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\begin{document}

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\section*{complex oriented cohomology theory}

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

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

[[!include cohomology - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{DefInTermsOfGeneralizedFirstChernClass}{In terms of generalized first Chern classes}\dotfill \pageref*{DefInTermsOfGeneralizedFirstChernClass} \linebreak
\noindent\hyperlink{InTermsOfGenera}{In terms of genera and $E_\infty$ orientations}\dotfill \pageref*{InTermsOfGenera} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{CohomologyRingOfBU1}{Cohomology ring of $B U(1)$}\dotfill \pageref*{CohomologyRingOfBU1} \linebreak
\noindent\hyperlink{FormalGroupLaw}{Formal group law}\dotfill \pageref*{FormalGroupLaw} \linebreak
\noindent\hyperlink{TheCohomologyRingOfBUn}{Cohomology ring of $B U(n)$}\dotfill \pageref*{TheCohomologyRingOfBUn} \linebreak
\noindent\hyperlink{canonical_orientation_on_complex_vector_bundles}{Canonical orientation on complex vector bundles}\dotfill \pageref*{canonical_orientation_on_complex_vector_bundles} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{complex oriented cohomology theory} is a [[generalized (Eilenberg-Steenrod) cohomology theory]] which is [[orientation in generalized cohomology|oriented]] on all [[complex vector bundles]]. Examples include [[ordinary cohomology]], complex [[topological K-theory]], [[elliptic cohomology]] and [[cobordism cohomology]].

The collection of all complex oriented cohomology theories turns out to be parameterized over the [[moduli stack of formal group laws]]. The [[stratification]] of this stack by the [[height of a formal group|height of formal group]] leads to the stratification of complex oriented cohomology theory by ``[[chromatic level]]'', a perspective also known as \emph{[[chromatic homotopy theory]]}.

For more detailed introduction see at \emph{[[Introduction to Cobordism and Complex Oriented Cohomology]]}.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{DefInTermsOfGeneralizedFirstChernClass}{}\subsubsection*{{In terms of generalized first Chern classes}}\label{DefInTermsOfGeneralizedFirstChernClass}

Write $\mathbb{C}P^\infty \simeq B U(1) \simeq K(\mathbb{Z},2)$ for the inifinite [[complex projective space]], equivalently the [[classifying space]] for [[circle group]]-[[principal bundles]] (an [[Eilenberg-MacLane space]]); write $S^2$ for the [[2-sphere]] and write

\begin{displaymath}
i \;\colon\; S^2 \longrightarrow B U(1)
\end{displaymath}
for a representative of $1 \in \mathbb{Z} \simeq  \pi_2(B U(1))$, classifying the [[universal complex line bundle]]. Regard both $S^2$ and $B U(1)$ as [[pointed homotopy types]] and take $i$ to be a pointed morphism.

Let $E^\bullet$ be a [[multiplicative cohomology theory]], i.e. a [[functor]] $X \mapsto \pi_\bullet[X,E]$ for $E$ a [[ring spectrum]]. Write $\tilde E^\bullet$ for the corresponding [[reduced cohomology]] on [[pointed topological spaces]], such that for any pointed space $X$ there is a canonical [[direct sum]] decomposition (\href{generalized+%28Eilenberg-Steenrod%29+cohomology#UnreducedCohomologyIsReducedPlusPointValue}{this prop.})

\begin{displaymath}
E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast)
  \,.
\end{displaymath}
By the [[suspension isomorphism]] there is an identification

\begin{displaymath}
\tilde E^2(S^2) \simeq \tilde E^0(S^0) \simeq E^0(\ast) \simeq \pi_0(E)
\end{displaymath}
with the [[commutative ring]] underlying $E$. Write $1 \in \pi_0(E)$ for the multiplicative identity element in this ring.

\begin{defn}
\label{ComplexOrientedCohomologyTheory}\hypertarget{ComplexOrientedCohomologyTheory}{}
A [[multiplicative cohomology theory]] $E$ is \emph{complex orientable} if 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}
\hypertarget{InTermsOfGenera}{}\subsubsection*{{In terms of genera and $E_\infty$ orientations}}\label{InTermsOfGenera}

Complex orientation in the \hyperlink{Definition}{above} sense is indeed universal [[MU]]-[[orientation in generalized cohomology]]:

\begin{prop}
\label{ComplexOrientationIsMUAlgebraStructure}\hypertarget{ComplexOrientationIsMUAlgebraStructure}{}
For $E$ a [[homotopy commutative ring spectrum]] then there is a [[bijection]] between complex orientations of $E$-cohomology and [[homotopy commutative ring spectrum]]-[[homomorphisms]] $MU \longrightarrow E$ out of [[MU]].

