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\begin{document}

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\section*{generalized homology}

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

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

[[!include cohomology - 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{reduced_homology}{Reduced homology}\dotfill \pageref*{reduced_homology} \linebreak
\noindent\hyperlink{unreduced_homology}{Unreduced homology}\dotfill \pageref*{unreduced_homology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{expression_by_ordinary_homology_via_atiyahhirzebruch_spectral_sequence}{Expression by ordinary homology via Atiyah-Hirzebruch spectral sequence}\dotfill \pageref*{expression_by_ordinary_homology_via_atiyahhirzebruch_spectral_sequence} \linebreak
\noindent\hyperlink{WhiteheadTheorem}{Whitehead theorem}\dotfill \pageref*{WhiteheadTheorem} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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{generalized homology theory} is a certain [[functor]] from suitable [[topological spaces]] to [[graded abelian groups]] which satisfies most, but not all, of the abstract properties of [[ordinary homology]] functors (e.g. [[singular homology]]).

By the [[Brown representability theorem]], under certain conditions every [[spectrum]] $K$ is the coefficient object of a [[generalized (Eilenberg-Steenrod) cohomology|generalized]] [[cohomology theory]] and [[Spanier-Whitehead duality|S-dually]] of a generalized homology theory. For $K = H R$ an [[Eilenberg-MacLane spectrum]] this reduces to [[homology|ordinary homology]].

See at \emph{[[generalized (Eilenberg-Steenrod) cohomology]]} for more.

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

\hypertarget{reduced_homology}{}\subsubsection*{{Reduced homology}}\label{reduced_homology}

Throughout, write [[Top]]${}_{CW}$ for the category of [[topological spaces]] [[homeomorphism|homeomorphic]] to [[CW-complexes]]. Write $Top^{\ast/}_{CW}$ for the corresponding category of [[pointed topological spaces]].

Recall that [[colimits]] in $Top^{\ast/}$ are computed as colimits in $Top$ after adjoining the base point and its inclusion maps to the given diagram

\begin{example}
\label{WedgeSumAsCoproduct}\hypertarget{WedgeSumAsCoproduct}{}
The [[coproduct]] in [[pointed topological spaces]] is the \emph{[[wedge sum]]}, denoted $\vee_{i \in I} X_i$.

\end{example}
Write

\begin{displaymath}
\Sigma \coloneqq S^1 \wedge (-)
  \;\colon\;
  Top^{\ast/}_{CW} \longrightarrow Top^{\ast/}_{CW}
\end{displaymath}
for the [[reduced suspension]] functor.

Write $Ab^{\mathbb{Z}}$ for the category of [[integer]]-[[graded abelian groups]].

\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]], example \ref{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}
\hypertarget{unreduced_homology}{}\subsubsection*{{Unreduced homology}}\label{unreduced_homology}

In the following a \emph{pair} $(X,A)$ refers to a [[subspace]] inclusion of [[topological spaces]] ([[CW-complexes]]) $A \hookrightarrow X$. 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{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}
\begin{defn}
\label{AlternativeFormulationOfExcisionAxiom}\hypertarget{AlternativeFormulationOfExcisionAxiom}{}
The excision axiom in def. \ref{GeneralizedHomologyTheory} is equivalent to the following statement:

For all $A,B \hookrightarrow X$ with $X = A \cup B$, 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(A, A \cap B)
    \overset{\simeq}{\longrightarrow}
  E_\bullet(X, B)
  \,.
\end{displaymath}
\end{defn}
(e.g \hyperlink{Switzer75}{Switzer 75, 7.2, 7.5})

\begin{proof}
First consider the statement under the condition that $X = Int(A) \cup Int(B)$.

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}
This shows the statement for the special case that $X = Int(A)\cup Int(U)$. The general statement reduces to this by finding a suitable homotopy equivalence to a slightly larger covering pair (e.g \hyperlink{Switzer75}{Switzer 75, 7.5}).

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

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

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

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

\begin{displaymath}
\bar \delta 
    \;\colon \;
  E_{q}(X,Y)
   \stackrel{\delta}{\longrightarrow}
  E_{q-1}(Y)
   \longrightarrow
  E_{q-1}(Y,Z) 
  \,.
\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})

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

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

\hypertarget{expression_by_ordinary_homology_via_atiyahhirzebruch_spectral_sequence}{}\subsubsection*{{Expression by ordinary homology via Atiyah-Hirzebruch spectral sequence}}\label{expression_by_ordinary_homology_via_atiyahhirzebruch_spectral_sequence}

The [[Atiyah-Hirzebruch spectral sequence]] serves to express generalized homology $E_\bullet$ in terms of ordinary homology with coefficients in $E_\bullet(\ast)$.

