapDefiningLim1} corresponding to the [[tower]]

\begin{displaymath}
Hom(A_\bullet,B)
  =
  (  \cdots \to Hom(A_2,B) \to Hom(A_1,B) \to  Hom(A_0,B) )
  \,.
\end{displaymath}
Hence truncating this long sequence by forming kernel and cokernel of $\partial$, respectively, it becomes the short exact sequence in question.

\end{proof}
\hypertarget{ThomTheorem}{}\subsubsection*{{Bordism and Thom's theorem}}\label{ThomTheorem}

\textbf{Idea.} By the [[Pontryagin-Thom collapse]] construction \hyperlink{ThomSpectra}{above}, there is an assignment

\begin{displaymath}
n Manifolds \longrightarrow \pi_n(M O)
\end{displaymath}
which sends [[disjoint union]] and [[Cartesian product]] of manifolds to sum and product in the [[ring]] of [[stable homotopy groups]] of the [[Thom spectrum]]. One finds then that two manifolds map to the same element in the [[stable homotopy groups]] $\pi_\bullet(M O)$ of the universal [[Thom spectrum]] precisely if they are connected by a [[bordism]]. The [[bordism]]-classes $\Omega_\bullet^O$ of manifolds form a [[commutative ring]] under [[disjoint union]] and [[Cartesian product]], called the \emph{[[bordism ring]]}, and Pontrjagin-Thom collapse produces a ring [[homomorphism]]

\begin{displaymath}
\Omega_\bullet^O \longrightarrow \pi_\bullet(M O)
  \,.
\end{displaymath}
[[Thom's theorem]] states that this homomorphism is an [[isomorphism]].

More generally, for $\mathcal{B}$ a multiplicative [[(B,f)-structure]], def. \ref{BfStructure}, there is such an identification

\begin{displaymath}
\Omega_\bullet^{\mathcal{B}} \simeq \pi_\bullet(M \mathcal{B})
\end{displaymath}
between the ring of $\mathcal{B}$-cobordism classes of manifolds with $\mathcal{B}$-structure and the [[stable homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]].

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, 1.5})

\hypertarget{bordism}{}\paragraph*{{Bordism}}\label{bordism}

Throughout, let $\mathcal{B}$ be a multiplicative [[(B,f)-structure]] (def. \ref{BfStructure}).

\begin{defn}
\label{NegativeOfManifoldWithBfStructure}\hypertarget{NegativeOfManifoldWithBfStructure}{}
Write $I \coloneqq [0,1]$ for the standard interval, regarded as a [[smooth manifold]] [[manifold with boundary|with boundary]]. For $c \in \mathbb{R}_+$ Consider its embedding

\begin{displaymath}
e \;\colon\; I \hookrightarrow \mathbb{R}\oplus \mathbb{R}_{\geq 0}
\end{displaymath}
as the arc

\begin{displaymath}
e \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2
  \,,
\end{displaymath}
where $(e_1, e_2)$ denotes the canonical [[linear basis]] of $\mathbb{R}^2$, and equipped with the structure of a manifold with normal [[framing]] structure (example \ref{ExamplesOfBfStructures}) by equipping it with the canonical framing

\begin{displaymath}
fr \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2
\end{displaymath}
of its [[normal bundle]].

Let now $\mathcal{B}$ be a [[(B,f)-structure]] (def. \ref{BfStructure}). Then for $X \overset{i}{\hookrightarrow}\mathbb{R}^k$ any embedded manifold with $\mathcal{B}$-structure $\hat g \colon X \to B_{k-n}$ on its [[normal bundle]] (def. \ref{ManifoldWithBfStructure}), define its \textbf{negative} or \textbf{orientation reversal} $-(X,i,\hat g)$ of $(X,i, \hat g)$ to be the restriction of the structured manifold

\begin{displaymath}
(X \times I \overset{(i,e)}{\hookrightarrow} \mathbb{R}^{k+2}, \hat g \times fr)
\end{displaymath}
to $t = 1$.

\end{defn}
\begin{defn}
\label{BordismRelation}\hypertarget{BordismRelation}{}
Two closed manifolds of [[dimension]] $n$ equipped with normal $\mathcal{B}$-structure $(X_1, i_1, \hat g_1)$ and $(X_2,i_2,\hat g_2)$ (\href{Introduction+to+Stable+homotopy+theory+--+S#ManifoldWithBfStructure}{def.}) are called \textbf{bordant} if there exists a [[manifold with boundary]] $W$ of dimension $n+1$ equipped with $\mathcal{B}$-strcuture $(W,i_W, \hat g_W)$ if its [[boundary]] with $\mathcal{B}$-structure restricted to that boundary is the [[disjoint union]] of $X_1$ with the negative of $X_2$, according to def. \ref{NegativeOfManifoldWithBfStructure}

\begin{displaymath}
\partial(W,i_W,\hat g_W)
    \simeq
  (X_1, i_1, \hat g_1)
    \sqcup
  -(X_2, i_2, \hat g_2)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{BordismIsAnEquivalenceRelation}\hypertarget{BordismIsAnEquivalenceRelation}{}
The [[relation]] of $\mathcal{B}$-[[bordism]] (def. \ref{BordismRelation}) is an [[equivalence relation]].

Write $\Omega^\mathcal{B}_{\bullet}$ for the $\mathbb{N}$-graded set of $\mathcal{B}$-bordism classes of $\mathcal{B}$-manifolds.

\end{prop}
\begin{prop}
\label{BordismGroupAndBordismRing}\hypertarget{BordismGroupAndBordismRing}{}
Under [[disjoint union]] of manifolds, then the set of $\mathcal{B}$-bordism equivalence classes of def. \ref{BordismIsAnEquivalenceRelation} becomes an $\mathbb{Z}$-graded [[abelian group]]

\begin{displaymath}
\Omega^{\mathcal{B}}_\bullet \in Ab^{\mathbb{Z}}
\end{displaymath}
(that happens to be concentrated in non-negative degrees). This is called the \textbf{$\mathcal{B}$-bordism group}.

Moreover, if the [[(B,f)-structure]] $\mathcal{B}$ is multiplicative (def. \ref{BfStructure}), then [[Cartesian product]] of manifolds followed by the multiplicative composition operation of $\mathcal{B}$-structures makes the $\mathcal{B}$-bordism ring into a [[commutative ring]], called the \textbf{$\mathcal{B}$-bordism ring}.

\begin{displaymath}
\Omega^{\mathcal{B}}_\bullet \in CRing^{\mathbb{Z}}
  \,.
\end{displaymath}
\end{prop}
e.g. (\hyperlink{Kochmann96}{Kochmann 96, prop. 1.5.3})

\hypertarget{SectionThomTheorem}{}\paragraph*{{Thom's theorem}}\label{SectionThomTheorem}

Recall that the [[Pontrjagin-Thom construction]] (def. \ref{PontrjaginThomConstruction}) associates to an embbeded manifold $(X,i,\hat g)$ with normal $\mathcal{B}$-structure (def. \ref{ManifoldWithBfStructure}) an element in the [[stable homotopy group]] $\pi_{dim(X)}(M \mathcal{B})$ of the universal $\mathcal{B}$-[[Thom spectrum]] in degree the dimension of that manifold.

\begin{lemma}
\label{PontrjaginThomIsRingHomomorphims}\hypertarget{PontrjaginThomIsRingHomomorphims}{}
For $\mathcal{B}$ be a multiplicative [[(B,f)-structure]] (def. \ref{BfStructure}), the $\mathcal{B}$-[[Pontrjagin-Thom construction]] (def. \ref{PontrjaginThomConstruction}) is compatible with all the relations involved to yield a graded [[ring]] [[homomorphism]]

\begin{displaymath}
\xi
  \;\colon\;
  \Omega^{\mathcal{B}}_\bullet
    \longrightarrow
  \pi_\bullet(M \mathcal{B})
\end{displaymath}
from the $\mathcal{B}$-[[bordism ring]] (def. \ref{BordismGroupAndBordismRing}) to the [[stable homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]] equipped with the ring structure induced from the canonical [[ring spectrum]] structure (def. \ref{UniversalThomSpectrumForBfStructure}).

\end{lemma}
\begin{proof}
By prop. \ref{PontrjaginThomConstructionGivesWellDefinedStableHomotopyGroup} the underlying function of sets is well-defined before dividing out the bordism relation (def. \ref{BordismRelation}). To descend this further to a function out of the set underlying the bordism ring, we need to see that the Pontrjagin-Thom construction respects the bordism relation. But the definition of bordism is just so as to exhibit under $\xi$ a [[left homotopy]] of representatives of homotopy groups.

Next we need to show that it is

\begin{enumerate}%
\item a group homomorphism;


\item a ring homomorphism.



\end{enumerate}
Regarding the first point:

The element 0 in the [[cobordism group]] is represented by the empty manifold. It is clear that the Pontrjagin-Thom construction takes this to the trivial stable homotopy now.

Given two $n$-manifolds with $\mathcal{B}$-structure, we may consider an embedding of their [[disjoint union]] into some $\mathbb{R}^{k}$ such that the [[tubular neighbourhoods]] of the two direct summands do not intersect. There is then a map from two copies of the [[n-cube|k-cube]], glued at one face

\begin{displaymath}
\Box^k \underset{\Box^{k-1}}{\sqcup} \Box^k
  \longrightarrow
  \mathbb{R}^k
\end{displaymath}
such that the first manifold with its tubular neighbourhood sits inside the image of the first cube, while the second manifold with its tubular neighbourhood sits indide the second cube. After applying the Pontryagin-Thom construction to this setup, each cube separately maps to the image under $\xi$ of the respective manifold, while the union of the two cubes manifestly maps to the sum of the resulting elements of homotopy groups, by the very definition of the group operation in the homotopy groups (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyGroupsOftopologicalSpaces}{def.}). This shows that $\xi$ is a group homomorphism.

Regarding the second point:

The element 1 in the [[cobordism ring]] is represented by the manifold which is the point. Without restriction we may consoder this as embedded into $\mathbb{R}^0$, by the identity map. The corresponding [[normal bundle]] is of [[rank]] 0 and hence (by remark \ref{ThomSpaceForRankZeroBundle}) its [[Thom space]] is $S^0$, the [[0-sphere]]. Also $V^{\mathcal{B}}_0$ is the rank-0 vector bundle over the point, and hence $(M \mathcal{B})_0 \simeq S^0$ (by def. \ref{UniversalThomSpectrumForBfStructure}) and so $\xi(\ast) \colon (S^0 \overset{\simeq}{\to} S^0)$ indeed represents the unit element in $\pi_\bullet(M\mathcal{B})$.

Finally regarding respect for the ring product structure: for two manifolds with stable normal $\mathcal{B}$-structure, represented by embeddings into $\mathbb{R}^{k_i}$, then the normal bundle of the embedding of their [[Cartesian product]] is the [[direct sum of vector bundles]] of the separate normal bundles bulled back to the product manifold. In the notation of prop. \ref{ThomSpaceOfExternalProductOfVectorBundles} there is a diagram of the form

\begin{displaymath}
\itexarray{
    \nu_1 \boxtimes \nu_2
      &\overset{\hat e_1 \boxtimes \hat e_2}{\longrightarrow}&
    V^{\mathcal{B}}_{n_1} \boxtimes V^{\mathcal{B}}_{n_2}
      &\overset{\kappa_{n_1,n_2}}{\longrightarrow}&
    V^{\mathcal{B}}_{n_1 + n_2}
    \\
    \downarrow &(pb)& \downarrow &(pb)& \downarrow
    \\
    X_1 \times X_2
      &\underset{\hat g_1 \times \hat g_2}{\longrightarrow}&
    B_{k_1-n_1} \times B_{k_2-n_2}
      &\underset{\mu_{k_1-n_1,k_2-n_2}}{\longrightarrow}&
    B_{k_1 + k_2 - n_1 - n_2}
  }
  \,.
\end{displaymath}
To the Pontrjagin-Thom construction of the product manifold is by definition the top composite in the diagram

\begin{displaymath}
\itexarray{
    S^{n_1 +n_2 + (k_1 + k_2 - n_1 - n_2)}
      &\overset{}{\longrightarrow}&
    Th(\nu_1 \boxtimes \nu_2)
      &\overset{Th(\hat e_1 \boxtimes \hat e_2)}{\longrightarrow}&
    Th(V^{\mathcal{B}}_{k_1-n_1} \boxtimes V^{\mathcal{B}}_{k_2-n_2})
      &\overset{\kappa_{k_1-n_1, k_2-n_2}}{\longrightarrow}&
    Th(V^{\mathcal{B}}_{k_1 + k_2 - n_1 - n_2})
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\simeq}}
      &&
    \downarrow^{\mathrlap{\simeq}}
      &&
    \downarrow^{\mathrlap{=}}
    \\
    S^{n_1 + (k_1 - n_1)}
      \wedge
    S^{n_2 + (k_2 - n_2)}
      &\overset{}{\longrightarrow}&
    Th(\nu_1) \wedge Th(\nu_2)
      &\overset{Th(\hat e_1)\wedge Th(\hat e_2)}{\longrightarrow}&
    Th(V^{\mathcal{B}}_1) \wedge Th(V^{\mathcal{B}}_2)
      &\overset{\kappa_{k_1-n_1, k_2-n_2}}{\longrightarrow}&
    Th(V^{\mathcal{B}}_{k_1 + k_2 - n_1 - n_2})
   }
   \,,
\end{displaymath}
which hence is equivalently the bottom composite, which in turn manifestly represents the product of the separate PT constructions in $\pi_\bullet(M\mathcal{B})$.

\end{proof}
\begin{theorem}
\label{MorphismFromCobordismRingToStableHomotopyGroupsOfThomSpectrumIsIso}\hypertarget{MorphismFromCobordismRingToStableHomotopyGroupsOfThomSpectrumIsIso}{}
The ring homomorphsim in lemma \ref{PontrjaginThomIsRingHomomorphims} is an [[isomorphism]].

\end{theorem}
Due to (\href{Thom+theorem#Thom54}{Thom 54}, \href{Thom+theorem#Pontrjagin55}{Pontrjagin 55}). See for instance (\hyperlink{Kochmann96}{Kochmann 96, theorem 1.5.10}).

\begin{proof}
Observe that given the result $\alpha \colon S^{n+(k-n)} \to Th(V_{k-n})$ of the Pontrjagin-Thom construction map, the original manifold $X \overset{i}{\hookrightarrow} \mathbb{R}^k$ may be recovered as this [[pullback]]:

\begin{displaymath}
\itexarray{
    X &\overset{i}{\longrightarrow}& S^{n + (k-n)}
    \\
    {}^{\mathllap{g_i}}\downarrow &(pb)& \downarrow^{\mathrlap{\alpha}}
    \\
    B O(k-n) &\longrightarrow& Th(V^{B O}_{k-n})
  }
  \,.
\end{displaymath}
To see this more explicitly, break it up into pieces:

\begin{displaymath}
\itexarray{
   X &\longrightarrow& X_+ &\hookrightarrow& S^{n + (k-n)}
   \\
   \downarrow &(pb)& \downarrow &(pb)& \downarrow
    \\
    X &\longrightarrow& X_+ \simeq Th(X) &\overset{Th(0)}{\longrightarrow}& Th(\nu_i)
    \\
    \downarrow &(pb)& \downarrow &(pb)& \downarrow
    \\
    B_{k-n}
      &\longrightarrow&
    (B_{k-n})_+ \simeq Th(B_{k-n})
    &\underset{Th(0)}{\longrightarrow}& Th(V^{\mathcal{B}}_{k-n})
    \\
    \downarrow &(pb)& \downarrow &(pb)& \downarrow
    \\
    B O(k-n)
      &\longrightarrow&
    (B O(k-n))_+ \simeq Th(B O(k-n))
      &\longrightarrow&
    Th(V^{B O}_{k-n})
  }
  \,.
\end{displaymath}
Moreover, since the [[n-spheres]] are [[compact topological spaces]], and since the [[classifying space]] $B O(n)$, and hence its universal Thom space, is a [[sequential colimit]] over [[relative cell complex]] inclusions, the right vertical map factors through some finite stage (by \href{Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes}{this lemma}), the manifold $X$ is equivalently recovered as a pullback of the form

\begin{displaymath}
\itexarray{
    X &\longrightarrow& S^{n + (k-n)}
    \\
    {}^{\mathllap{g_i}}\downarrow &(pb)& \downarrow
    \\
    Gr_{k-n}(\mathbb{R}^k)
      &\overset{i}{\longrightarrow}&
    Th( V_{k-n}(\mathbb{R}^k) \underset{O(k-n)}{\times} \mathbb{R}^{k-n})
  }
  \,.
\end{displaymath}
(Recall that $V^{\mathcal{B}}_{k-n}$ is our notation for the [[universal vector bundle]] with $\mathcal{B}$-structure, while $V_{k-n}(\mathbb{R}^k)$ denotes a [[Stiefel manifold]].)

The idea of the proof now is to use this property as the blueprint of the construction of an [[inverse]] $\zeta$ to $\xi$: given an element in $\pi_{n}(M \mathcal{B})$ represented by a map as on the right of the above diagram, try to define $X$ and the structure map $g_i$ of its normal bundle as the pullback on the left.

The technical problem to be overcome is that for a general continuous function as on the right, the pullback has no reason to be a smooth manifold, and for two reasons:

\begin{enumerate}%
\item the map $S^{n+(k-n)} \to Th(V_{k-n})$ may not be smooth around the image of $i$;


\item even if it is smooth around the image of $i$, it may not be [[transversal map|transversal]] to $i$, and the intersection of two non-transversal smooth functions is in general still not a smooth manifold.



\end{enumerate}
The heart of the proof is in showing that for any $\alpha$ there are small homotopies relating it to an $\alpha'$ that is both smooth around the image of $i$ and transversal to $i$.

The first condition is guaranteed by [[Sard's theorem]], the second by [[Thom's transversality theorem]].