\end{prop}
(\hyperlink{Hopkins99}{Hopkins 99, section 4}, \hyperlink{LurieLecture}{Lurie, lecture 6, theorem 8})

See at \emph{[[universal complex orientation on MU]]}.

\begin{remark}
\label{}\hypertarget{}{}
One might hope that the above universal property can be refined to say that that if $E$ is an $E_\infty$-ring spectrum, then complex orientations of $E$ are in bijection with $E_\infty$ ring maps $MU \to E$. This is not the case; Ando classified $H_\infty$ ring maps $MU \to E$ in \hyperlink{Ando92}{his thesis} and in particular showed that not every complex orientation gives rise to an $H_\infty$ ring map out of $MU$. More recently, \hyperlink{HopkinsLawson16}{Hopkins and Lawson} have classified the further structure constituting an $E_\infty$ ring map out of $MU$.

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Examples of complex orientable cohomology theories:

\begin{example}
\label{}\hypertarget{}{}
\textbf{([[ordinary cohomology]])}

For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]], the ordinary [[first Chern class]]

\begin{displaymath}
c_1 \in H^2(B U(1), \mathbb{Z})
\end{displaymath}
defines a complex orientation of $H\mathbb{Z}$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
\textbf{([[topological K-theory]])}

For $E = KU$ complex [[topological K-theory]], then the class of the image of the the [[universal complex line bundle]] $\mathcal{O}(1)$ in [[reduced K-theory]] is a complex orientation.

The induced [[formal group law]] (by prop. \ref{ComplexOrientedCohomologyTheoryFormalGroupLaw}) is the [[multiplicative formal group law]].

For details see at \emph{[[topological K-theory]]} the section \emph{\href{topological+K-theory#ComplexOrientationAndFormalGroupLaw}{Complex orientation and Formal group law}}.

\end{example}
\begin{example}
\label{ComplexCobordism}\hypertarget{ComplexCobordism}{}
\textbf{(complex cobordism)}

For $E = MU$ [[complex cobordism cohomology theory]], the canonical map

\begin{displaymath}
B U(1) \stackrel{\simeq}{\to} MU(1) \to MU
\end{displaymath}
defines a complex orientation.

\end{example}
\begin{example}
\label{BrownPeterson}\hypertarget{BrownPeterson}{}
[[Brown-Peterson cohomology]] $E = B P^\bullet$.

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{CohomologyRingOfBU1}{}\subsubsection*{{Cohomology ring of $B U(1)$}}\label{CohomologyRingOfBU1}

\begin{prop}
\label{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}\hypertarget{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}{}
Given a complex oriented cohomology theory $(E^\bullet, c^E_1)$ according to def. \ref{ComplexOrientedCohomologyTheory}, then there are [[isomorphisms]] of [[graded rings]]

\begin{enumerate}%
\item $E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]$

(between the $E$-[[cohomology ring]] of $B U(1)$ and the [[formal power series]] (but see remark \ref{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}) in one generator of even degree over the $E$-[[cohomology ring]] of the point);


\item $E^\bullet(B U(1) \times B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E \otimes 1 , 1 \otimes c_1^E ] ]$.



\end{enumerate}
\end{prop}
\begin{proof}
We may realize the [[classifying space]] $B U(1)$ as the infinite [[complex projective space]] $\mathbb{C}P^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n$ (\href{classifying+space#RealComplexProjectiveSpaceAsGrassmannian}{exmpl.}). There is a standard [[CW-complex]]-structure on the [[classifying space]] $\mathbb{C}P^\infty$, given by inductively identifying $\mathbb{C}P^{n+1}$ with the result of attaching a single $2n$-cell to $\mathbb{C}P^n$ (\href{complex+projective+space#CellComplexStructureOnComplexProjectiveSpace}{prop.}). 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]] 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 (\href{complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace}{prop.}) the [[ordinary cohomology]] with [[integer]] [[coefficients]] of 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]] (\href{complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace}{prop.}), 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 (\href{multiplicative+cohomology+theory#LinearityOfDifferentialsInSerreAHSSForMultiplicativeCohomologyTheory}{prop.}), 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 \href{spectral+sequence#ExtensionProblem}{extension problem} 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}
\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]] $E$ 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 (\href{complex+projective+space+OrdinaryCohomologyOfComplexProjectiveSpace}{prop.})

\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{FormalGroupLaw}{}\subsubsection*{{Formal group law}}\label{FormalGroupLaw}

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 Hausdorff space]] (notably a [[CW-complex]], since all [[CW-complexes are paracompact Hausdorff spaces]]), 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]] (an abelian [[H-group]]).