\hypertarget{WhiteheadTheorem}{}\subsubsection*{{Whitehead theorem}}\label{WhiteheadTheorem}

\begin{prop}
\label{}\hypertarget{}{}
Let $\phi \colon E \longrightarrow F$ be a morphism of reduced generalized (co-)homology functors, def. \ref{ReducedGeneralizedHomology} (a [[natural transformation]]) such that its component

\begin{displaymath}
\phi(S^0) \colon E(S^0) \longrightarrow F(S^0)
\end{displaymath}
on the [[0-sphere]] is an [[isomorphism]]. Then $\phi(X)\colon E(X)\to F(X)$ is an [[isomorphism]] for $X$ any [[CW-complex]] with a [[finite number]] of cells. If both $E$ and $F$ satisfy the [[wedge axiom]], then $\phi(X)$ is an isomorphism for $X$ any [[CW-complex]], not necessarily finite.

\end{prop}
For $E$ and $F$ [[ordinary cohomology]]/[[ordinary homology]] functors a proof of this is in (\hyperlink{EilenbergSteenrod52}{Eilenberg-Steenrod 52, section III.10}). From this the general statement follows (e.g. \hyperlink{Kochman96}{Kochman 96, theorem 3.4.3, corollary 4.2.8}) via the [[natural transformation|naturality]] of the [[Atiyah-Hirzebruch spectral sequence]] (the classical result gives that $\phi$ induces an isomorphism between the second pages of the AHSSs for $E$ and $F$). A complete proof of the general result is also given as (\hyperlink{Switzer75}{Switzer 75, theorem 7.55, theorem 7.67})

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

\begin{itemize}%
\item [[stable homotopy homology theory]] is the homology theory represented by the [[sphere spectrum]]


\item [[ordinary homology]] is the homology theory represented by an [[Eilenberg-MacLane spectrum]]


\item [[bordism homology theory]] is the homology theory represented by a [[Thom spectrum]];



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

\begin{itemize}%
\item [[generalized cohomology]]


\item [[Kronecker pairing]]


\item [[Atiyah-Hirzebruch spectral sequence]]


\item [[Steenrod algebra]]


\item [[phantom map]]


\item [[bivariant cohomology]]


\item [[Bousfield equivalence]]


\item [[Hurewicz homomorphism]], [[Boardman homomorphism]]



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

(For more see the references at \emph{[[generalized (Eilenberg-Steenrod) cohomology]]}.)

Original articles include

\begin{itemize}%
\item [[George Whitehead]], \emph{Generalized homology theories} (1961) (\href{http://www.maths.ed.ac.uk/~aar/papers/gww9.pdf}{pdf})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Stanley Kochmann]], section 3.4 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Robert Switzer]], chapter 7 (and 8-12) of \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.


\item [[Stefan Schwede]], chapter II, section 6 of \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item [[Friedrich Bauer]], \emph{Classifying spectra for generalized homology theories} Annali di Maternatica pura ed applicata (IV), Vol. CLXIV (1993), pp. 365-399


\item [[Friedrich Bauer]], \emph{Remarks on universal coefficient theorems for generalized homology theories} Quaestiones Mathematicae Volume 9, Issue 1 \& 4, 1986, Pages 29 - 54



\end{itemize}
A general construction of homologies by ``geometric cycles'' similar to the [[Baum-Douglas geometric cycles]] for [[K-homology]] is discussed in

\begin{itemize}%
\item S. Buoncristiano, C. P. Rourke and B. J. Sanderson, \emph{A geometric approach to homology theory}, Cambridge Univ. Press, Cambridge, Mass. (1976)

\end{itemize}
Further generalization of this to [[bivariant cohomology theories]] is in

\begin{itemize}%
\item Martin Jakob, \emph{Bivariant theories for smooth manifolds}, Applied Categorical Structures 10 no. 3 (2002)

\end{itemize}
[[!redirects generalized homology theory]] [[!redirects generalized homology theories]]

[[!redirects homology theory]] [[!redirects homology theories]]

[[!redirects generalized homologies]]

[[!redirects generalised homology]] [[!redirects generalised homologies]]



\end{document}