(\ldots{})

\end{proof}
\hypertarget{ThomIsomorphism}{}\subsubsection*{{Thom isomorphism}}\label{ThomIsomorphism}

\textbf{Idea.} If a [[vector bundle]] $E \stackrel{p}{\longrightarrow} X$ of [[rank]] $n$ carries a cohomology class $\omega \in H^n(Th(E),R)$ that looks fiberwise like a [[volume form]] -- a [[Thom class]] -- then the operation of pulling back from base space and then forming the [[cup product]] with this [[Thom class]] is an [[isomorphism]] on (reduced) cohomology

\begin{displaymath}
( (-) \cup \omega) \circ p^\ast
  \;\colon\;
  H^\bullet(X,R) 
   \stackrel{\simeq}{\longrightarrow} 
  \tilde H^{\bullet+n}(Th(E),R)
  \,.
\end{displaymath}
This is the \emph{[[Thom isomorphism]]}. It follows from the [[Serre spectral sequence]] (or else from the [[Leray-Hirsch theorem]]). A closely related statement gives the \emph{[[Thom-Gysin sequence]]}.

In the special case that the vector bundle is trivial of rank $n$, then its [[Thom space]] coincides with the $n$-fold [[suspension]] of the base space (example \ref{ThomSpaceConstructionReducingToSuspension}) and the Thom isomorphism coincides with the [[suspension isomorphism]]. In this sense the Thom isomorphism may be regarded as a \emph{twisted suspension isomorphism}.

We need this below to compute (co)homology of universal Thom spectra $M U$ in terms of that of the [[classifying spaces]] $B U$.

Composed with pullback along the [[Pontryagin-Thom collapse map]], the Thom isomorphism produces maps in cohomology that covariantly follow the underlying maps of spaces. These ``[[Umkehr maps]]'' have the interpretation of [[fiber integration]] against the Thom class.

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, 2.6})

\hypertarget{thomgysin_sequence}{}\paragraph*{{Thom-Gysin sequence}}\label{thomgysin_sequence}

The \emph{[[Thom-Gysin sequence]]} is a type of [[long exact sequence in cohomology]] induced by a [[spherical fibration]] and expressing the [[cohomology groups]] of the total space in terms of those of the base plus correction. The sequence may be obtained as a corollary of the [[Serre spectral sequence]] for the given fibration. It induces, and is induced by, the [[Thom isomorphism]].

\begin{prop}
\label{ThomGysinSequence}\hypertarget{ThomGysinSequence}{}
Let $R$ be a [[commutative ring]] and let

\begin{displaymath}
\itexarray{
    S^n &\longrightarrow& E
    \\
    && \downarrow^{\mathrlap{\pi}}
    \\
    && B
  }
\end{displaymath}
be a [[Serre fibration]] over a [[simply connected topological space|simply connected]] [[CW-complex]] with typical [[fiber]] (\href{Introduction+to+Stable+homotopy+theory+--+P#FibersOfSerreFibrations}{exmpl.}) the [[n-sphere]].

Then there exists an element $c \in H^{n+1}(E; R)$ (in the [[ordinary cohomology]] of the total space with [[coefficients]] in $R$, called the \textbf{Euler class} of $\pi$) such that the [[cup product]] operation $c \cup (-)$ sits in a [[long exact sequence]] of [[cohomology groups]] of the form

\begin{displaymath}
\cdots
   \to
   H^k(B; R)
     \stackrel{\pi^\ast}{\longrightarrow}
   H^k(E; R)
    \stackrel{}{\longrightarrow}
   H^{k-n}(B;R)
   \stackrel{c \cup (-)}{\longrightarrow}
   H^{k+1}(B; R)
   \to
   \cdots
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Switzer75}{Switzer 75, section 15.30}, \hyperlink{Kochmann96}{Kochman 96, corollary 2.2.6})

\begin{proof}
Under the given assumptions there is the corresponding [[Serre spectral sequence]]

\begin{displaymath}
E_2^{s,t}
    \;=\;
  H^s(B; H^t(S^n;R))
    \;\Rightarrow\;
  H^{s+t}(E; R)
  \,.
\end{displaymath}
Since the [[ordinary cohomology]] of the [[n-sphere]] [[fiber]] is concentrated in just two degees

\begin{displaymath}
H^t(S^n; R)
  =
  \left\{
    \itexarray{
      R & for \; t= 0 \; and \; t = n
      \\
      0 & otherwise
    }
  \right.
\end{displaymath}
the only possibly non-vanishing terms on the $E_2$ page of this spectral sequence, and hence on all the further pages, are in bidegrees $(\bullet,0)$ and $(\bullet,n)$:

\begin{displaymath}
E^{\bullet,0}_2 \simeq H^\bullet(B; R)
  \,,
  \;\;\;\;
  and
  \;\;\;
  E^{\bullet,n}_2 \simeq H^\bullet(B; R)
  \,.
\end{displaymath}
As a consequence, since the differentials $d_r$ on the $r$th page of the Serre spectral sequence have bidegree $(r+1,-r)$, the only possibly non-vanishing differentials are those on the $(n+1)$-page of the form

\begin{displaymath}
\itexarray{
     E_{n+1}^{\bullet,n} & \simeq & H^\bullet(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow
     \\
     E_{n+1}^{\bullet+n+1,0} & \simeq &  H^{\bullet+n+1}(B;R)
  }
  \,.
\end{displaymath}
Now since the [[coefficients]] $R$ is a [[ring]], the [[Serre spectral sequence]] is [[multiplicative spectral sequence|multiplicative]] under [[cup product]] and the [[differential]] is a [[derivation]] (of total degree 1) with respect to this product. (See at \emph{\href{multiplicative+spectral+sequence#AHSSForMultiplicativeCohomology}{multiplicative spectral sequence -- Examples -- AHSS for multiplicative cohomology}}.)

To make use of this, write

\begin{displaymath}
\iota \coloneqq 1 \in H^0(B;R) \stackrel{\simeq}{\longrightarrow} E_{n+1}^{0,n}
\end{displaymath}
for the unit in the [[cohomology ring]] $H^\bullet(B;R)$, but regarded as an element in bidegree $(0,n)$ on the $(n+1)$-page of the spectral sequence. (In particular $\iota$ does \emph{not} denote the unit in bidegree $(0,0)$, and hence $d_{n+1}(\iota)$ need not vanish; while by the [[derivation]] property, it does vanish on the actual unit $1 \in H^0(B;R) \simeq E_{n+1}^{0,0}$.)

Write

\begin{displaymath}
c
    \coloneqq
  d_{n+1}(\iota)
   \;\;
  \in E_{n+1}^{n+1,0}
    \stackrel{\simeq}{\longrightarrow}
  H^{n+1}(B; R)
\end{displaymath}
for the image of this element under the differential. We will show that this is the Euler class in question.

To that end, notice that every element in $E_{n+1}^{\bullet,n}$ is of the form $\iota \cdot b$ for $b\in E_{n+1}^{\bullet,0} \simeq H^\bullet(B;R)$.

(Because the [[multiplicative spectral sequence|multiplicative structure]] gives a group homomorphism $\iota \cdot(-) \colon H^\bullet(B;R) \simeq E_{n+1}^{0,0} \to E^{0,n}_{n+1} \simeq H^\bullet(B;R)$, which is an isomorphism because the product in the spectral sequence does come from the [[cup product]] in the [[cohomology ring]], see for instance (\hyperlink{Kochmann96}{Kochman 96, first equation in the proof of prop. 4.2.9}), and since hence $\iota$ does act like the unit that it is in $H^\bullet(B;R)$).

Now since $d_{n+1}$ is a graded [[derivation]] and vanishes on $E_{n+1}^{\bullet,0}$ (by the above degree reasoning), it follows that its action on any element is uniquely fixed to be given by the product with $c$:

\begin{displaymath}
\begin{aligned}
    d_{n+1}(\iota \cdot b)
      & =
    d_{n+1}(\iota) \cdot b + (-1)^{n}\, \iota \cdot \underset{= 0}{\underbrace{d_{n+1}(b)}}
    \\
    & = c \cdot b
  \end{aligned}
  \,.
\end{displaymath}
This shows that $d_{n+1}$ is identified with the cup product operation in question:

\begin{displaymath}
\itexarray{
     E_{n+1}^{s,n} & \simeq & H^s(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}}
     \\
     E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R)
  }
  \,.
\end{displaymath}
In summary, the non-vanishing entries of the $E_\infty$-page of the spectral sequence sit in [[exact sequences]] like so

\begin{displaymath}
\itexarray{
     0
     \\
     \downarrow
     \\
     E_\infty^{s,n}
     \\
     {}^{\mathllap{ker(d_{n+1})}}\downarrow
     \\
     E_{n+1}^{s,n} & \simeq & H^s(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}}
     \\
     E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R)
     \\
     {}^{\mathllap{coker(d_{n+1})}}\downarrow
     \\
     E_\infty^{s+n+1,0}
     \\
     \downarrow
     \\
     0
  }
  \,.
\end{displaymath}
Finally observe (lemma \ref{ImplicationsOfSparesnessOfSSSForSphericalFibration}) that due to the sparseness of the $E_\infty$-page, there are also [[short exact sequences]] of the form

\begin{displaymath}
0 \to E_\infty^{s,0} \longrightarrow H^s(E; R) \longrightarrow  E_\infty^{s-n,n} \to 0
  \,.
\end{displaymath}
Concatenating these with the above exact sequences yields the desired [[long exact sequence]].

\end{proof}
\begin{lemma}
\label{ImplicationsOfSparesnessOfSSSForSphericalFibration}\hypertarget{ImplicationsOfSparesnessOfSSSForSphericalFibration}{}
Consider a cohomology [[spectral sequence]] converging to some [[filtered object|filtered]] [[graded abelian group]] $F^\bullet C^\bullet$ such that

\begin{enumerate}%
\item $F^0 C^\bullet = C^\bullet$;


\item $F^{s} C^{\lt s} = 0$;


\item $E_\infty^{s,t} = 0$ unless $t = 0$ or $t = n$,



\end{enumerate}
for some $n \in \mathbb{N}$, $n \geq 1$. Then there are [[short exact sequences]] of the form

\begin{displaymath}
0 \to E_\infty^{s,0} \overset{}{\longrightarrow} C^s \longrightarrow  E_\infty^{s-n,n} \to 0
  \,.
\end{displaymath}
\end{lemma}
(e.g. \hyperlink{Switzer75}{Switzer 75, p. 356})

\begin{proof}
By definition of convergence of a spectral sequence, the $E_{\infty}^{s,t}$ sit in [[short exact sequences]] of the form

\begin{displaymath}
0 \to F^{s+1}C^{s+t} \overset{i}{\longrightarrow} F^s C^{s+t} \longrightarrow E_\infty^{s,t} \to 0
  \,.
\end{displaymath}
So when $E_\infty^{s,t} = 0$ then the morphism $i$ above is an [[isomorphism]].

We may use this to either shift away the filtering degree

\begin{itemize}%
\item if $t \geq n$ then $F^s C^{s+t} = F^{(s-1)+1}C^{(s-1)+(t+1)} \underoverset{\simeq}{i^{s-1}}{\longrightarrow} F^0 C^{(s-1)+(t+1)} = F^0 C^{s+t} \simeq C^{s+t}$;

\end{itemize}
or to shift away the offset of the filtering to the total degree:

\begin{itemize}%
\item if $0 \leq t-1 \leq n-1$ then $F^{s+1}C^{s+t} = F^{s+1}C^{(s+1)+(t-1)}
\underoverset{\simeq}{i^{-(t-1)}}{\longrightarrow} F^{s+t}C^{(s+1)+(t-1)} = F^{s+t}C^{s+t}$

\end{itemize}
Moreover, by the assumption that if $t \lt 0$ then $F^{s}C^{s+t} = 0$, we also get

\begin{displaymath}
F^{s}C^{s} \simeq E_\infty^{s,0}
  \,.
\end{displaymath}
In summary this yields the vertical isomorphisms

\begin{displaymath}
\itexarray{
  0
    &\to&
  F^{s+1}C^{s+n}
    &\longrightarrow&
  F^{s}C^{s+n}
    &\longrightarrow&
  E_\infty^{s,n}
    &\to&
  0
  \\
  &&
    {}^{\mathllap{i^{-(n-1)}}}\downarrow^{\mathrlap{\simeq}}
  &&
    {}^{\mathllap{i^{s-1}}}\downarrow^{\mathrlap{\simeq}}
  &&
    \downarrow^{\mathrlap{=}}
  \\
  0
    &\to&
  F^{s+n}C^{s+n}
  \simeq
  E_\infty^{s+n,0}
    &\longrightarrow&
  C^{s+n}
    &\longrightarrow&
  E_\infty^{s,n}
    &\to&
  0
  }
\end{displaymath}
and hence with the top sequence here being exact, so is the bottom sequence.

\end{proof}
\hypertarget{OrientationAndFiberIntegration}{}\subsubsection*{{Orientation in generalized cohomology}}\label{OrientationAndFiberIntegration}

\textbf{Idea.} From the way the [[Thom isomorphism]] via a [[Thom class]] works in [[ordinary cohomology]] (as \hyperlink{ThomIsomorphism}{above}), one sees what the general concept of [[orientation in generalized cohomology]] and of [[fiber integration in generalized cohomology]] is to be.

Specifically we are interested in [[complex oriented cohomology]] theories $E$, characterized by an orientation class on infinity [[complex projective space]] $\mathbb{C}P^\infty$ (def. \ref{ComplexProjectiveSpace}), the [[classifying space]] for [[complex line bundles]], which restricts to a generator on $S^2 \hookrightarrow \mathbb{C}P^\infty$.

(Another important application is given by taking $E =$ [[KU]] to be [[topological K-theory]]. Then [[orientation in generalized cohomology|orientation]] is [[spin{\tt \symbol{94}}c structure]] and fiber integration with coefficients in $E$ is [[fiber integration in K-theory]]. This is classical \emph{[[index theory]]}.)

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, section 4.3}, \hyperlink{Adams74}{Adams 74, part III, section 10}, \hyperlink{Lurie10}{Lurie 10, lecture 5})

\begin{itemize}%
\item [[Riccardo Pedrotti]], \emph{Complex oriented cohomology -- Orientation in generalized cohomology}, 2016 ([[PedrotticECohomology2018.pdf:file]])

\end{itemize}
$\,$

\hypertarget{universal_orientation}{}\paragraph*{{Universal $E$-orientation}}\label{universal_orientation}

\begin{defn}
\label{EOrientationOfAVectorBundle}\hypertarget{EOrientationOfAVectorBundle}{}
Let $E$ be a [[multiplicative cohomology theory]] (def. \ref{MultiplicativeCohomologyTheory}) and let $V \to X$ be a topological [[vector bundle]] of [[rank]] $n$. Then an \textbf{$E$-[[orientation in generalized cohomology|orientation]]} or \textbf{$E$-[[Thom class]]} on $V$ is an element of degree $n$

\begin{displaymath}
u \in \tilde E^n(Th(V))
\end{displaymath}
in the [[reduced cohomology|reduced]] $E$-[[cohomology ring]] of the [[Thom space]] (def. \ref{ThomSpace}) of $V$, such that for every point $x \in X$ its restriction $i_x^* u$ along

\begin{displaymath}
i_x
    \;\colon\;
  S^n \simeq Th(\mathbb{R}^n) \overset{Th(e_x)}{\longrightarrow} Th(V)
\end{displaymath}
(for $\mathbb{R}^n \overset{fib_x}{\hookrightarrow} V$ the [[fiber]] of $V$ over $x$) is a \emph{generator}, in that it is of the form

\begin{displaymath}
i^\ast u = \epsilon \cdot \gamma_n
\end{displaymath}
for

\begin{itemize}%
\item $\epsilon \in \tilde E^0(S^0)$ a [[unit]] in $E^\bullet$;


\item $\gamma_n \in \tilde E^n(S^n)$ the image of the multiplicative unit under the [[suspension isomorphism]] $\tilde E^0(S^0) \stackrel{\simeq}{\to}\tilde E^n(S^n)$.



\end{itemize}
\end{defn}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, def. 4.3.4})

\begin{remark}
\label{}\hypertarget{}{}
Recall that a \emph{[[(B,f)-structure]]} $\mathcal{B}$ (def. \ref{BfStructure}) is a system of [[Serre fibrations]] $B_n \overset{f_n}{\longrightarrow} B O(n)$ over the [[classifying spaces]] for [[orthogonal structure]] equipped with maps

\begin{displaymath}
g_{n,n+1} \;\colon\; B_n \longrightarrow B_{n+1}
\end{displaymath}
covering the canonical inclusions of classifying spaces. For instance for $G_n \to O(n)$ a compatible system of [[topological group]] [[homomorphisms]], then the $(B,f)$-structure given by the [[classifying spaces]] $B G_n$ (possibly suitably resolved for the maps $B G_n \to B O(n)$ to become Serre fibrations) defines \emph{[[G-structure]]}.

Given a $(B,f)$-structure, then there are the [[pullbacks]] $V^{\mathcal{B}}_n \coloneqq f_n^\ast (E O(n)\underset{O(n)}{\times}\mathbb{R}^n)$ of the [[universal vector bundles]] over $B O(n)$, which are the \emph{universal vector bundles equipped with $(B,f)$-structure}

\begin{displaymath}
\itexarray{
    V^{\mathcal{B}}_n &\longrightarrow& E O(n)\underset{O(n)}{\times} \mathbb{R}^n
    \\
    \downarrow &(pb)& \downarrow
    \\
    B_n & \underset{f_n}{\longrightarrow} & B O(n)
  }
  \,.
\end{displaymath}
Finally recall that there are canonical morphisms (\href{Thom+spectrum#PullbackOfUniversalOnBundleUnderCoordinateRestriction}{prop.})

\begin{displaymath}
\phi_n
    \;\colon\;
  \mathbb{R} \oplus V^{\mathcal{B}}_n
    \longrightarrow
  V^{\mathcal{B}}_{n+1}
\end{displaymath}
\end{remark}
\begin{defn}
\label{EOrientationOfABfStructure}\hypertarget{EOrientationOfABfStructure}{}
Let $E$ be a [[multiplicative cohomology theory]] and let $\mathcal{B}$ be a multiplicative [[(B,f)-structure]]. Then a \textbf{universal $E$-orientation for vector bundles with $\mathcal{B}$-structure} is an $E$-orientation, according to def. \ref{EOrientationOfAVectorBundle}, for each rank-$n$ universal vector bundle with $\mathcal{B}$-structure:

\begin{displaymath}
\xi_n
    \in
  \tilde E^n(Th(E_n^{\mathcal{B}}))
  \;\;\;\;
  \forall n \in \mathbb{N}
\end{displaymath}
such that these are compatible in that

\begin{enumerate}%
\item for all $n \in \mathbb{N}$ then

\begin{displaymath}
\xi_n = \phi_n^\ast \xi_{n+1}
  \,,
\end{displaymath}
where

\begin{displaymath}
\xi_n
   \in
  \tilde E^n(Th(V_n))
    \simeq
  \tilde E^{n+1}(\Sigma Th(V_n))
     \simeq
  \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))
\end{displaymath}
(with the first isomorphism is the [[suspension isomorphism]] of $E$ and the second exhibiting the [[homeomorphism]] of Thom spaces $Th(\mathbb{R} \oplus V)\simeq \Sigma Th(V)$ (prop. \ref{SuspensionOfThomSpaces}) and where

\begin{displaymath}
\phi_n^\ast
  \;\colon\;
   \tilde E^{n+1}(Th(V_{n+1}))
   \longrightarrow
  \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))
\end{displaymath}
is pullback along the canonical $\phi_n \colon \mathbb{R}\oplus V_n \to V_{n+1}$ (prop. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction}).