\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]] (\href{complex+projective+space#CellComplexStructureOnComplexProjectiveSpace}{prop.}) it is cofibrant in the [[standard model structure on topological spaces]], 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 an [[H-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]] (\href{formal+group#Commutative1DimFormalGroupLaw}{exmpl.}) 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}
\begin{example}
\label{}\hypertarget{}{}
The [[formal group law]] of [[complex cobordism cohomology theory]], example \ref{ComplexCobordism} is [[universal property|universal]] in that for every [[commutative ring]] $R$ there is a [[natural bijection]]

\begin{displaymath}
CRing(MU^\bullet, R)
  \simeq
  FormalGroupLaws_{/R}
  \,.
\end{displaymath}
$MU^\bullet$ is the \emph{[[Lazard ring]]}.

This is \emph{[[Milnor-Quillen's theorem on MU]]} (involving [[Lazard's theorem]]).

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The [[formal group law]] of [[Brown-Peterson cohomology theory]], example \ref{BrownPeterson} is [[universal property|universal]] for $p$-local cohomology theories in that $\mathbb{G}_{B P}$ is universal among $p$-local, [[p-typical formal group laws]].

\end{example}
\hypertarget{TheCohomologyRingOfBUn}{}\subsubsection*{{Cohomology ring of $B U(n)$}}\label{TheCohomologyRingOfBUn}

\begin{prop}
\label{CohomologyRingOfBUn}\hypertarget{CohomologyRingOfBUn}{}
For $E$ a complex oriented cohomology theory and $n \in \mathbb{N}$, restriction along the canonical map

\begin{displaymath}
(B U(1))^n  \longrightarrow B U(n)
\end{displaymath}
induces an [[isomorphism]]

\begin{displaymath}
E^\bullet(B U(n))
    \stackrel{\simeq}{\longrightarrow}
  (\pi_\bullet E)[ [ c^E_1, \cdots, c^E_n ] ]
  \simeq
  E^\bullet((B U(1))^n)^{\Sigma_n}
  \hookrightarrow
  E^\bullet((B U(1))^n)
  \simeq
  (\pi_\bullet E)[ [(c_1^E)_1, \cdots (c_1^E)_n ] ]  
  \,,
\end{displaymath}
of $E^\bullet(B U(n))$ with the [[cyclic group]]-[[invariants]] in $E^\bullet((B U(1))^n)$, hence with the [[power series]] ring in the [[elementary symmetric polynomials]] $c_i^E$ (the [[generalized Chern classes]]) in the $c_1^E$-s (the generalized first Chern classes of prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}).

\end{prop}
Use \href{ordinary+homology+spectra+split#WhenGeneralizedHomologySpectraSplit}{this proposition} to reduce to the situation for ordinary [[Chern classes]]. (e.g. \hyperlink{LurieLecture}{Lurie 10, lecture 4})

\hypertarget{canonical_orientation_on_complex_vector_bundles}{}\subsubsection*{{Canonical orientation on complex vector bundles}}\label{canonical_orientation_on_complex_vector_bundles}

The follows says that complex oriented cohomology theories in the sense of def. \ref{ComplexOrientedCohomologyTheory}, indeed canonically have an [[orientation in generalized cohomology]] for the ([[spherical fibration]] of) any [[complex vector bundle]].

For more details see at \emph{[[universal complex orientation on MU]]}.

\begin{prop}
\label{ThomSpaceOfZetan}\hypertarget{ThomSpaceOfZetan}{}
For $E$ any [[cohomology theory]] and $n \in \mathbb{N}$, $n \geq 1$, there is a canonical [[isomorphism]] of [[relative cohomology]]

\begin{displaymath}
E^\bullet(B U(n), B U(n-1))
  \simeq
  E^\bullet( B( \zeta_n), S( \zeta_n) )
  \,,
\end{displaymath}
where $\zeta_n \coloneqq E U(n) \underset{U(n)}{\times} \mathbb{R}^{2n}$ is the [[universal complex vector bundle]].

\end{prop}
\begin{proof}
Observe that the [[sphere bundle]] $S(\zeta_n) \to B U(n)$ of the [[universal complex vector bundle]] is equivalently the canonical map $B U(n-1) \to B U(n)$.

This follows form the fact that $S^{2n-1} \simeq U(n)/U(n-1)$ and that hence the [[unit sphere]] bundle is equivalently the quotient of the $U(n)$-[[universal principal bundle]] by $U(n-1)$

\begin{displaymath}
\itexarray{
    U(n) &\longrightarrow& \ast
    \\
    && \downarrow
    \\
    && B U(n)
  }
  \;\;\;\; \stackrel{}{\mapsto}\;\;\;\;
  \itexarray{
     S^{2n-1} \simeq & U(n)/U(n-1) &\longrightarrow& B U(n-1) 
     \\
     & && \downarrow 
     \\
     & &&  B U(n)
  }
  \,.
\end{displaymath}
The unit ball bundle $B(\zeta_n)$ is weakly equivalent to $B U(n)$, and under this identification the map $S(\zeta_n) \to B(\zeta_n)$ is equivalent to $B U(n-1) \to B U(n)$.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For $E$ a complex oriented cohomology theory, its $n$th [[generalized Chern class]] $c^E_n$, prop. \ref{CohomologyRingOfBUn}, identified as an element of $E^\bullet(B(\zeta_n), S(\zeta_n))$ via prop. \ref{ThomSpaceOfZetan}, is a [[Thom class]].

\end{prop}
(e.g. \hyperlink{LurieLecture}{Lurie 10, lecture 5, prop. 6})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[real oriented cohomology theory]]


\item [[generalized (Eilenberg-Steenrod) cohomology theory]]


\item [[multiplicative cohomology theory]]


\item [[orientation in generalized cohomology]]


\item [[complex cobordism cohomology theory]]


\item [[Landweber exact functor theorem]]


\item [[chromatic homotopy theory]]



\end{itemize}
[[!include chromatic tower examples - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Textbook accounts include

\begin{itemize}%
\item [[Frank Adams]], part II, section 2 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Stanley Kochmann]], section 4.3 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
Introduction includes

\begin{itemize}%
\item \emph{[[Introduction to Cobordism and Complex Oriented Cohomology]]}


\item [[Riccardo Pedrotti]], \emph{Complex oriented cohomology, generalized orientation and Thom isomorphism}, 2016, 2018 ([[PedrotticECohomology2018.pdf:file]])



\end{itemize}
The perspective of [[chromatic homotopy theory]] originates in

\begin{itemize}%
\item [[Mike Hopkins]], \emph{[[Complex oriented cohomology theories and the language of stacks]]}, 1999 course notes (\href{http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf}{pdf})

\end{itemize}
and is further developed in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, Lecture notes, 2010 (\href{http://www.math.harvard.edu/~lurie/252x.html}{web})

Lecture 4 \emph{Complex-oriented cohomology theories} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf}{pdf})

Lecture 6 \emph{MU and complex orientations} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf}{pdf})



\end{itemize}
See also the references at \emph{\href{equivariant+cohomology#InComplexOrientedGeneralizedCohomologyTheory}{equivariant cohomology -- References -- Complex oriented cohomology}}-.

A comparison between complex orientations and $H_\infty$ ring maps out of $MU$ was given in

\begin{itemize}%
\item [[Matthew Ando]] \emph{Operations in complex-oriented cohomology theories related to subgroups of formal groups} (\href{https://dspace.mit.edu/bitstream/handle/1721.1/13230/26903638-MIT.pdf?sequence=2}{PhD Thesis})

\end{itemize}
More recent developments include

\begin{itemize}%
\item [[Michael Hopkins]], [[Tyler Lawson]], \emph{Strictly commutative complex orientation theory} (\href{http://arxiv.org/abs/1603.00047}{arXiv:1603.00047})

\end{itemize}
[[!redirects complex oriented cohomology theories]]

[[!redirects complex oriented cohomology]]

[[!redirects complex-oriented cohomology theory]] [[!redirects complex-oriented cohomology theories]]

[[!redirects complex orientable cohomology theory]] [[!redirects complex orientable cohomology theories]]



\end{document}