\item for all $n_1, n_2 \in \mathbb{N}$ then

\begin{displaymath}
\xi_{n+1} \cdot \xi_{n+2} = \xi_{n_1 + n_2}
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{prop}
\label{UniversalEOrientationsAreEquivalentlyMorphismsOfRingSpectra}\hypertarget{UniversalEOrientationsAreEquivalentlyMorphismsOfRingSpectra}{}
A universal $E$-orientation, in the sense of def. \ref{EOrientationOfABfStructure}, for vector bundles with [[(B,f)-structure]] $\mathcal{B}$, is equivalently (the homotopy class of) a homomorphism of [[ring spectra]]

\begin{displaymath}
\xi \;\colon\; M\mathcal{B} \longrightarrow E
\end{displaymath}
from the universal $\mathcal{B}$-[[Thom spectrum]] to a spectrum which via the [[Brown representability theorem]] (theorem \ref{BrownRepresentabilityForTraditionalBrownFunctors}) represents the given [[generalized (Eilenberg-Steenrod) cohomology theory]] $E$ (and which we denote by the same symbol).

\end{prop}
\begin{proof}
The [[Thom spectrum]] $M\mathcal{B}$ has a standard structure of a [[CW-spectrum]]. Let now $E$ denote a [[sequential spectrum|sequential]] [[Omega-spectrum]] representing the multiplicative cohomology theory of the same name. Since, in the standard [[model structure on topological sequential spectra]], [[CW-spectra]] are cofibrant (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CellSpectraAreCofibrantInModelStructureOnTopologicalSequentialSpectra}{prop.}) and Omega-spectra are fibrant (\href{Introduction+to+Stable+homotopy+theory+--+1-1#StableModelStructureOnSequentialSpectraIsModelCategory}{thm.}) we may represent all morphisms in the [[stable homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#TheStableHomotopyCategory}{def.}) by actual morphisms

\begin{displaymath}
\xi
   \;\colon\;
  M \mathcal{B}
    \longrightarrow
  E
\end{displaymath}
of sequential spectra (due to \href{Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory}{this lemma}).

Now by definition (\href{Introduction+to+Stable+homotopy+theory+--+1-1#SequentialSpectra}{def.}) such a homomorphism is precissely a sequence of base-point preserving [[continuous functions]]

\begin{displaymath}
\xi_n
    \;\colon\;
  (M\mathcal{B})_n
   =
  Th(V_n^{\mathcal{B}})
    \longrightarrow
  E_n
\end{displaymath}
for $n \in \mathbb{N}$, such that they are compatible with the structure maps $\sigma_n$ and equivalently with their $(S^1 \wedge(-)\dashv Maps(S^1,-)_\ast)$-[[adjuncts]] $\tilde \sigma_n$, in that these diagrams commute:

\begin{displaymath}
\itexarray{
    S^1 \wedge Th(V^{\mathcal{B}}_n)
     &\overset{S^1 \wedge \xi_n}{\longrightarrow}&
    S^1 \wedge E_n
    \\
    {}^{\mathllap{\sigma^{M\mathcal{B}}_n}}\downarrow
      &&
    \downarrow^{\mathrlap{\sigma^E_n}}
    \\
    Th(V^{\mathcal{B}}_{n+1})
      &\underset{\xi_{n+1}}{\longrightarrow}&
    E_{n+1}
  }
  \;\;\;\;\;\;\;\;\;
  \leftrightarrow
  \;\;\;\;\;\;\;\;\;
  \itexarray{
     Th(V^{\mathcal{B}}_n)
       &\overset{\xi_n}{\longrightarrow}&
     E_n
     \\
     {}^{\mathllap{\tilde \sigma^{M\mathcal{B}}_n}}\downarrow
       &&
     \downarrow^{\mathrlap{\tilde \sigma^E_n}}
     \\
     Maps(S^1,Th(V^{\mathcal{B}}_{n+1}))
       &\underset{Maps(S^1,\xi_{n+1})_\ast}{\longrightarrow}&
     Maps(S^1, E_{n+1})_{\ast}
  }
\end{displaymath}
for all $n \in \mathbb{N}$.

First of all this means (via the identification given by the [[Brown representability theorem]], see prop. \ref{AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum}, that the components $\xi_n$ are equivalently representatives of elements in the [[cohomology groups]]

\begin{displaymath}
\xi_n \in \tilde E^n(Th(V^{\mathcal{B}}_n))
\end{displaymath}
(which we denote by the same symbol, for brevity).

Now by the definition of universal [[Thom spectra]] (def. \ref{UniversalThomSpectrum}, def. \ref{UniversalThomSpectrumForBfStructure}), the structure map $\sigma_n^{M\mathcal{B}}$ is just the map $\phi_n \colon \mathbb{R}\oplus Th(V^{\mathcal{B}}_n)\to Th(V_{n+1}^{\mathcal{B}})$ from above.

Moreover, by the [[Brown representability theorem]], the [[adjunct]] $\tilde \sigma_n^E \circ \xi_n$ (on the right) of $\sigma^E_n \circ S^1 \wedge \xi_n$ (on the left) is what represents (again by prop. \ref{AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum}) the image of

\begin{displaymath}
\xi_n \in E^n(Th(V^{\mathcal{B}}_n))
\end{displaymath}
under the [[suspension isomorphism]]. Hence the [[commutative square|commutativity]] of the above squares is equivalently the first compatibility condition from def. \ref{EOrientationOfABfStructure}: $\xi_n \simeq \phi_n^\ast \xi_{n+1}$ in $\tilde E^{n+1}(Th(\mathbb{R}\oplus V_n^{\mathcal{B}}))$

Next, $\xi$ being a homomorphism of [[ring spectra]] means equivalently (we should be modelling $M\mathcal{B}$ and $E$ as [[structured spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#DiagramSpectra}{here.}) to be more precise on this point, but the conclusion is the same) that for all $n_1, n_2\in \mathbb{N}$ then

\begin{displaymath}
\itexarray{
    Th(V_{n_1}^{\mathcal{B}}) \wedge Th(V_{n_2}^{\mathcal{B}})
      &\overset{}{\longrightarrow}&
    Th(V_{n_1 + n_2})
    \\
    {}^{\mathllap{\xi_{n_1} \wedge \xi_{n_2}}}\downarrow
      &&
    \downarrow^{\mathrlap{\xi_{n_1 + n_2}}}
    \\
    E_{n_1} \wedge E_{n_2}
      &\underset{\cdot}{\longrightarrow}&
    E_{n_1 + n_2}
  }
  \,.
\end{displaymath}
This is equivalently the condition $\xi_{n_1} \cdot \xi_{n_2} \simeq \xi_{n_1  + n_2}$.

Finally, since $M\mathcal{B}$ is a [[ring spectrum]], there is an essentially unique multiplicative homomorphism from the [[sphere spectrum]]

\begin{displaymath}
\mathbb{S}
    \overset{e}{\longrightarrow}
  M\mathcal{B}
  \,.
\end{displaymath}
This is given by the component maps

\begin{displaymath}
e_n
    \;\colon\;
  S^n
    \simeq
  Th(\mathbb{R}^n)
    \longrightarrow
  Th(V_{n}^{\mathcal{B}})
\end{displaymath}
that are induced by including the fiber of $V_{n}^{\mathcal{B}}$.

Accordingly the composite

\begin{displaymath}
\mathbb{S}
    \overset{e}{\longrightarrow}
  M\mathcal{B}
    \overset{\xi}{\longrightarrow}
  E
\end{displaymath}
has as components the restrictions $i^\ast \xi_n$ appearing in def. \ref{EOrientationOfAVectorBundle}. At the same time, also $E$ is a ring spectrum, hence it also has an essentially unique multiplicative morphism $\mathbb{S} \to E$, which hence must agree with $i^\ast \xi$, up to homotopy. If we represent $E$ as a [[symmetric ring spectrum]], then the canonical such has the required property: $e_0$ is the identity element in degree 0 (being a unit of an ordinary ring, by definition) and hence $e_n$ is necessarily its image under the suspension isomorphism, due to compatibility with the structure maps and using the above analysis.

\end{proof}
\hypertarget{complex_projective_space}{}\paragraph*{{Complex projective space}}\label{complex_projective_space}

For the fine detail of the discussion of [[complex oriented cohomology theories]] \hyperlink{ComplexOrientatioon}{below}, we recall basic facts about [[complex projective space]].

Complex projective space $\mathbb{C}P^n$ is the [[projective space]] $\mathbb{A}P^n$ for $\mathbb{A} = \mathbb{C}$ being the [[complex numbers]] (and for $n \in \mathbb{N}$), a [[complex manifold]] of complex [[dimension]] $n$ (real dimension $2n$). Equivalently, this is the complex [[Grassmannian]] $Gr_1(\mathbb{C}^{n+1})$ (def. \ref{RealAndComplexGrassmannian}). For the special case $n = 1$ then $\mathbb{C}P^1 \simeq S^2$ is the [[Riemann sphere]].

As $n$ ranges, there are natural inclusions

\begin{displaymath}
\ast = \mathbb{C}P^0
    \hookrightarrow
  \mathbb{C}P^1
    \hookrightarrow
  \mathbb{C}P^2
    \hookrightarrow
  \mathbb{C}P^3
    \hookrightarrow
  \cdots
  \,.
\end{displaymath}
The [[sequential colimit]] over this sequence is the infinite complex projective space $\mathbb{C}P^\infty$. This is a model for the [[classifying space]] $B U(1)$ of [[circle principal bundles]]/[[complex line bundles]] (an [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$).

\begin{defn}
\label{ComplexProjectiveSpace}\hypertarget{ComplexProjectiveSpace}{}
For $n \in \mathbb{N}$, then \textbf{complex $n$-dimensional complex projective space} is the [[complex manifold]] (often just regarded as its underlying [[topological space]]) defined as the [[quotient]]

\begin{displaymath}
\mathbb{C}P^n \coloneqq (\mathbb{C}^{n+1}-\{0\})/_\sim
\end{displaymath}
of the [[Cartesian product]] of $(n+1)$-copies of the [[complex plane]], with the origin removed, by the [[equivalence relation]]

\begin{displaymath}
(z \sim w) \Leftrightarrow (z = \kappa \cdot w)
\end{displaymath}
for some $\kappa \in \mathbb{C} - \{0\}$ and using the canonical multiplicative [[action]] of $\mathbb{C}$ on $\mathbb{C}^{n+1}$.

The canonical inclusions

\begin{displaymath}
\mathbb{C}^{n+1} \hookrightarrow \mathbb{C}^{n+2}
\end{displaymath}
induce canonical inclusions

\begin{displaymath}
\mathbb{C}P^n \hookrightarrow \mathbb{C}P^{n+1}
  \,.
\end{displaymath}
The [[sequential colimit]] over this sequence of inclusions is the \emph{infinite complex projective space}

\begin{displaymath}
\mathbb{C}P^\infty
    \coloneqq
  \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n
  \,.
\end{displaymath}
\end{defn}
The following equivalent characterizations are immediate but useful:

\begin{prop}
\label{ComplexProjectiveSpaceAsGrassmannian}\hypertarget{ComplexProjectiveSpaceAsGrassmannian}{}
For $n \in \mathbb{N}$ then complex projective space, def. \ref{ComplexProjectiveSpace}, is equivalently the complex [[Grassmannian]]

\begin{displaymath}
\mathbb{C}P^n \simeq Gr_1(\mathbb{C}^{n+1})
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{ComplexProjectiveSpaceAsS1Quotient}\hypertarget{ComplexProjectiveSpaceAsS1Quotient}{}
For $n \in \mathbb{N}$ then complex projective space, def. \ref{ComplexProjectiveSpace}, is equivalently

\begin{enumerate}%
\item the [[coset]]

\begin{displaymath}
\mathbb{C}P^n \simeq U(n+1)/(U(n) \times U(1))
  \,,
\end{displaymath}

\item the quotient of the [[n-sphere|(2n+1)-sphere]] by the [[circle group]] $S^1 \simeq \{ \kappa  \in \mathbb{C}| {\vert \kappa \vert} = 1\}$



\end{enumerate}
\begin{displaymath}
\mathbb{C}P^n \simeq S^{2n+1}/S^1
    \,.
\end{displaymath}
\end{prop}
\begin{proof}
To see the second characterization from def. \ref{ComplexProjectiveSpace}:

With ${\vert -\vert} \colon \mathbb{C}^{n} \longrightarrow \mathbb{R}$ the standard [[norm]], then every element $\vec z \in \mathbb{C}^{n+1}$ is identified under the defining equivalence relation with

\begin{displaymath}
\frac{1}{\vert \vec z\vert}\vec z \in S^{2n-1} \hookrightarrow \mathbb{C}^{n+1}
\end{displaymath}
lying on the unit $(2n-1)$-sphere. This fixes the action of $\mathbb{C}-0$ up to a remaining action of complex numbers of unit [[absolute value]]. These form the [[circle group]] $S^1$.

The first characterization follows via prop. \ref{ComplexProjectiveSpaceAsGrassmannian} from the general discusion at \emph{[[Grassmannian]]}. With this the second characterization follows also with the [[coset]] identification of the $(2n+1)$-sphere: $S^{2n+1} \simeq U(n+1)/U(n)$ (\href{unitary+group#nSphereAsUnitaryCosetSpace}{exmpl.}).

\end{proof}
\begin{prop}
\label{CellComplexStructureOnComplexProjectiveSpace}\hypertarget{CellComplexStructureOnComplexProjectiveSpace}{}
There is a [[CW-complex]] structure on complex projective space $\mathbb{C}P^n$ (def. \ref{ComplexProjectiveSpace}) for $n \in \mathbb{N}$, given by [[induction]], where $\mathbb{C}P^{n+1}$ arises from $\mathbb{C}P^n$ by attaching a single cell of dimension $2(n+1)$ with attaching map the [[projection]] $S^{2n+1} \longrightarrow \mathbb{C}P^n$ from prop. \ref{ComplexProjectiveSpaceAsS1Quotient}:

\begin{displaymath}
\itexarray{
     S^{2n+1} &\longrightarrow& S^{2n+1}/S^1 \simeq \mathbb{C}P^n
     \\
     {}^{\mathllap{\iota_{2n+2}}}\downarrow
       &(po)& \downarrow
     \\
     D^{2n+2} &\longrightarrow& \mathbb{C}P^{n+1}
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Given homogenous coordinates $(z_0, z_1, \cdots, z_n, z_{n+1}, z_{n+2}) \in \mathbb{C}^{n+2}$ for $\mathbb{C}P^{n+1}$, let

\begin{displaymath}
\phi \coloneqq -arg(z_{n+2})
\end{displaymath}
be the [[phase]] of $z_{n+2}$. Then under the equivalence relation defining $\mathbb{C}P^{n+1}$ these coordinates represent the same element as

\begin{displaymath}
\frac{1}{\vert \vec z\vert}(e^{i \phi} z_0, e^{i \phi}z_1,\cdots, e^{i \phi}z_{n+1}, r)
  \,,
\end{displaymath}
where

\begin{displaymath}
r = {\vert z_{n+2}\vert}\in [0,1) \subset \mathbb{C}
\end{displaymath}
is the [[absolute value]] of $z_{n+2}$. Representatives $\vec z'$ of this form (${\vert \vec z' \vert = 1}$ and $z'_{n+2} \in [0,1]$) parameterize the [[n-disk|2n+2-disk]] $D^{2n+2}$ ($2n+3$ real parameters subject to the one condition that the sum of their norm squares is unity) with [[boundary]] the $(2n+1)$-sphere at $r = 0$. The only remaining part of the action of $\mathbb{C}-\{0\}$ which fixes the form of these representatives is $S^1$ acting on the elements with $r = 0$ by phase shifts on the $z_0, \cdots, z_{n+1}$. The quotient of this remaining action on $D^{2(n+1)}$ identifies its boundary $S^{2n+1}$-sphere with $\mathbb{C}P^{n}$, by prop. \ref{ComplexProjectiveSpaceAsS1Quotient}.

\end{proof}
\begin{prop}
\label{OrdinaryCohomologyOfComplexProjectiveSpace}\hypertarget{OrdinaryCohomologyOfComplexProjectiveSpace}{}
For $A \in$ [[Ab]] any [[abelian group]], then the [[ordinary homology]] [[homology groups|groups]] of complex projective space $\mathbb{C}P^n$ with [[coefficients]] in $A$ are

\begin{displaymath}
H_k(\mathbb{C}P^n,A)\simeq
  \left\{
    \itexarray{
       A & for \; k \;even\; and \; k \leq 2n
       \\
       0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Similarly the [[ordinary cohomology]] [[cohomology groups|groups]] of $\mathbb{C}P^n$ is

\begin{displaymath}
H^k(\mathbb{C}P^n,A)
   \simeq
  \left\{
    \itexarray{
       A & for \; k \;even\; and \; k \leq 2n
       \\
       0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Moreover, if $A$ carries the structure of a [[ring]] $R = (A, \cdot)$, then under the [[cup product]] the [[cohomology ring]] of $\mathbb{C}P^n$ is the the [[graded ring]]

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, R)
   \simeq
  R[c_1] / (c_1^{n+1})
\end{displaymath}
which is the [[quotient]] of the [[polynomial ring]] on a single generator $c_1$ in degree 2, by the relation that identifies [[cup products]] of more than $n$-copies of the generator $c_1$ with zero.

Finally, the [[cohomology ring]] of the infinite-dimensional complex projective space is the [[formal power series ring]] in one generator:

\begin{displaymath}
H^\bullet(\mathbb{C}P^\infty, R)
   \simeq
  R[ [ c_1 ] ]
  \,.
\end{displaymath}
(Or else the [[polynomial ring]] $R[c_1]$, see remark \ref{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory})

\end{prop}
\begin{proof}
First consider the case that the coefficients are the [[integers]] $A = \mathbb{Z}$.

Since $\mathbb{C}P^n$ admits the structure of a [[CW-complex]] by prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, we may compute its [[ordinary homology]] equivalently as its [[cellular homology]] (\href{Introduction+to+Stable+homotopy+theory+--+I#CelluarEquivalentToSingularFromSpectralSequence}{thm.}). By definition (\href{cellular+homology#CellularChainComplex}{defn.}) this is the [[chain homology]] of the chain complex of [[relative homology]] groups

\begin{displaymath}
\cdots
    \overset{\partial_{cell}}{\longrightarrow}
  H_{q+2}((\mathbb{C}P^n)_{q+2}, (\mathbb{C}P^n)_{q+1})
   \overset{\partial_{cell}}{\longrightarrow}
  H_{q+1}((\mathbb{C}P^n)_{q+1}, (\mathbb{C}P^n)_{q})
   \overset{\partial_{cell}}{\longrightarrow}
  H_{q}((\mathbb{C}P^n)_{q}, (\mathbb{C}P^n)_{q-1})
    \overset{\partial_{cell}}{\longrightarrow}
  \cdots
  \,,
\end{displaymath}
where $(-)_q$ denotes the $q$th stage of the [[CW-complex]]-structure. Using the CW-complex structure provided by prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, then there are cells only in every second degree, so that

\begin{displaymath}
(\mathbb{C}P^n)_{2k+1} = (\mathbb{C}P)_{2k}
\end{displaymath}
for all $k \in \mathbb{N}$. It follows that the cellular chain complex has a zero group in every second degree, so that all differentials vanish. Finally, since prop. \ref{CellComplexStructureOnComplexProjectiveSpace} says that $(\mathbb{C}P^n)_{2k+2}$ arises from $(\mathbb{C}P^n)_{2k+1} = (\mathbb{C}P^n)_{2k}$ by attaching a single $2k+2$-cell it follows that (by passage to [[reduced homology]])

\begin{displaymath}
H_{2k}(\mathbb{C}P^n, \mathbb{Z})
    \simeq
  \tilde H_{2k}(S^{2k})((\mathbb{C}P^n)_{2k}/(\mathbb{C}P^n)_{2k-1})
   \simeq
  \tilde H_{2k}(S^{2k})
    \simeq
  \mathbb{Z}
  \,.
\end{displaymath}
This establishes the claim for ordinary homology with integer coefficients.

In particular this means that $H_q(\mathbb{C}P^n, \mathbb{Z})$ is a [[free abelian group]] for all $q$. Since free abelian groups are the [[projective objects]] in [[Ab]] (\href{projective+object#ProjectiveObjectsInAbAreFreeGroups}{prop.}) it follows (with the discussion at \emph{[[derived functors in homological algebra]]}) that the [[Ext]]-groups vanishe:

\begin{displaymath}
Ext^1(H_q(\mathbb{C}P^n, \mathbb{Z}),A) = 0
\end{displaymath}
and the [[Tor]]-groups vanishes:

\begin{displaymath}
Tor_1(H_q(\mathbb{C}P^n), A) = 0
  \,.
\end{displaymath}
With this, the statement about homology and cohomology groups with general coefficients follows with the [[universal coefficient theorem]] for ordinary homology (\href{universal+coefficient+theorem#TheoremInOrdinaryHomology}{thm.}) and for ordinary cohomology (\href{universal+coefficient+theorem#OrdinaryStatementInCohomology}{thm.}).

Finally to see the action of the [[cup product]]: by definition this is the composite

\begin{displaymath}
\cup
  \;\colon\;
  H^\bullet(\mathbb{C}P^n, R)
  \otimes
  H^\bullet(\mathbb{C}P^n, R)
    \longrightarrow
  H^\bullet(\mathbb{C}P^n \times \mathbb{C}P^n , R)
    \overset{\Delta^\ast}{\longrightarrow}
  H^\bullet(\mathbb{C}P^n,R)
\end{displaymath}
of the ``cross-product'' map that appears in the [[Kunneth theorem]], and the pullback along the [[diagonal]] $\Delta\colon \mathbb{C}P^n \to \mathbb{C}P^n \times \mathbb{C}P^n$.

Since, by the above, the groups $H^{2k}(\mathbb{C}P^n,R) \simeq R[2k]$ and $H^{2k+1}(\mathbb{C}P^n,R) = 0$ are free and finitely generated, the [[Kunneth theorem]] in ordinary cohomology applies (\href{K&#252;nneth+theorem#KunnethInOrdinaryCohomology}{prop.}) and says that the cross-product map above is an isomorphism. This shows that under cup product pairs of generators are sent to a generator, and so the statement $H^\bullet(\mathbb{C}P^n , R)\simeq R[c_1](c_1^{n+1})$ follows.

This also implies that the projection maps

\begin{displaymath}
H^\bullet((\mathbb{C}P^\infty)_{2n+2}, R)
  =
  H^\bullet(\mathbb{C}P^{n+1}, R)
  \to
  H^\bullet(\mathbb{C}P^{n+}, R)
  =
  H^\bullet((\mathbb{C}P^\infty)_{2n}, R)
\end{displaymath}
are all [[epimorphisms]]. Therefore this sequence satisfies the [[Mittag-Leffler condition]] (def. \ref{MittagLefflerCondition}, example \ref{MittagLefflerSatisfiedInParticularForTowerOfSurjections}) and therefore the [[Milnor exact sequence]] for cohomology (prop. \ref{MilorSequenceForReducedCohomologyOnCWComplex}) implies the last claim to be proven:

\begin{displaymath}
\begin{aligned}
    H^\bullet(\mathbb{C}P^\infty, R)
    &
     \simeq
    H^\bullet( \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n , R)
   \\
    &\simeq \underset{\longrightarrow}{\lim}_n H^\bullet(\mathbb{C}P^n, R)
   \\
   &\simeq \underset{\longrightarrow}{\lim}_n ( R [c_1^E] / ((c_1)^{n+1})  )
   \\
   & \simeq R[ [ c_1 ] ]
   \,,
  \end{aligned}
\end{displaymath}
where the last step is \href{formal+scheme#FormalPowerSeries}{this prop.}.

\end{proof}
\begin{remark}
\label{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}\hypertarget{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}{}
There is in general a choice to be made in interpreting the [[cohomology groups]] of a [[multiplicative cohomology theory]] $E$ (def. \ref{MultiplicativeCohomologyTheory}) as a [[ring]]:

a priori $E^\bullet(X)$ is a sequence

\begin{displaymath}
\{E^n(X)\}_{n \in \mathbb{Z}}
\end{displaymath}
of [[abelian groups]], together with a system of group homomorphisms

\begin{displaymath}
E^{n_1}(X) \otimes E^{n_2}(X)
    \longrightarrow
  E^{n_1 + n_2}(X)
  \,,
\end{displaymath}
one for each pair $(n_1,n_2) \in \mathbb{Z}\times\mathbb{Z}$.

In turning this into a single [[ring]] by forming [[formal sums]] of elements in the groups $E^n(X)$, there is in general the choice of whether allowing formal sums of only finitely many elements, or allowing arbitrary formal sums.

In the former case the ring obtained is the [[direct sum]]

\begin{displaymath}
\oplus_{n \in \mathbb{N}} E^n(X)
\end{displaymath}
while in the latter case it is the [[Cartesian product]]

\begin{displaymath}
\prod_{n \in \mathbb{N}} E^n (X)
  \,.
\end{displaymath}
These differ in general. For instance if $E$ is [[ordinary cohomology]] with [[integer]] [[coefficients]] and $X$ is infinite [[complex projective space]] $\mathbb{C}P^\infty$, then (prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace}))

\begin{displaymath}
E^n(X)
  =
  \left\{
    \itexarray{
      \mathbb{Z} & n \; even
      \\
      0 & otherwise
    }
  \right.
\end{displaymath}
and the product operation is given by

\begin{displaymath}
E^{2{n_1}}(X)\otimes E^{2 n_2}(X)
    \longrightarrow
  E^{2(n_1 + n_2)}(X)
\end{displaymath}
for all $n_1, n_2$ (and zero in odd degrees, necessarily). Now taking the [[direct sum]] of these, this is the [[polynomial ring]] on one generator (in degree 2)

\begin{displaymath}
\oplus_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z}[c_1]
  \,.
\end{displaymath}
But taking the [[Cartesian product]], then this is the [[formal power series ring]]

\begin{displaymath}
\prod_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z} [ [ c_1 ] ]
  \,.
\end{displaymath}
A priori both of these are sensible choices. The former is the usual choice in traditional [[algebraic topology]]. However, from the point of view of regarding [[ordinary cohomology]] theory as a [[multiplicative cohomology theory]] right away, then the second perspective tends to be more natural:

The cohomology of $\mathbb{C}P^\infty$ is naturally computed as the [[inverse limit]] of the cohomolgies of the $\mathbb{C}P^n$, each of which unambiguously has the ring structure $\mathbb{Z}[c_1]/((c_1)^{n+1})$. So we may naturally take the limit in the [[category]] of [[commutative rings]] right away, instead of first taking it in $\mathbb{Z}$-indexed sequences of abelian groups, and then looking for ring structure on the result. But the limit taken in the category of rings gives the [[formal power series ring]] (see \href{formal+scheme#FormalPowerSeries}{here}).

See also for instance remark 1.1. in [[Jacob Lurie]]: \emph{[[A Survey of Elliptic Cohomology]]}.

\end{remark}
\hypertarget{ComplexOrientatioon}{}\paragraph*{{Complex orientation}}\label{ComplexOrientatioon}

\begin{defn}
\label{ComplexOrientedCohomologyTheories}\hypertarget{ComplexOrientedCohomologyTheories}{}
A [[multiplicative cohomology theory]] $E$ (def. \ref{MultiplicativeCohomologyTheory}) is called \textbf{complex orientable} if the the following equivalent conditions hold

\begin{enumerate}%
\item The morphism

\begin{displaymath}
i^\ast \;\colon\; E^2(B U(1)) \longrightarrow E^2(S^2)
\end{displaymath}
is [[surjection|surjective]].


\item The morphism

\begin{displaymath}
\tilde i^\ast \;\colon\; \tilde E^2(B U(1)) \longrightarrow \tilde E^2(S^2)
  \simeq \pi_0(E)
\end{displaymath}
is [[surjection|surjective]].


\item The element $1 \in \pi_0(E)$ is in the [[image]] of the morphism $\tilde i^\ast$.



\end{enumerate}
A \textbf{complex orientation} on a [[multiplicative cohomology theory]] $E^\bullet$ is an element

\begin{displaymath}
c_1^E \in \tilde E^2(B U(1))
\end{displaymath}
(the ``first [[generalized Chern class]]'') such that

\begin{displaymath}
i^\ast c^E_1 = 1 \in \pi_0(E)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Since $B U(1) \simeq K(\mathbb{Z},2)$ is the [[classifying space]] for [[complex line bundles]], it follows that a complex orientation on $E^\bullet$ induces an $E$-[[generalized Chern class|generalization]] of the [[first Chern class]] which to a [[complex line bundle]] $\mathcal{L}$ on $X$ classified by $\phi \colon X \to B U(1)$ assigns the class $c_1(\mathcal{L}) \coloneqq \phi^\ast c_1^E$. This construction extends to a general construction of $E$-[[Chern classes]].

\end{remark}
\begin{prop}
\label{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}\hypertarget{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}{}
Given a [[complex oriented cohomology theory]] $(E^\bullet, c^E_1)$ (def. \ref{ComplexOrientedCohomologyTheories}), then there is an [[isomorphism]] of [[graded rings]]

\begin{displaymath}
E^\bullet(\mathbb{C}P^\infty) \simeq E^\bullet(\ast)[ [ c_1^E ] ]
\end{displaymath}
between the $E$-[[cohomology ring]] of infinite-dimensional complex projective space (def. \ref{ComplexProjectiveSpace}) and the [[formal power series]] (see remark \ref{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}) in one generator of even degree over the $E$-[[cohomology ring]] of the point.

\end{prop}
\begin{proof}
Using the [[CW-complex]]-structure on $\mathbb{C}P^\infty$ from prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, given by inductively identifying $\mathbb{C}P^{n+1}$ with the result of attaching a single $2n$-cell to $\mathbb{C}P^n$. With this structure, the unique 2-cell inclusion $i \;\colon\; S^2 \hookrightarrow \mathbb{C}P^\infty$ is identified with the canonical map $S^2 \to B U(1)$.

Then consider the [[Atiyah-Hirzebruch spectral sequence]] (prop. \ref{AHSSExistence}) for the $E$-cohomology of $\mathbb{C}P^n$.

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \;\Rightarrow\;
  E^\bullet(\mathbb{C}P^n)
  \,.
\end{displaymath}
Since, by prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace}, the [[ordinary cohomology]] with [[integer]] [[coefficients]] of complex projective space is

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, \mathbb{Z}) \simeq \mathbb{Z}[c_1]/((c_1)^{n+1})
  \,,
\end{displaymath}
where $c_1$ represents a unit in $H^2(S^2, \mathbb{Z})\simeq \mathbb{Z}$, and since similarly the [[ordinary homology]] of $\mathbb{C}P^n$ is a [[free abelian group]], hence a [[projective object]] in abelian groups (\href{projective+object#ProjectiveObjectsInAbAreFreeGroups}{prop.}), the [[Ext]]-group vanishes in each degree ($Ext^1(H_n(\mathbb{C}P^n), E^\bullet(\ast)) = 0$) and so the [[universal coefficient theorem]] (\href{universal+coefficient+theorem#OrdinaryStatementInCohomology}{prop.}) gives that the second page of the spectral sequence is

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \simeq
  E^\bullet(\ast)[ c_1 ] / (c_1^{n+1})
  \,.
\end{displaymath}
By the standard construction of the [[Atiyah-Hirzebruch spectral sequence]] (\href{Atiyah&#8211;Hirzebruch+spectral+sequence#ConstructionByFilteringTheBaseSpace}{here}) in this identification the element $c_1$ is identified with a generator of the [[relative cohomology]]

\begin{displaymath}
E^2((\mathbb{C}P^n)_2, (\mathbb{C}P^n)_1)
  \simeq
  \tilde E^2(S^2)
\end{displaymath}
(using, by the above, that this $S^2$ is the unique 2-cell of $\mathbb{C}P^n$ in the standard cell model).

This means that $c_1$ is a permanent cocycle of the spectral sequence (in the kernel of all differentials) precisely if it arises via restriction from an element in $E^2(\mathbb{C}P^n)$ and hence precisely if there exists a complex orientation $c_1^E$ on $E$. Since this is the case by assumption on $E$, $c_1$ is a permanent cocycle. (For the fully detailed argument see (\hyperlink{Pedrotti16}{Pedrotti 16})).

The same argument applied to all elements in $E^\bullet(\ast)[c]$, or else the $E^\bullet(\ast)$-linearity of the differentials (prop. \ref{LinearityOfDifferentialsInSerreAHSSForMultiplicativeCohomologyTheory}), implies that all these elements are permanent cocycles.

Since the AHSS of a [[multiplicative cohomology theory]] is a [[multiplicative spectral sequence]] (\href{multiplicative+cohomology+theory#AHSSForMultiplicativeCohomologyTheoryIsMultiplicative}{prop.}) this implies that the differentials in fact vanish on all elements of $E^\bullet(\ast) [c_1] / (c_1^{n+1})$, hence that the given AHSS collapses on the second page to give

\begin{displaymath}
\mathcal{E}_\infty^{\bullet,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1^{E} ] / ((c_1^E)^{n+1})
\end{displaymath}
or in more detail:

\begin{displaymath}
\mathcal{E}_\infty^{p,\bullet}
  \simeq
  \left\{
    \itexarray{
      E^\bullet(\ast) & \text{if}\; p \leq  2n \; and\; even
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Moreover, since therefore all $\mathcal{E}_\infty^{p,\bullet}$ are [[free modules]] over $E^\bullet(\ast)$, and since the filter stage inclusions $F^{p+1} E^\bullet(X) \hookrightarrow F^{p}E^\bullet(X)$ are $E^\bullet(\ast)$-[[module]] homomorphisms (\href{multiplicative+cohomology+theory#RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}{prop.}) the extension problem (remark \ref{ExtensionProblemForSpectralSequences}) trivializes, in that all the [[short exact sequences]]

\begin{displaymath}
0 \to F^{p+1}E^{p+\bullet}(X) \longrightarrow F^{p}E^{p+\bullet}(X) \longrightarrow \mathcal{E}_\infty^{p,\bullet} \to 0
\end{displaymath}
[[split exact sequence|split]] (since the [[Ext]]-group $Ext^1_{E^\bullet(\ast)}(\mathcal{E}_\infty^{p,\bullet},-) = 0$ vanishes on the [[free module]], hence [[projective module]] $\mathcal{E}_\infty^{p,\bullet}$).

In conclusion, this gives an isomorphism of graded rings

\begin{displaymath}
E^\bullet(\mathbb{C}P^n)
  \simeq
  \underset{p}{\oplus} \mathcal{E}_\infty^{p,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1 ] / ((c_1^{E})^{n+1})
  \,.
\end{displaymath}
A first consequence is that the projection maps

\begin{displaymath}
E^\bullet((\mathbb{C}P^\infty)_{2n+2})
  =
  E^\bullet(\mathbb{C}P^{n+1})
  \to
  E^\bullet(\mathbb{C}P^{n+})
  =
  E^\bullet((\mathbb{C}P^\infty)_{2n})
\end{displaymath}
are all [[epimorphisms]]. Therefore this sequence satisfies the [[Mittag-Leffler condition]] (\href{lim1#MittagLefflerCondition}{def.}, \href{lim1#MittagLefflerSatisfiedInParticularForTowerOfSurjections}{exmpl.}) and therefore the [[Milnor exact sequence]] for generalized cohomology (\href{lim1#MilorSequenceForReducedCohomologyOnCWComplex}{prop.}) finally implies the claim:

\begin{displaymath}
\begin{aligned}
    E^\bullet(B U(1))
     & \simeq
    E^\bullet(\mathbb{C}P^\infty)
    \\
    & \simeq
    E^\bullet( \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n )
   \\
    &\simeq \underset{\longrightarrow}{\lim}_n E^\bullet(\mathbb{C}P^n)
   \\
   &\simeq \underset{\longrightarrow}{\lim}_n ( E^\bullet(\ast) [c_1^E] / ((c_1^E)^{n+1})  )
   \\
   & \simeq E^\bullet(\ast)[ [ c_1^E ] ]
   \,,
  \end{aligned}
\end{displaymath}
where the last step is \href{formal+scheme#FormalPowerSeries}{this prop.}.

\end{proof}
$\,$

\hypertarget{ComplexOrientedCohomologyTheory}{}\subsection*{{Complex oriented cohomology}}\label{ComplexOrientedCohomologyTheory}

\textbf{Idea.} Given the concept of [[orientation in generalized cohomology]] as \hyperlink{OrientationAndFiberIntegration}{above}, it is clearly of interest to consider [[cohomology theories]] $E$ such that there exists an [[orientation in generalized cohomology|orientation]]/[[Thom class]] on the [[universal vector bundle]] over any [[classifying space]] $B G$ (or rather: on its induced [[spherical fibration]]), because then \emph{all} $G$-[[associated bundle|associated]] [[vector bundles]] inherit an orientation.

Considering this for $G = U(n)$ the [[unitary groups]] yields the concept of \emph{[[complex oriented cohomology theory]]}.

It turns out that a complex orientation on a generalized cohomology theory $E$ in this sense is already given by demanding that there is a suitable generalization of the [[first Chern class]] of [[complex line bundles]] in $E$-cohomology. By the [[splitting principle]], this already implies the existence of [[generalized Chern classes]] ([[Conner-Floyd Chern classes]]) of all degrees, and these are the required universal generalized [[Thom classes]].

Where the ordinary [[first Chern class]] in [[ordinary cohomology]] is simply additive under [[tensor product]] of [[complex line bundles]], one finds that the composite of generalized first Chern classes is instead governed by more general commutative [[formal group laws]]. This phenomenon governs much of the theory to follow.

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, section 4.3}, \hyperlink{Lurie10}{Lurie 10, lectures 1-10}, \hyperlink{Adams74}{Adams 74, Part I, Part II}, \hyperlink{Pedrotti16}{Pedrotti 16}).

\hypertarget{ChernClasses}{}\subsubsection*{{Chern classes}}\label{ChernClasses}

\textbf{Idea}. In particular [[ordinary cohomology]] [[HR]] is canonically a [[complex oriented cohomology theory]]. The behaviour of general [[Conner-Floyd Chern classes]] to be discussed \hyperlink{ConnerFloydChernClasses}{below} follows closely the behaviour of the ordinary [[Chern classes]].

An ordinary [[Chern class]] is a [[characteristic class]] of [[complex vector bundles]], and since there is the [[classifying space]] $B U$ of complex vector bundles, the \emph{universal} Chern classes are those of the [[universal complex vector bundle]] over the [[classifying space]] $B U$, which in turn are just the [[ordinary cohomology]] classes in $H^\bullet(B U)$

These may be computed inductively by iteratively applying to the [[spherical fibrations]]

\begin{displaymath}
S^{2n-1} \longrightarrow B U(n-1) \longrightarrow B U(n)
\end{displaymath}
the [[Thom-Gysin exact sequence]], a special case of the [[Serre spectral sequence]].

Pullback of Chern classes along the canonical map $(B U(1))^n \longrightarrow B U(n)$ identifies them with the [[elementary symmetric polynomials]] in the [[first Chern class]] in $H^2(B U(1))$. This is the \emph{[[splitting principle]]}.

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, section 2.2 and 2.3}, \hyperlink{Switzer75}{Switzer 75, section 16}, \hyperlink{Lurie10}{Lurie 10, lecture 5, prop. 6})

$\,$

\hypertarget{existence}{}\paragraph*{{Existence}}\label{existence}

\begin{prop}
\label{GeneratorsOfCohomologyOfBunChernClasses}\hypertarget{GeneratorsOfCohomologyOfBunChernClasses}{}
The [[cohomology ring]] of the [[classifying space]] $B U(n)$ (for the [[unitary group]] $U(n)$) is the [[polynomial ring]] on generators $\{c_k\}_{k = 1}^{n}$ of degree 2, called the \emph{Chern classes}

\begin{displaymath}
H^\bullet(B U(n), \mathbb{Z})
    \simeq
  \mathbb{Z}[c_1, \cdots, c_n]
  \,.
\end{displaymath}
Moreover, for $B i \colon B U(n_1) \longrightarrow BU(n_2)$ the canonical inclusion for $n_1 \leq n_2 \in \mathbb{N}$, then the induced pullback map on cohomology

\begin{displaymath}
(B i)^\ast
     \;\colon\;
   H^\bullet(B U(n_2))
     \longrightarrow
   H^\bullet(B U(n_1))
\end{displaymath}
is given by

\begin{displaymath}
(B i)^\ast(c_k)
  \;=\;
  \left\{
    \itexarray{
      c_k & for \; 1 \leq k \leq n_1
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, theorem 2.3.1})

\begin{proof}
For $n = 1$, in which case $B U(1) \simeq \mathbb{C}P^\infty$ is the infinite [[complex projective space]], we have by prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace}

\begin{displaymath}
H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ]
  \,,
\end{displaymath}
where $c_1$ is the [[first Chern class]]. From here we proceed by [[induction]]. So assume that the statement has been shown for $n-1$.

Observe that the canonical map $B U(n-1) \to B U(n)$ has as [[homotopy fiber]] the [[n-sphere|(2n-1)sphere]] (prop. \ref{SphereFibrationOverInclusionOfClassifyingSpaces}) hence there is a [[homotopy fiber sequence]] of the form

\begin{displaymath}
S^{2n-1} \longrightarrow B U(n-1) \longrightarrow B U(n)
  \,.
\end{displaymath}
Consider the induced [[Thom-Gysin sequence]] (prop. \ref{ThomGysinSequence}).

In odd degrees $2k+1 \lt 2n$ it gives the [[exact sequence]]

\begin{displaymath}
\cdots
   \to
  H^{2k}(B U(n-1))
   \longrightarrow
  \underset{\simeq 0}{\underbrace{H^{2k+1-2n}(B U(n))}}
   \longrightarrow
  H^{2k+1}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}}
  \to
  \cdots
  \,,
\end{displaymath}
where the right term vanishes by induction assumption, and the middle term since [[ordinary cohomology]] vanishes in negative degrees. Hence

\begin{displaymath}
H^{2k+1}(B U(n)) \simeq 0 \;\;\; for \; 2k+1 \lt 2n
\end{displaymath}
Then for $2k+1 \gt 2n$ the Thom-Gysin sequence gives

\begin{displaymath}
\cdots
   \to
  H^{2k+1-2n}(B U(n))
   \longrightarrow
  H^{2k+1}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}}
  \to
  \cdots
  \,,
\end{displaymath}
where again the right term vanishes by the induction assumption. Hence [[exact sequence|exactness]] now gives that

\begin{displaymath}
H^{2k+1-2n}(B U(n))
   \overset{}{\longrightarrow}
  H^{2k+1}(B U(n))
\end{displaymath}
is an [[epimorphism]], and so with the previous statement it follows that

\begin{displaymath}
H^{2k+1}(B U(n)) \simeq 0
\end{displaymath}
for all $k$.

Next consider the Thom Gysin sequence in degrees $2k$

\begin{displaymath}
\cdots
   \to
  \underset{\simeq 0}{\underbrace{H^{2k-1}(B U(n-1))}}
   \longrightarrow
  H^{2k-2n}(B U(n))
   \longrightarrow
  H^{2k}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  H^{2k}(B U(n-1))
   \longrightarrow
  \underset{\simeq 0}{\underbrace{H^{2k +1 - 2n}(B U(n))}}
  \to
  \cdots
  \,.
\end{displaymath}
Here the left term vanishes by the induction assumption, while the right term vanishes by the previous statement. Hence we have a [[short exact sequence]]

\begin{displaymath}
0
   \to
  H^{2k-2n}(B U(n))
   \longrightarrow
  H^{2k}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  H^{2k}(B U(n-1))
    \to
  0
\end{displaymath}
for all $k$. In degrees $\bullet\leq 2n$ this says

\begin{displaymath}
0
   \to
  \mathbb{Z}
   \overset{c_n \cup (-)}{\longrightarrow}
  H^{\bullet \leq 2n}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n}
    \to
  0
\end{displaymath}
for some [[Thom class]] $c_n \in H^{2n}(B U(n))$, which we identify with the next Chern class.

Since [[free abelian groups]] are [[projective objects]] in [[Ab]], their [[extensions]] are all split (the [[Ext]]-group out of them vanishes), hence the above gives a [[direct sum]] decomposition

\begin{displaymath}
\begin{aligned}
    H^{\bullet \leq 2n}(B U(n))
    & \simeq
    (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \oplus \mathbb{Z}\langle 2n\rangle
    \\
    & \simeq
    (\mathbb{Z}[c_1, \cdots, c_{n}])_{\bullet \leq 2n}
  \end{aligned}
  \,.
\end{displaymath}
Now by another induction over these short exact sequences, the claim follows.

\end{proof}
\hypertarget{splitting_principle}{}\paragraph*{{Splitting principle}}\label{splitting_principle}

\begin{lemma}
\label{FromBUnTOBU1nPullbackInCohomologyIsInjective}\hypertarget{FromBUnTOBU1nPullbackInCohomologyIsInjective}{}
For $n \in \mathbb{N}$ let $\mu_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on [[ordinary cohomology]]

\begin{displaymath}
\mu^\ast_n
  \;\colon\;
   H^\bullet( B U(n); \mathbb{Z} )
   \longrightarrow
   H^\bullet( B U(1)^n; \mathbb{Z} )
\end{displaymath}
is a [[monomorphism]].

\end{lemma}
A \textbf{proof} of lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective} via analysis of the [[Serre spectral sequence]] of $U(n)/U(1)^n \to B U(1)^n \to B U(n)$ is indicated in (\hyperlink{Kochmann96}{Kochmann 96, p. 40}). A proof via [[Becker-Gottlieb transfer|transfer]] of the [[Euler class]] of $U(n)/U(1)^n$ is indicated at \emph{[[splitting principle]]} (\href{splitting+principle#InjectivityOfPullbackInCohomologyToBT}{here}).

\begin{prop}
\label{SplittingPrincipleForChernClasses}\hypertarget{SplittingPrincipleForChernClasses}{}
For $k \leq n \in \mathbb{N}$ let $B i_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on [[ordinary cohomology]] is of the form

\begin{displaymath}
(B i_n)^\ast
  \;\colon\;
  \mathbb{Z}[c_1, \cdots, c_k]
  \longrightarrow
  \mathbb{Z}[(c_1)_1,\cdots (c_1)_n]
\end{displaymath}
and sends the $k$th Chern class $c_k$ (def. \ref{GeneratorsOfCohomologyOfBunChernClasses}) to the $k$th [[elementary symmetric polynomial]] in the $n$ copies of the [[first Chern class]]:

\begin{displaymath}
(B i_n)^\ast
    \;\colon\;
  c_k \mapsto \sigma_k( (c_1)_1, \cdots, (c_1)_n )
  \coloneqq
  \underset{1 \leq i_1 \leq \cdots \leq i_k \leq n}{\sum}
   (c_1)_{i_1} \cdots (c_1)_{i_n}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
First consider the case $n = 1$.

The [[classifying space]] $B U(1)$ (def. \ref{EOn}) is equivalently the infinite [[complex projective space]] $\mathbb{C}P^\infty$. Its [[ordinary cohomology]] is the [[polynomial ring]] on a single generator $c_1$, the [[first Chern class]] (prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace})

\begin{displaymath}
H^\bullet(B U(1))
    \simeq
  \mathbb{Z}[ c_1 ]
  \,.
\end{displaymath}
Moreover, $B i_1$ is the identity and the statement follows.

Now by the [[Künneth theorem]] for ordinary cohomology (\href{K%C3%BCnneth+theorem#KunnethInOrdinaryCohomology}{prop.}) the cohomology of the [[Cartesian product]] of $n$ copies of $B U(1)$ is the [[polynomial ring]] in $n$ generators

\begin{displaymath}
H^\bullet(B U(1)^n)
   \simeq
  \mathbb{Z}[(c_1)_1, \cdots, (c_1)_n]
  \,.
\end{displaymath}
By prop. \ref{GeneratorsOfCohomologyOfBunChernClasses} the domain of $(B i_n)^\ast$ is the [[polynomial ring]] in the Chern classes $\{c_i\}$, and by the previous statement the codomain is the polynomial ring on $n$ copies of the first Chern class

\begin{displaymath}
(B i_n)^\ast
    \;\colon\;
   \mathbb{Z}[ c_1, \cdots, c_n ]
     \longrightarrow
   \mathbb{Z}[ (c_1)_1, \cdots, (c_1)_n ]
  \,.
\end{displaymath}
This allows to compute $(B i_n)^\ast(c_k)$ by [[induction]]:

Consider $n \geq 2$ and assume that $(B i_{n-1})^\ast_{n-1}(c_k) = \sigma_k((c_1)_1, \cdots, (c_1)_{(n-1)})$. We need to show that then also $(B i_n)^\ast(c_k) = \sigma_k((c_1)_1,\cdots, (c_1)_n)$.

Consider then the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    B U(1)^{n-1}
      &\overset{ B i_{n-1} }{\longrightarrow}&
    B U(n-1)
    \\
    {}^{\mathllap{B j_{\hat t}}}\downarrow
      &&
    \downarrow^{\mathrlap{B i_{\hat t}}}
    \\
    B U(1)^n
     &\underset{B i_n}{\longrightarrow}&
    B U(n)
  }
\end{displaymath}
where both vertical morphisms are induced from the inclusion

\begin{displaymath}
\mathbb{C}^{n-1} \hookrightarrow \mathbb{C}^n
\end{displaymath}
which omits the $t$th coordinate.

Since two embeddings $i_{\hat t_1}, i_{\hat t_2} \colon U(n-1) \hookrightarrow U(n)$ differ by [[conjugation]] with an element in $U(n)$, hence by an [[inner automorphism]], the maps $B i_{\hat t_1}$ and $B_{\hat i_{t_2}}$ are [[homotopy|homotopic]], and hence $(B i_{\hat t})^\ast = (B i_{\hat n})^\ast$, which is the morphism from prop. \ref{GeneratorsOfCohomologyOfBunChernClasses}.

By that proposition, $(B i_{\hat t})^\ast$ is the identity on $c_{k \lt n}$ and hence by induction assumption

\begin{displaymath}
\begin{aligned}
    (B i_{n-1})^\ast (B i_{\hat t})^\ast c_{k \lt n}
    &=
    (B i_{n-1})^\ast c_{k \lt n}
    \\
    =
    \sigma_k( (c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n )
  \end{aligned}
  \,.
\end{displaymath}
Since pullback along the left vertical morphism sends $(c_1)_t$ to zero and is the identity on the other generators, this shows that

\begin{displaymath}
(B i_n)^\ast(c_{k \lt n})
    \simeq
   \sigma_{k\lt n}((c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n)
   \;\;
   mod (c_1)_t
  \,.
\end{displaymath}
This implies the claim for $k \lt n$.

For the case $k = n$ the commutativity of the diagram and the fact that the right map is zero on $c_n$ by prop. \ref{GeneratorsOfCohomologyOfBunChernClasses} shows that the element $(B j_{\hat t})^\ast (B i_n)^\ast c_n = 0$ for all $1 \leq t \leq n$. But by lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective} the morphism $(B i_n)^\ast$, is injective, and hence $(B i_n)^\ast(c_n)$ is non-zero. Therefore for this to be annihilated by the morphisms that send $(c_1)_t$ to zero, for all $t$, the element must be proportional to all the $(c_1)_t$. By degree reasons this means that it has to be the product of all of them

\begin{displaymath}
\begin{aligned}
    (B i_n)^{\ast}(c_n)
     & = (c_1)_1 \otimes (c_1)_2 \otimes \cdots \otimes (c_1)_n
     \\
     & = \sigma_n( (c_1)_1, \cdots, (c_1)_n )
  \end{aligned}
  \,.
\end{displaymath}
This completes the induction step, and hence the proof.

\end{proof}
\begin{prop}
\label{WhitneySumChernClasses}\hypertarget{WhitneySumChernClasses}{}
For $k\leq n \in \mathbb{N}$, consider the canonical map

\begin{displaymath}
\mu_{k,n-k}
    \;\colon\;
  B U(k) \times B U(n-k)
    \longrightarrow
  B U(n)
\end{displaymath}
(which classifies the [[Whitney sum]] of [[complex vector bundles]] of [[rank]] $k$ with those of rank $n-k$). Under pullback along this map the universal [[Chern classes]] (prop. \ref{GeneratorsOfCohomologyOfBunChernClasses}) are given by

\begin{displaymath}
(\mu_{k,n-k})^\ast(c_t)
  \;=\;
  \underoverset{i = 0}{t}{\sum} c_i \otimes c_{t-i}
  \,,
\end{displaymath}
where we take $c_0 = 1$ and $c_j = 0 \in H^\bullet(B U(r))$ if $j \gt r$.

So in particular

\begin{displaymath}
(\mu_{k,n-k})^\ast(c_n) \;=\; c_k \otimes c_{n-k}
  \,.
\end{displaymath}
\end{prop}
e.g. (\hyperlink{Kochmann96}{Kochmann 96, corollary 2.3.4})

\begin{proof}
Consider the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    H^\bullet( B U(n) )
      &\overset{\mu_{k,n-k}^\ast}{\longrightarrow}&
    H^\bullet( B U(k) ) \otimes H^\bullet( B U(n-k) )
    \\
    {}^{\mathllap{\mu_k^\ast}}\downarrow
      &&
    \downarrow^{\mathrlap{ \mu_{k}^\ast \otimes \mu_{n-k}^\ast }}
    \\
    H^\bullet( B U(1)^n )
     &\simeq&
    H^\bullet( B U(1)^k ) \otimes H^\bullet( B U(1)^{n-k} )
  }
  \,.
\end{displaymath}
This says that for all $t$ then

\begin{displaymath}
\begin{aligned}
    (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t)
    & =
    \mu^\ast_n(c_t)
    \\
    & = \sigma_t((c_1)_1, \cdots, (c_1)_n)
  \end{aligned}
  \,,
\end{displaymath}
where the last equation is by prop. \ref{SplittingPrincipleForChernClasses}.

Now the [[elementary symmetric polynomial]] on the right decomposes as required by the left hand side of this equation as follows:

\begin{displaymath}
\sigma_t((c_1)_1, \cdots, (c_1)_n)
    \;=\;
  \underoverset{r = 0}{t}{\sum}
  \sigma_r((c_1)_1, \cdots, (c_1)_{n-k})
  \cdot
  \sigma_{t-r}( (c_1)_{n-k+1}, \cdots, (c_1)_n )
  \,,
\end{displaymath}
where we agree with $\sigma_q((c_1)_1, \cdots, (c_1)_p) = 0$ if $q \gt p$. It follows that

\begin{displaymath}
(\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t)
  =
  (\mu_k^\ast \otimes \mu_{n-k}^\ast)
  \left(
    \underoverset{r=0}{t}{\sum} c_r \otimes c_{t-r}
  \right)
  \,.
\end{displaymath}
Since $(\mu_k^\ast \otimes \mu_{n-k}^\ast)$ is a monomorphism by lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective}, this implies the claim.

\end{proof}
\hypertarget{ConnerFloydChernClasses}{}\subsubsection*{{Conner-Floyd Chern classes}}\label{ConnerFloydChernClasses}

\textbf{Idea.} For $E$ a [[complex oriented cohomology theory]], then the generators of the $E$-[[cohomology groups]] of the [[classifying space]] $B U$ are called the \emph{[[Conner-Floyd Chern classes]]}, in $E^\bullet(B U)$.

Using basic properties of the classifying space $B U(1)$ via its incarnation as the infinite [[complex projective space]] $\mathbb{C}P^\infty$, one finds that the [[Atiyah-Hirzebruch spectral sequences]]

\begin{displaymath}
H^p(\mathbb{C}P^n, \pi_q(E)) \Rightarrow H^\bullet(\mathbb{C}P^n)
\end{displaymath}
collapse right away, and that the [[inverse system]] which they form satisfies the [[Mittag-Leffler condition]]. Accordingly the [[Milnor exact sequence]] gives that the ordinary [[first Chern class]] $c_1$ generates, over $\pi_\bullet(E)$, all Conner-Floyd classes over $B U(1)$:

\begin{displaymath}
E^\bullet(B U(1)) \simeq \pi_\bullet(E) [ [ c_1 ] ]
  \,.
\end{displaymath}
This is the key input for the discussion of [[formal group laws]] \hyperlink{FormalGroupLaws}{below}.

Combining the [[Atiyah-Hirzebruch spectral sequence]] with the [[splitting principle]] as for ordinary Chern classes \hyperlink{ChernClasses}{above} yields, similarly, that in general Conner-Floyd classes are generated, over $\pi_\bullet(E)$, from the ordinary Chern classes.

Finally one checks that Conner-Floyd classes canonically serve as [[Thom classes]] for $E$-cohomology of the [[universal complex vector bundle]], thereby showing that [[complex oriented cohomology theories]] are indeed canonically [[orientation in generalized cohomology|oriented]] on ([[spherical fibrations]] of) [[complex vector bundles]].

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, section 4.3} \hyperlink{Adams74}{Adams 74, part I.4, part II.2 II.4, part III.10}, \hyperlink{Lurie10}{Lurie 10, lecture 5})

\begin{prop}
\label{ConnerFloyedClasses}\hypertarget{ConnerFloyedClasses}{}
Given a [[complex oriented cohomology theory]] $E$ with complex orientation $c_1^E$, then the $E$-[[generalized cohomology]] of the [[classifying space]] $B U(n)$ is freely generated over the [[graded commutative ring]] $\pi_\bullet(E)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum}{prop.}) by classes $c_k^E$ for $0 \leq \leq n$ of degree $2k$, these are called the \textbf{[[Conner-Floyd-Chern classes]]}

\begin{displaymath}
E^\bullet(B U(n))
    \;\simeq\;
  \pi_\bullet(E)[ [ c_1^E, c_2^E, \cdots, c_n^E ] ]
  \,.
\end{displaymath}
Moreover, pullback along the canonical inclusion $B U(n) \to B U(n+1)$ is the identity on $c_k^E$ for $k \leq n$ and sends $c_{n+1}^E$ to zero.

For $E$ being [[ordinary cohomology]], this reduces to the ordinary [[Chern classes]] of prop. \ref{GeneratorsOfCohomologyOfBunChernClasses}.

\end{prop}
For details see (\hyperlink{Pedrotti16}{Pedrotti 16, prop. 3.1.14}).

\hypertarget{FormalGroupLaws}{}\subsubsection*{{Formal group laws of first CF-Chern classes}}\label{FormalGroupLaws}

\textbf{Idea.} The [[classifying space]] $B U(1)$ for [[complex line bundles]] is a [[homotopy type]] canonically equipped with commutative group structure ([[infinity-group]]-structure), corresponding to the [[tensor product]] of [[complex line bundles]]. By the above, for $E$ a [[complex oriented cohomology theory]] the first [[Conner-Floyd Chern class]] of these complex line bundles generates the $E$-cohomology of $B U(1)$, it follows that the [[cohomology ring]] $E^\bullet(B U(1)) \simeq \pi_\bullet(E)[ [ c_1 ] ]$ behaves like the ring of $\pi_\bullet(E)$-valued functions on a 1-dimensional commutative [[formal group]] equipped with a canonical [[coordinate]] function $c_1$. This is called a \emph{[[formal group law]]} over the [[graded commutative ring]] $\pi_\bullet(E)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum}{prop.}).

On abstract grounds it follows that there exists a commutative ring $L$ and a universal (1-dimensional commutative) formal group law $\ell$ over $L$. This is called the \emph{[[Lazard ring]]}. [[Lazard's theorem]] identifies this ring concretely: it turns out to simply be the [[polynomial ring]] on generators in every even degree.

Further below this has profound implications on the structure theory for complex oriented cohomology. The [[Milnor-Quillen theorem on MU]] identifies the Lazard ring as the cohomology ring of the [[Thom spectrum]] [[MU]], and then the [[Landweber exact functor theorem]], implies that there are lots of complex oriented cohomology theories.

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, section 4.4}, \hyperlink{Lurie10}{Lurie 10, lectures 1 and 2})

\hypertarget{formal_group_laws}{}\paragraph*{{Formal group laws}}\label{formal_group_laws}

\begin{defn}
\label{AdicRing}\hypertarget{AdicRing}{}
An (commutative) [[adic ring]] is a ([[commutative ring|commutative]]) [[topological ring]] $A$ and an ideal $I \subset A$ such that

\begin{enumerate}%
\item the [[topological space|topology]] on $A$ is the $I$-[[adic topology]];


\item the canonical morphism

\begin{displaymath}
A \longrightarrow \underset{\longleftarrow}{\lim}_n (A/I^n)
\end{displaymath}
to the [[limit]] over [[quotient rings]] by powers of the ideal is an [[isomorphism]].



\end{enumerate}
A [[homomorphism]] of adic rings is a ring [[homomorphism]] that is also a [[continuous function]] (hence a function that preserves the filtering $A \supset \cdots \supset A/I^2 \supset A/I$). This gives a category $AdicRing$ and a subcategory $AdicCRing$ of commutative adic rings.

The [[opposite category]] of $AdicRing$ (on [[Noetherian rings]]) is that of affine [[formal schemes]].

Similarly, for $R$ any fixed [[commutative ring]], then adic rings under $R$ are \emph{adic $R$-algebras}. We write $Adic A Alg$ and $Adic A CAlg$ for the corresponding categories.

\end{defn}
\begin{example}
\label{PowerSeriesAlgebra}\hypertarget{PowerSeriesAlgebra}{}
For $R$ a [[commutative ring]] and $n \in \mathbb{N}$ then the [[formal power series ring]]

\begin{displaymath}
R[ [ x_1, x_2, \cdots, x_n ] ]
\end{displaymath}
in $n$ [[variables]] with [[coefficients]] in $R$ and equipped with the ideal

\begin{displaymath}
I = (x_1, \cdots , x_n)
\end{displaymath}
is an adic ring (def. \ref{AdicRing}).

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
There is a [[fully faithful functor]]

\begin{displaymath}
AdicRing \hookrightarrow ProRing
\end{displaymath}
from [[adic rings]] (def. \ref{AdicRing}) to [[pro-rings]], given by

\begin{displaymath}
(A,I) \mapsto ( (A/I^{\bullet}))
  \,,
\end{displaymath}
i.e. for $A,B \in AdicRing$ two adic rings, then there is a [[natural isomorphism]]

\begin{displaymath}
Hom_{AdicRing}(A,B)
    \simeq
   \underset{\longleftarrow}{\lim}_{n_2}
   \underset{\longrightarrow}{\lim}_{n_1}
   Hom_{Ring}(A/I^{n_1},B/I^{n_2})
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{GroupObjectFormalGroupLaw}\hypertarget{GroupObjectFormalGroupLaw}{}
For $R \in CRing$ a [[commutative ring]] and for $n \in \mathbb{N}$, a \textbf{formal group law} of dimension $n$ over $R$ is the structure of a [[group object]] in the category $Adic R CAlg^{op}$ from def. \ref{AdicRing} on the object $R [ [x_1, \cdots ,x_n] ]$ from example \ref{PowerSeriesAlgebra}.

Hence this is a morphism

\begin{displaymath}
\mu
  \;\colon\;
  R[ [ x_1, \cdots, x_n ] ]
   \longrightarrow
  R [ [ x_1, \cdots, x_n, \, y_1, \cdots, y_n ] ]
\end{displaymath}
in $Adic R CAlg$ satisfying unitality, associativity.

This is a \textbf{commutative formal group law} if it is an abelian group object, hence if it in addition satisfies the corresponding commutativity condition.

\end{defn}
This is equivalently a set of $n$ power series $F_i$ of $2n$ variables $x_1,\ldots,x_n,y_1,\ldots,y_n$ such that (in notation $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)$, $F(x,y) = (F_1(x,y),\ldots,F_n(x,y))$)

\begin{displaymath}
F(x,F(y,z))=F(F(x,y),z)
\end{displaymath}
\begin{displaymath}
F_i(x,y) = x_i+y_i+\,\,higher\,\,order\,\,terms
\end{displaymath}
\begin{example}
\label{Commutative1DimFormalGroupLaw}\hypertarget{Commutative1DimFormalGroupLaw}{}
A 1-dimensional commutative formal group law according to def. \ref{GroupObjectFormalGroupLaw} is equivalently a [[formal power series]]

\begin{displaymath}
\mu(x,y)
  =
  \underset{i,j \geq 0}{\sum} a_{i,j} x^i y^j
\end{displaymath}
(the image under $\mu$ in $R[ [ x,y ] ]$ of the element $t \in R [ [ t ] ]$) such that

\begin{enumerate}%
\item (unitality)

\begin{displaymath}
\mu(x,0) = x$ and $\mu(0,x) = x
  \,;
\end{displaymath}

\item (associativity)

\begin{displaymath}
\mu(x,\mu(y,z)) = \mu(\mu(x,y),z)
  \,;
\end{displaymath}

\item (commutativity)

\begin{displaymath}
\mu(x,y) = \mu(y,x)
  \,.
\end{displaymath}


\end{enumerate}
The first condition means equivalently that

\begin{displaymath}
a_{i,0}
    =
  \left\{
    \itexarray{
      1 & if i = 0
      \\
      0 & otherwise
    }
  \right.
  \;\;\;\;\,,
  \;\;\;\;\;
  a_{0,i}
    =
  \left\{
    \itexarray{
      1 & if i = 0
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Hence $\mu$ is necessarily of the form

\begin{displaymath}
\mu(x,y)
   \;=\;
   x + y + \underset{i,j \geq 1}{\sum} a_{i,j} x^i y^j
  \,.
\end{displaymath}
The existence of inverses is no extra condition: by [[induction]] on the index $i$ one finds that there exists a unique

\begin{displaymath}
\iota(x) = \underset{i \geq 1}{\sum} \iota(x)_i x^i
\end{displaymath}
such that

\begin{displaymath}
\mu(x,iota(x)) = x
  \;\;\;\,,
  \;\;\;
  \mu(\iota(x),x) = x
  \,.
\end{displaymath}
Hence 1-dimensional formal group laws over $R$ are equivalently [[monoids]] in $Adic R CAlg^{op}$ on $R[ [ x ] ]$.

\end{example}
\hypertarget{formal_group_laws_from_complex_orientation}{}\paragraph*{{Formal group laws from complex orientation}}\label{formal_group_laws_from_complex_orientation}

Let again $B U(1)$ be the [[classifying space]] for [[complex line bundles]], modeled, in particular, by infinite [[complex projective space]] $\mathbb{C}P^\infty)$.

\begin{lemma}
\label{BU1HomotopyGroupStructure}\hypertarget{BU1HomotopyGroupStructure}{}
There is a [[continuous function]]

\begin{displaymath}
\mu
    \;\colon\;
  \mathbb{C}P^\infty \times \mathbb{C}P
    \longrightarrow
  \mathbb{C}P^\infty
\end{displaymath}
which represents the [[tensor product of line bundles]] in that under the defining equivalence, and for $X$ any [[paracompact topological space]], then

\begin{displaymath}
\itexarray{
     [X, \mathbb{C}P^\infty \times \mathbb{C}P^\infty]
     &\simeq&
     \mathbb{C}LineBund(X)_{/\sim} \times \mathbb{C}LineBund(X)_{/\sim}
     \\
     {}^{\mathllap{[X,\mu]}}\downarrow
       &&
     \downarrow^{\mathrlap{\otimes}}
     \\
     [X,\mathbb{C}P^\infty]
       &\simeq&
     \mathbb{C}LineBund(X)_{/\sim}
  }
  \,,
\end{displaymath}
where $[-,-]$ denotes the [[hom-sets]] in the (Serre-Quillen-)[[classical homotopy category]] and $\mathbb{C}LineBund(X)_{/\sim}$ denotes the set of [[isomorphism classes]] of [[complex line bundles]] on $X$.

Together with the canonical point inclusion $\ast \to \mathbb{C}P^\infty$, this makes $\mathbb{C}P^\infty$ an [[abelian group|abelian]] [[group object]] in the [[classical homotopy category]].

\end{lemma}
\begin{proof}
By the [[Yoneda lemma]] (the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]]) there exists such a morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \longrightarrow \mathbb{C}P^\infty$ in the [[classical homotopy category]]. But since $\mathbb{C}P^\infty$ admits the structure of a [[CW-complex]] (prop. \ref{CellComplexStructureOnComplexProjectiveSpace})) it is cofibrant in the [[standard model structure on topological spaces]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopQuillenModelStructure}{thm.}), as is its [[Cartesian product]] with itself (\href{CW+complex#ClosureOfCWComplexesUnderCartesianProduct}{prop.}). Since moreover all spaces are fibrant in the [[classical model structure on topological spaces]], it follows (by \href{Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory}{this lemma}) that there is an actual [[continuous function]] representing that morphism in the homotopy category.

That this gives the structure of an [[abelian group|abelian]] [[group object]] now follows via the [[Yoneda lemma]] from the fact that each $\mathbb{C}LineBund(X)_{/\sim}$ has the structure of an [[abelian group]] under [[tensor product of line bundles]], with the [[trivial bundle|trivial]] line bundle (wich is classified by maps factoring through $\ast \to \mathbb{C}P^\infty$) being the neutral element, and that this group structure is [[natural transformation|natural]] in $X$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The space $B U(1) \simeq \mathbb{C}P^\infty$ has in fact more structure than that of a homotopy group from lemma \ref{BU1HomotopyGroupStructure}. As an object of the [[homotopy theory]] represented by the [[classical model structure on topological spaces]], it is a \emph{[[2-group]]}, a [[truncated object in an (infinity,1)-category|1-truncated]] [[infinity-group]].

\end{remark}
\begin{prop}
\label{ComplexOrientedCohomologyTheoryFormalGroupLaw}\hypertarget{ComplexOrientedCohomologyTheoryFormalGroupLaw}{}
Let $(E, c_1^E)$ be a [[complex oriented cohomology theory]]. Under the identification

\begin{displaymath}
E^\bullet(\mathbb{C}P^\infty)
   \simeq
  \pi_\bullet(E)[ [ c^E_1 ] ]
  \;\;\;\,,
  \;\;\;
  E^\bullet(\mathbb{C}P^\infty \times \mathbb{C}P^\infty)
   \simeq
  \pi_\bullet(E)[ [ c^E_1 \otimes 1 , \, 1 \otimes c^E_1 ] ]
\end{displaymath}
from prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}, the operation

\begin{displaymath}
\pi_\bullet(E)
  [ [ c^E_1 ] ]
    \simeq
  E^\bullet(\mathbb{C}P^\infty)
    \longrightarrow
  E^\bullet( \mathbb{C}P^\infty \times \mathbb{C}P^\infty )
    \simeq
  \pi_\bullet(E)[ [ c_1^E \otimes 1, 1 \otimes c_1^E ] ]
\end{displaymath}
of pullback in $E$-cohomology along the maps from lemma \ref{BU1HomotopyGroupStructure} constitutes a 1-dimensional graded-commutative [[formal group law]] (example \ref{Commutative1DimFormalGroupLaw})over the [[graded commutative ring]] $\pi_\bullet(E)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum}{prop.}). If we consider $c_1^E$ to be in degree 2, then this formal group law is compatibly graded.

\end{prop}
\begin{proof}
The associativity and commutativity conditions follow directly from the respective properties of the map $\mu$ in lemma \ref{BU1HomotopyGroupStructure}. The grading follows from the nature of the identifications in prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
That the grading of $c_1^E$ in prop. \ref{ComplexOrientedCohomologyTheoryFormalGroupLaw} is in negative degree is because by definition

\begin{displaymath}
\pi_\bullet(E) = E_\bullet = E^{-\bullet}
\end{displaymath}
(\href{Introduction+to+Stable+homotopy+theory+--+1-2#EMHomology}{rmk.}).

Under different choices of orientation, one obtains different but isomorphic formal group laws.

\end{remark}
\hypertarget{the_universal_1d_commutative_formal_group_law_and_lazards_theorem}{}\paragraph*{{The universal 1d commutative formal group law and Lazard's theorem}}\label{the_universal_1d_commutative_formal_group_law_and_lazards_theorem}

It is immediate that there exists a ring carrying a universal formal group law. For observe that for $\underset{i,j}{\sum} a_{i,j} x_1^i x_1^j$ an element in a [[formal power series]] algebra, then the condition that it defines a [[formal group law]] is equivalently a sequence of polynomial [[equations]] on the [[coefficients]] $a_k$. For instance the commutativity condition means that

\begin{displaymath}
a_{i,j} = a_{j,i}
\end{displaymath}
and the unitality constraint means that

\begin{displaymath}
a_{i 0}
  =
  \left\{
    \itexarray{
      1 & if \; i = 1
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Similarly associativity is equivalently a condition on combinations of triple products of the coefficients. It is not necessary to even write this out, the important point is only that it is some polynomial equation.

This allows to make the following definition

\begin{defn}
\label{LazardRing}\hypertarget{LazardRing}{}
The \textbf{[[Lazard ring]]} is the [[graded commutative ring]] generated by elenebts $a_{i j}$ in degree $2(i+j-1)$ with $i,j \in \mathbb{N}$

\begin{displaymath}
L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below)
\end{displaymath}
quotiented by the relations

\begin{enumerate}%
\item $a_{i j} = a_{j i}$


\item $a_{10} = a_{01} = 1$; $\forall i \neq 1: a_{i 0} = 0$


\item the obvious associativity relation



\end{enumerate}
for all $i,j,k$.

The \textbf{universal 1-dimensional commutative [[formal group law]]} is the formal power series with [[coefficients]] in the Lazard ring given by

\begin{displaymath}
\ell(x,y) \coloneqq \sum_{i,j} a_{i j} x^i y^j \in L[ [ x , y ] ]
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{LazardRing}\hypertarget{LazardRing}{}
The grading is chosen with regards to the formal group laws arising from [[complex oriented cohomology theories]] (\href{complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw}{prop.}) where the [[variable]] $x$ naturally has degree -2. This way

\begin{displaymath}
deg(a_{i j} x^i y^j)
  =
  deg(a_i,j) + i deg(x) + j deg(y)
  =
  -2
  \,.
\end{displaymath}
\end{remark}
The following is immediate from the definition:

\begin{prop}
\label{LazardRingIsUniversal}\hypertarget{LazardRingIsUniversal}{}
For every [[ring]] $R$ and 1-dimensional commutative [[formal group law]] $\mu$ over $R$ (example \ref{Commutative1DimFormalGroupLaw}), there exists a unique ring [[homomorphism]]

\begin{displaymath}
f \;\colon\; L \longrightarrow R
\end{displaymath}
from the [[Lazard ring]] (def. \ref{LazardRing}) to $R$, such that it takes the universal formal group law $\ell$ to $\mu$

\begin{displaymath}
f_\ast \ell = \mu
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
If the formal group law $\mu$ has coefficients $\{c_{i,j}\}$, then in order that $f_\ast \ell = \mu$, i.e. that

\begin{displaymath}
\underset{i,j}{\sum} f(a_{i,j}) x^i y^j
  =
  \underset{i,j}{\sum} c_{i,j} x^i y^j
\end{displaymath}
it must be that $f$ is given by

\begin{displaymath}
f(a_{i,j}) = c_{i,j}
\end{displaymath}
where $a_{i,j}$ are the generators of the Lazard ring. Hence it only remains to see that this indeed constitutes a ring homomorphism. But this is guaranteed by the vary choice of relations imposed in the definition of the Lazard ring.

\end{proof}
What is however highly nontrivial is this statement:

\begin{theorem}
\label{LazardTheorem}\hypertarget{LazardTheorem}{}
\textbf{([[Lazard's theorem]])}

The [[Lazard ring]] $L$ (def. \ref{LazardRing}) is [[isomorphism|isomorphic]] to a [[polynomial ring]]

\begin{displaymath}
L
  \simeq
  \mathbb{Z}[ t_1, t_2, \cdots ]
\end{displaymath}
in countably many generators $t_i$ in degree $2 i$.

\end{theorem}
\begin{remark}
\label{}\hypertarget{}{}
The [[Lazard theorem]] \ref{LazardTheorem} first of all implies, via prop. \ref{LazardRingIsUniversal}, that there exists an abundance of 1-dimensional formal group laws: given any ring $R$ then every choice of elements $\{t_i \in R\}$ defines a formal group law. (On the other hand, it is nontrivial to say which formal group law that is.)

Deeper is the fact expressed by the [[Milnor-Quillen theorem on MU]]: the Lazard ring in its polynomial incarnation of prop. \ref{LazardTheorem} is canonically identieif with the [[graded commutative ring]] $\pi_\bullet(M U)$ of [[stable homotopy groups]] of the universal complex [[Thom spectrum]] [[MU]]. Moreover:

\begin{enumerate}%
\item [[MU]] carries a [[universal complex orientation on MU|universal complex orientation]] in that for $E$ any [[homotopy commutative ring spectrum]] then homotopy classes of homotopy ring homomorphisms $M U \to E$ are in bijection to [[complex oriented cohomology theory|complex orientations]] on $E$;


\item every [[complex oriented cohomology|complex orientation]] on $E$ induced a 1-dimensional commutative [[formal group law]] (\href{complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw}{prop.})


\item under forming stable homtopy groups every ring spectrum homomorphism $M U \to E$ induces a ring homomorphism

\begin{displaymath}
L \simeq \pi_\bullet(M U) \longrightarrow \pi_\bullet(E)
\end{displaymath}
and hence, by the universality of $L$, a formal group law over $\pi_\bullet(E)$.



\end{enumerate}
This is the formal group law given by the above complex orientation.

Hence the universal group law over the Lazard ring is a kind of [[decategorification]] of the [[universal complex orientation on MU]].

\end{remark}
\hypertarget{ComplexCobordismCohomology}{}\subsubsection*{{Complex cobordism}}\label{ComplexCobordismCohomology}

\textbf{Idea.} There is a [[weak homotopy equivalence]] $\phi \colon B U(1)\stackrel{\simeq}{\longrightarrow} M U(1)$ between the [[classifying space]] for [[complex line bundles]] and the [[Thom space]] of the [[universal vector bundle|universal complex line bundle]]. This gives an element $\pi_\ast(c_1) \in M U^2(B U(1))$ in the [[complex cobordism cohomology]] of $B U(1)$ which makes the universal complex [[Thom spectrum]] [[MU]] become a [[complex oriented cohomology theory]].

This turns out to be a [[universal complex orientation on MU]]: for every other [[homotopy commutative ring spectrum]] $E$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum}{def.}) there is an equivalence between complex orientations on $E$ and homotopy classes of homotopy ring spectrum homomorphisms

\begin{displaymath}
\{M U \longrightarrow E\}_{/\simeq}
     \;\simeq\;
  \{complex\;orientations\;on\;E\}
  \,.
\end{displaymath}
Hence [[complex oriented cohomology theory]] is [[higher algebra]] over [[MU]].

\textbf{Literature.} (\hyperlink{Schwede12}{Schwede 12, example 1.18}, \hyperlink{Kochman96}{Kochman 96, section 1.4, 1.5, 4.4}, \hyperlink{Lurie10}{Lurie 10, lectures 5 and 6})

\hypertarget{connerfloydchern_classes_are_thom_classes}{}\paragraph*{{Conner-Floyd-Chern classes are Thom classes}}\label{connerfloydchern_classes_are_thom_classes}

We discuss that for $E$ a [[complex oriented cohomology theory]], then the $n$th universal [[Conner-Floyd-Chern class]] $c^E_n$ is in fact a universal [[Thom class]] for rank $n$ [[complex vector bundles]]. On the one hand this says that the choice of a [[complex oriented cohomology theory|complex orientation]] on $E$ indeed universally [[orientation in generalized cohomology|orients]] all [[complex vector bundles]]. On the other hand, we interpret this fact \hyperlink{ComplexOrientationAsRingSpectrumMaps}{below} as the [[unitality]] condition on a [[homomorphism]] of [[homotopy commutative ring spectra]] $M U \to E$ which represent that universal orienation.

\begin{lemma}
\label{SphereBundleBunminus1}\hypertarget{SphereBundleBunminus1}{}
For $n \in \mathbb{N}$, the [[fiber sequence]] (prop. \ref{SphereFibrationOverInclusionOfClassifyingSpaces})

\begin{displaymath}
\itexarray{
    S^{2n-1} &\longrightarrow& B U(n-1)
    \\
    && \downarrow
    \\
    && B U(n)
  }
\end{displaymath}
exhibits $B U(n-1)$ as the [[sphere bundle]] of the [[universal vector bundle|universal complex vector bundle]] over $B U(n)$.

\end{lemma}
\begin{proof}
When exhibited by a fibration, here the vertical morphism is equivalently the quotient map

\begin{displaymath}
(E U(n))/U(n-1) \longrightarrow (E U(n))/U(n)
\end{displaymath}
(by the proof of prop. \ref{SphereFibrationOverInclusionOfClassifyingSpaces}).

Now the [[universal principal bundle]] $E U(n)$ is (def. $\backslash$ref\{EOn)\}) equivalently the colimit

\begin{displaymath}
E U(n) \simeq \underset{\longrightarrow}{\lim}_k  U(k)/U(k-n)
  \,.
\end{displaymath}
Here each [[Stiefel manifold]]/[[coset spaces]] $U(k)/U(k-n)$ is equivalently the space of (complex) $n$-dimensional subspaces of $\mathbb{C}^k$ that are equipped with an orthonormal (hermitian) [[linear basis]]. The [[universal vector bundle]]

\begin{displaymath}
E U(n) \underset{U(n)}{\times} \mathbb{C}^n
    \simeq
  \underset{\longrightarrow}{\lim}_k  U(k)/U(k-n) \underset{U(n)}{\times} \mathbb{C}^n
\end{displaymath}
has as fiber precisely the linear span of any such choice of basis.

While the quotient $U(k)/(U(n-k)\times U(n))$ (the [[Grassmannian]]) divides out the entire choice of basis, the quotient $U(k)/(U(n-k) \times U(n-1))$ leaves the choice of precisly one unit vector. This is parameterized by the sphere $S^{2n-1}$ which is thereby identified as the unit sphere in the respective fiber of $E U(n) \underset{U(n)}{\times} \mathbb{C}^n$.

\end{proof}
In particular:

\begin{lemma}
\label{UniversalComplexLineBundleThomSpace}\hypertarget{UniversalComplexLineBundleThomSpace}{}
The canonical map from the [[classifying space]] $B U(1) \simeq \mathbb{C}P^\infty$ (the inifnity [[complex projective space]]) to the [[Thom space]] of the [[universal vector bundle|universal]] [[complex line bundle]] is a [[weak homotopy equivalence]]

\begin{displaymath}
B U(1)
    \overset{\in W_{cl}}{\longrightarrow}
  M U(1)
   \coloneqq
  Th( E U(1) \underset{U(1)}{\times} \mathbb{C})
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
Observe that the [[circle group]] $U(1)$ is naturally identified with the unit sphere in $\mathbb{C}$: $U (1) \simeq S(\mathbb{S})$. Therefore the sphere bundle of the universal complex line bundle is equivalently the $U(1)$-[[universal principal bundle]]

\begin{displaymath}
\begin{aligned}
    E U(1) \underset{U(1)}{\times} S(\mathbb{C})
    & \simeq
    E U(1) \underset{U(1)}{\times} U(1)
    \\
    & \simeq E U(1)
  \end{aligned}
  \,.
\end{displaymath}
But the [[universal principal bundle]] is [[contractible topological space|contractible]]

\begin{displaymath}
E U(1) \overset{\in W_{cl}}{\longrightarrow} \ast
  \,.
\end{displaymath}
(Alternatively this is the special case of lemma \ref{SphereBundleBunminus1} for $n = 0$.)

Therefore the [[Thom space]]

\begin{displaymath}
\begin{aligned}
    Th( E U(1) \underset{U(1)}{\times} \mathbb{B} )
    & \coloneqq
    D( E U(1) \underset{U(1)}{\times} \mathbb{B} )
    /
    S( E U(1) \underset{U(1)}{\times} \mathbb{B} )
    \\
    & \overset{\in W_{cl}}{\longrightarrow}
    D( E U(1) \underset{U(1)}{\times} \mathbb{B} )
    \\
    & \overset{\in W_{cl}}{\longrightarrow}
    B U(1)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{lemma}
\label{UniversalComplexVectorBundleThomSpace}\hypertarget{UniversalComplexVectorBundleThomSpace}{}
For $E$ a [[generalized (Eilenberg-Steenrod) cohomology]] theory, then the $E$-[[reduced cohomology]] of the [[Thom space]] of the complex [[universal vector bundle]] is equivalently the [[relative cohomology]] of $B U(n)$ relative $B U(n-1)$

\begin{displaymath}
\tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) )
    \;\simeq\;
  E^\bullet( B U(n), B U(n-1))
  \,.
\end{displaymath}
If $E$ is equipped with the structure of a [[complex oriented cohomology theory]] then

\begin{displaymath}
\tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) )
   \simeq
  c^E_n \cdot (\pi_\bullet(E))[ [ c^E_1, \cdots, c^E_n ] ]
  \,,
\end{displaymath}
where the $c_i$ are the universal $E$-[[Conner-Floyd-Chern classes]].

\end{lemma}
\begin{proof}
Regarding the first statement:

In view of lemma \ref{SphereBundleBunminus1} and using that the disk bundle is homotopy equivalent to the base space we have

\begin{displaymath}
\begin{aligned}
    \tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) )
    &
    =
    E^\bullet( D(E U(n) \underset{U(n)}{\times} \mathbb{C}^n), S(E U(n)
\underset{U(n)}{\times} \mathbb{C}^n) )
    \\
    & \simeq
    E^\bullet( E U(n), B U(n-1))
  \end{aligned}
  \,.
\end{displaymath}
Regarding the second statement: the Conner-Floyd classes freely generate the $E$-cohomology of $B U(n)$ for all $n$:

\begin{displaymath}
E^\bullet(B U(n))
  \simeq
  \pi_\bullet(E)[ [ c^E_1, \cdots, c^E_n ] ]
  \,.
\end{displaymath}
and the restriction morphism

\begin{displaymath}
E^\bullet(B U(n))
   \longrightarrow
  E^{\bullet}(B U(n-1))
\end{displaymath}
projects out $c_n^E$. Since this is in particular a surjective map, the [[relative cohomology]] $E^\bullet( B U(n), B U(n-1) )$ is just the [[kernel]] of this map.

\end{proof}
\begin{prop}
\label{ThomClassesCFClass}\hypertarget{ThomClassesCFClass}{}
Let $E$ be a [[complex oriented cohomology theory]]. Then the $n$th $E$-[[Conner-Floyd-Chern class]]

\begin{displaymath}
c^E_n \in \tilde E(Th( E U(n) \underset{U(n)}{\times} \mathbb{C}^n ))
\end{displaymath}
(using the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}) is a [[Thom class]] in that its restriction to the Thom space of any fiber is a suspension of a unit in $\pi_0(E)$.

\end{prop}
(\hyperlink{Lurie10}{Lurie 10, lecture 5, prop. 6})

\begin{proof}
Since $B U(n)$ is [[connected topological space|connected]], it is sufficient to check the statement over the base point. Since that fixed fiber is canonically isomorphic to the direct sum of $n$ complex lines, we may equivalently check that the restriction of $c^E_n$ to the pullback of the universal rank $n$ bundle along

\begin{displaymath}
i \colon B U(1)^n \longrightarrow B U(n)
\end{displaymath}
satisfies the required condition. By the [[splitting principle]], that restriction is the product of the $n$-copies of the first $E$-Conner-Floyd-Chern class

\begin{displaymath}
i^\ast c_n \simeq ( (c_1^E)_1 \cdots (c_1^E)_n )
  \,.
\end{displaymath}
Hence it is now sufficient to see that each factor restricts to a unit on the fiber, but that it precisely the condition that $c_1^E$ is a complex orientaton of $E$. In fact by def. \ref{StrictComplexOrientation} the restriction is even to $1 \in \pi_0(E)$.

\end{proof}
\hypertarget{ComplexOrientationAsRingSpectrumMaps}{}\paragraph*{{Complex orientation as ring spectrum maps}}\label{ComplexOrientationAsRingSpectrumMaps}

For the present purpose:

\begin{defn}
\label{StrictComplexOrientation}\hypertarget{StrictComplexOrientation}{}
For $E$ a [[generalized (Eilenberg-Steenrod) cohomology]] theory, then a \emph{[[complex oriented cohomology theory|complex orientation]]} on $E$ is a choice of element

\begin{displaymath}
c_1^E \in E^2(B U(1))
\end{displaymath}
in the cohomology of the [[classifying space]] $B U(1)$ (given by the infinite [[complex projective space]]) such that its image under the restriction map

\begin{displaymath}
\phi
  \;\colon\;
  \tilde E^2( B U(1) )
    \longrightarrow
  \tilde E^2 (S^2)
    \simeq
  \pi_0(E)
\end{displaymath}
is the unit

\begin{displaymath}
\phi(c_1^E) = 1
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Lurie10}{Lurie 10, lecture 4, def. 2})

\begin{remark}
\label{}\hypertarget{}{}
Often one just requires that $\phi(c_1^E)$ is \emph{a} [[unit]], i.e. an invertible element. However we are after identifying $c_1^E$ with the degree-2 component $M U(1) \to E_2$ of homtopy ring spectrum morphisms $M U \to E$, and by unitality these necessarily send $S^2 \to M U(1)$ to the unit $\iota_2 \;\colon\; S^2 \to E$ (up to homotopy).

\end{remark}
\begin{lemma}
\label{S2SpectrumMapFromComplexOrientation}\hypertarget{S2SpectrumMapFromComplexOrientation}{}
Let $E$ be a [[homotopy commutative ring spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum}{def.}) equipped with a [[complex oriented cohomology theory|complex orientation]] (def. \ref{StrictComplexOrientation}) represented by a map

\begin{displaymath}
c_1^E
  \;\colon\;
  B U(1)
    \longrightarrow
  E_2
  \,.
\end{displaymath}
Write $\{c^E_k\}_{k \in \mathbb{N}}$ for the induced [[Conner-Floyd-Chern classes]]. Then there exists a morphism of $S^2$-[[sequential spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#SequentialTSpectra}{def.})

\begin{displaymath}
M U \longrightarrow E
\end{displaymath}
whose component map $M U_{2n} \longrightarrow E_{2n}$ represents $c_n^E$ (under the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}), for all $n \in \mathbb{N}$.

\end{lemma}
\begin{proof}
Consider the standard model of [[MU]] as a sequential $S^2$-spectrum with component spaces the [[Thom spaces]] of the complex [[universal vector bundle]]

\begin{displaymath}
M U_{2n}
    \coloneqq
  Th( E U(n) \underset{}{\times} \mathbb{C}^n)
  \,.
\end{displaymath}
Notice that this is a [[CW-spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum}{def.}, \href{Thom+space#ThomSpaceCWStructure}{lemma}).

In order to get a homomorphism of $S^2$-[[sequential spectra]], we need to find representatives $f _{2n} \;\colon\; M U_{2n} \longrightarrow E_{2n}$ of $c^E_n$ (under the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}) such that all the squares

\begin{displaymath}
\itexarray{
    S^2 \wedge M U_{2n}
      &\overset{id \wedge f_{2n}}{\longrightarrow}&
    S^2 \wedge E_{2n}
    \\
    \downarrow
      &&
    \downarrow
    \\
    M U_{2(n+1)}
      &\underset{f_{2(n+1)}}{\longrightarrow}&
    E_{2n+1}
  }
\end{displaymath}
commute strictly (not just up to homotopy).

To begin with, pick a map

\begin{displaymath}
f_0 \;\colon\; M U_0 \simeq S^0 \longrightarrow E_0
\end{displaymath}
that represents $c_0 = 1$.

Assume then by [[induction]] that maps $f_{2k}$ have been found for $k \leq n$. Observe that we have a homotopy-commuting diagram of the form

\begin{displaymath}
\itexarray{
    S^2 \wedge M U_{2n}
      &\overset{id \wedge f_{2n}}{\longrightarrow}&
    S^2 \wedge E_{2n}
    \\
    \downarrow
      &\swArrow&
    \downarrow
    \\
    M U_{2} \wedge M U_{2 n}
      &\overset{c_1 \wedge c_{n}}{\longrightarrow}&
    E_2 \wedge E_{2n}
    \\
    \downarrow
      &\swArrow&
    \downarrow^{\mathrlap{\mu_{2,2n}}}
    \\
    M U_{2(n+1)}
      &\underset{c_{n+1}}{\longrightarrow}&
    E_{2(n+1)}
  }
  \,,
\end{displaymath}
where the maps denoted $c_k$ are any representatives of the Chern classes of the same name, under the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}. Here the homotopy in the top square exhibits the fact that $c_1^E$ is a complex orientation, while the homotopy in the bottom square exhibits the Whitney sum formula for Chern classes (prop. \ref{WhitneySumChernClasses})).

Now since $M U$ is a [[CW-spectrum]], the total left vertical morphism here is a (Serre-)cofibration, hence a [[Hurewicz cofibration]], hence satisfies the [[homotopy extension property]]. This means precisely that we may find a map $f_{2n+1} \colon M U_{2(n+1)} \longrightarrow E_{2(n+1)}$ homotopic to the given representative $c_{n+1}$ such that the required square commutes strictly.

\end{proof}
\begin{lemma}
\label{HRingSpectrumS2SpectrumMapFromComplexOrientation}\hypertarget{HRingSpectrumS2SpectrumMapFromComplexOrientation}{}
For $E$ a [[complex oriented cohomology theory|complex oriented]] [[homotopy commutative ring spectrum]], the morphism of spectra

\begin{displaymath}
c \;\colon\; M U \longrightarrow E
\end{displaymath}
that represents the complex orientation by lemma \ref{S2SpectrumMapFromComplexOrientation} is a [[homomorphism]] of [[homotopy commutative ring spectra]].

\end{lemma}
(\hyperlink{Lurie10}{Lurie 10, lecture 6, prop. 6})

\begin{proof}
The unitality condition demands that the diagram

\begin{displaymath}
\itexarray{
    \mathbb{S}
      &\overset{}{\longrightarrow}&
    M U
    \\
    & \searrow & \downarrow^{\mathrlap{c}}
    \\
    && E
  }
\end{displaymath}
commutes in the [[stable homotopy category]] $Ho(Spectra)$. In components this means that

\begin{displaymath}
\itexarray{
    S^{2n}
      &\overset{}{\longrightarrow}&
    M U_{2n}
    \\
    & \searrow & \downarrow^{\mathrlap{c_n}}
    \\
    && E_{2n}
  }
\end{displaymath}
commutes up to homotopy, hence that the restriction of $c_n$ to a fiber is the $2n$-fold suspension of the unit of $E_{2n}$. But this is the statement of prop. \ref{ThomClassesCFClass}: the Chern classes are universal Thom classes.

Hence componentwise all these triangles commute up to some homotopy. Now we invoke the [[Milnor sequence]] for generalized cohomology of spectra (prop. \ref{CohomologyOfSpectraMilnorSequence}). Observe that the [[tower]] of abelian groups $n \mapsto E^{n_1}(S^n)$ is actually constant ([[suspension isomorphism]]) hence trivially satisfies the [[Mittag-Leffler condition]] and so a homotopy of morphisms of spectra $\mathbb{S} \to E$ exists as soon as there are componentwise homotopies (cor. \ref{WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre}).

Next, the respect for the product demands that the square

\begin{displaymath}
\itexarray{
    M U \wedge M U
      &\overset{c \wedge c}{\longrightarrow}&
    E \wedge E
    \\
    \downarrow
      &&
    \downarrow
    \\
    M U
      &\underset{c}{\longrightarrow}&
    E
  }
\end{displaymath}
commutes in the [[stable homotopy category]] $Ho(Spectra)$. In order to rephrase this as a condition on the components of the ring spectra, regard this as happening in the [[homotopy category of a model category|homotopy category]] $Ho(OrthSpec(Top_{cg}))_{stable}$ of the [[model structure on orthogonal spectra]], which is [[equivalence of categories|equivalent]] to the [[stable homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#SequentialSpectraQuillenEquivalence}{thm.}).

Here the derived [[symmetric monoidal smash product of spectra]] is given by [[Day convolution]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#SsymModuleSymmetricSpectra}{def.}) and maps out of such a product are equivalently as in the above diagram is equivalent (\href{Introduction+to+Stable+homotopy+theory+--+1-2#DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}{cor.}) to a suitably equivariant collection diagrams of the form

\begin{displaymath}
\itexarray{
    M U_{2 n_1} \wedge M U_{2 n_2}
      &\overset{c_{n_1} \wedge c_{n_2}}{\longrightarrow}&
    E_{2 n_1} \wedge E_{2 n_2}
    \\
    \downarrow
      &&
    \downarrow
    \\
    M U_{2(n_1 + n_2)}
      &\underset{c_{(n_1 + n_2)}}{\longrightarrow}&
   E_{2 (n_1 + n_2)}
  }
  \,,
\end{displaymath}
where on the left we have the standard pairing operations for $M U$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#OrthogonalComplexThomSpectrum}{def.}) and on the right we have the given pairing on $E$.

That this indeed commutes up to homotopy is the Whitney sum formula for Chern classes (\href{Chern+class#WhitneySumChernClasses}{prop.}).

Hence again we have componentwise homotopies. And again the relevant [[Mittag-Leffler condition]] on $n \mapsto E^{n-1}((MU \wedge MU)_n)$-holds, by the nature of the universal [[Conner-Floyd classes]], prop. \ref{ConnerFloyedClasses}. Therefore these componentwise homotopies imply the required homotopy of morphisms of spectra (cor. \ref{WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre}).

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
Let $E$ be a [[homotopy commutative ring spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum}{def.}). Then the map

\begin{displaymath}
(M U \overset{c}{\longrightarrow} E)
  \;\mapsto\;
  (B U(1) \simeq M U_{2} \overset{c_1}{\longrightarrow} E_2)
\end{displaymath}
which sends a homomorphism $c$ of [[homotopy commutative ring spectra]] to its component map in degree 2, interpreted as a class on $B U(1)$ via lemma \ref{UniversalComplexLineBundleThomSpace}, constitutes a [[bijection]] from homotopy classes of homomorphisms of homotopy commutative ring spectra to complex orientations (def. \ref{StrictComplexOrientation}) on $E$.

\end{theorem}
(\hyperlink{Lurie10}{Lurie 10, lecture 6, theorem 8})

\begin{proof}
By lemma \ref{S2SpectrumMapFromComplexOrientation} and lemma \ref{HRingSpectrumS2SpectrumMapFromComplexOrientation} the map is surjective, hence it only remains to show that it is injective.

So let $c, c' \colon M U \to E$ be two morphisms of homotopy commutative ring spectra that have the same restriction, up to homotopy, to $c_1 \simeq c_1'\colon M U_2 \simeq B U(1)$. Since both are homotopy ring spectrum homomophisms, the restriction of their components $c_n, c'_n \colon M U_{2n} \to E_{2 n}$ to $B U(1)^{\wedge^n}$ is a product of $c_1 \simeq c'_1$, hence $c_n$ becomes homotopic to $c_n'$ after this restriction. But by the [[splitting principle]] this restriction is injective on cohomology classes, hence $c_n$ itself ist already homotopic to $c'_n$.

It remains to see that these homotopies may be chosen compatibly such as to form a single homotopy of maps of spectra

\begin{displaymath}
f \;\colon\; M U \wedge I_+ \longrightarrow E
  \,,
\end{displaymath}
This follows due to the existence of the [[Milnor exact sequence|Milnor]] [[short exact sequence]] from prop. \ref{CohomologyOfSpectraMilnorSequence}:

\begin{displaymath}
0
    \to
  \underset{\longleftarrow}{\lim}^1_n E^{-1}( \Sigma^{-2n} M U_{2n} )
    \longrightarrow
  E^0(M U)
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n E^0( \Sigma^{-2n} M U_{2n} )
    \to
  0
  \,.
\end{displaymath}
Here the [[Mittag-Leffler condition]] (def. \ref{MittagLefflerCondition}) is clearly satisfied (by prop. \ref{ConnerFloyedClasses} and lemma \ref{UniversalComplexVectorBundleThomSpace} all relevant maps are epimorphisms, hence the condition is satisfied by example \ref{MittagLefflerSatisfiedInParticularForTowerOfSurjections}). Hence the [[lim{\tt \symbol{94}}1]]-term vanishes (prop. \ref{Lim1VanihesUnderMittagLeffler}), and so by exactness the canonical morphism

\begin{displaymath}
E^0(M U)
    \overset{\simeq}{\longrightarrow}
  \underset{\longleftarrow}{\lim}_n E^0( \Sigma^{-2n} M U_{2n} )
\end{displaymath}
is an [[isomorphism]]. This says that two homotopy classes of morphisms $M U \to E$ are equal precisely already if all their component morphisms are homotopic (represent the same cohomology class).

\end{proof}
\hypertarget{HomologyOfMU}{}\subsubsection*{{Homology of $M U$}}\label{HomologyOfMU}

\textbf{Idea.} Since, by the above, every [[complex oriented cohomology theory]] $E$ is indeed [[orientation in generalized cohomology|oriented]] over [[complex vector bundles]], there is a [[Thom isomorphism]] which reduces the computation of the $E$-[[homology of MU]], $E_\bullet(M U)$ to that of the [[classifying space]] $B U$. The homology of $B U$, in turn, may be determined by the duality with its cohomology ([[universal coefficient theorem]]) via [[Kronecker pairing]] and the induced duality of the corresponding [[Atiyah-Hirzebruch spectral sequences]] (prop. \ref{AHSSPairing}) from the Conner-Floyd classes \hyperlink{ConnerFloydChernClasses}{above}. Finally, via the [[Hurewicz homomorphism]]/[[Boardman homomorphism]] the homology of $M U$ gives a first approximation to the [[homotopy groups]] of [[MU]].

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, section 2.4, 4.3}, \hyperlink{Lurie10}{Lurie 10, lecture 7})

\hypertarget{QuillenTheoremOnMU}{}\subsubsection*{{Milnor-Quillen theorem on $M U$}}\label{QuillenTheoremOnMU}

\textbf{Idea.} From the computation of the [[homology of MU]] \hyperlink{HomologyOfMU}{above} and applying the [[Boardman homomorphism]], one deduces that the [[stable homotopy groups]] $\pi_\bullet(MU)$ of [[MU]] are finitely generated. This implies that it is suffient to compute them over the [[p-adic integers]] for all primes $p$. Using the [[change of rings theorem]], this finally is obtained from inspection of the filtration in the $H\mathbb{F}_p$-[[Adams spectral sequence]] for $MU$. This is Milnor's theorem wich together with [[Lazard's theorem]] shows that there is an isomorphism of rings $L \simeq \pi_\bullet(M U)$ with the [[Lazard ring]]. Finally [[Quillen's theorem on MU]] says that this isomorphism is exhibited by the universal ring homomorphism $L \longrightarrow \pi_\bullet(M U)$ which classifies the universal complex orientation on $M U$.

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, section 4.4}, \hyperlink{Lurie10}{Lurie 10, lecture 10})

\hypertarget{landweber_exact_functor_theorem}{}\subsubsection*{{Landweber exact functor theorem}}\label{landweber_exact_functor_theorem}

\textbf{Idea.} By the above, every [[complex oriented cohomology theory]] induces a [[formal group law]] from its first [[Conner-Floyd Chern class]]. Moreover, [[Quillen's theorem on MU]] together with [[Lazard's theorem]] say that the [[cohomology ring]] $\pi_\bullet(M U)$ of [[complex cobordism cohomology]] [[MU]] is the classifying ring for formal group laws.

The \emph{[[Landweber exact functor theorem]]} says that, conversely, forming the [[tensor product]] of [[complex cobordism cohomology theory]] ([[MU]]) with a [[Landweber exactness|Landweber exact]] [[ring]] via some [[formal group law]] yields a [[cohomology theory]], hence a [[complex oriented cohomology theory]].

\textbf{Literature.} (\hyperlink{Lurie10}{Lurie 10, lectures 15,16})

\hypertarget{outlook_geometry_of_}{}\subsection*{{Outlook: Geometry of $Spec(MU)$}}\label{outlook_geometry_of_}

The grand conclusion of [[Quillen's theorem on MU]] (\hyperlink{QuillenTheoremOnMU}{above}): [[complex oriented cohomology theory]] is essentially the [[spectral geometry]] over $Spec(M U)$, and the latter is a kind of derived version of the [[moduli stack of formal groups]] (1-dimensional commutative).

\begin{itemize}%
\item [[Landweber-Novikov theorem]]


\item [[Adams-Quillen theorem]]


\item [[Adams-Novikov spectral sequence]]



\end{itemize}
(\ldots{})

\textbf{Literature.} (\hyperlink{Kochman96}{Kochman 96, sections 4.5-4.7 and section 5}, \hyperlink{Lurie10}{Lurie 10, lectures 12-14})

$\,$

$\,$


\vspace{.5em} \hrule \vspace{.5em}
$\,$

\hypertarget{References}{}\subsection*{{References}}\label{References}

We follow in outline the textbook

\begin{itemize}%
\item [[Stanley Kochman]], chapters I - IV of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996

\end{itemize}
For some basics in [[algebraic topology]] see also

\begin{itemize}%
\item [[Robert Switzer]], \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

\end{itemize}
Specifically for \textbf{S.1) Generalized cohomology} a neat account is in:

\begin{itemize}%
\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})

\end{itemize}
For \textbf{S.2) Cobordism theory} an efficient collection of the highlights is in

\begin{itemize}%
\item [[Cary Malkiewich]], \emph{Unoriented cobordism and $M O$}, 2011 (\href{http://math.uiuc.edu/~cmalkiew/cobordism.pdf}{pdf})

\end{itemize}
except that it omits proof of the [[Leray-Hirsch theorem]]/[[Serre spectral sequence]] and that of the [[Thom isomorphism]], but see the references there and see (\hyperlink{Kochman96}{Kochman 96}, \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, section 11.7}) for details.

For \textbf{S.3) Complex oriented cohomology} besides (\hyperlink{Kochman96}{Kochman 96, chapter 4}) have a look at

\begin{itemize}%
\item [[Frank Adams]], \emph{[[Stable homotopy and generalized homology]]}, Chicago Lectures in mathematics, 1974

\end{itemize}
and

\begin{itemize}%
\item [[Jacob Lurie]], lectures 1-10 of \emph{[[Chromatic Homotopy Theory]]}, 2010

\end{itemize}
See also

\begin{itemize}%
\item [[Stefan Schwede]], \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})

\end{itemize}
[[!redirects Introduction to Stable homotopy theory -- S]]



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