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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{singular homology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra} [[!include homological algebra - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{BasicExamples}{Basic examples}\dotfill \pageref*{BasicExamples} \linebreak \noindent\hyperlink{HomologyOfDisksAndSpheres}{Homology of cells: disks and spheres}\dotfill \pageref*{HomologyOfDisksAndSpheres} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{HomotopyInvariant}{Homotopy invariance}\dotfill \pageref*{HomotopyInvariant} \linebreak \noindent\hyperlink{RelationToHomotopyGroups}{Relation to homotopy groups}\dotfill \pageref*{RelationToHomotopyGroups} \linebreak \noindent\hyperlink{RelationToRelativeHomology}{Relation to relative homology}\dotfill \pageref*{RelationToRelativeHomology} \linebreak \noindent\hyperlink{relation_to_generalized_homology}{Relation to generalized homology}\dotfill \pageref*{relation_to_generalized_homology} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{examples_and_applications}{Examples and applications}\dotfill \pageref*{examples_and_applications} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{singular homology} of a [[topological space]] $X$ is the [[simplicial homology]] of its [[singular simplicial complex]]: a \textbf{singular $n$-[[chain]]} on $X$ is a [[formal linear combination]] of [[singular simplices]] $\sigma : \Delta^n \to X$, and a singular $n$-[[cycle]] is such a chain such that its oriented [[boundary]] in $X$ vanishes. Two singular chains are \textbf{homologous} if they differ by a boudary. The \textbf{singular homology} of $X$ in degree $n$ is the group of $n$-cycles modulo those that are boundaries. Singular homology of a topological space conincide with its [[ordinary homology]] as defined more abstractly (see at [[generalized homology theory]]). (Here ``singular'' refers to the contrast with [[cellular homology]], referring to the fact that a [[simplex]] $\Delta_{top} \to X$ in the [[singular simplicial complex]] is not required to be a [[topological embedding]], but may be a ``singular map'', such as for instance a [[constant function]].) \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} Let $X \in$ [[Top]] be [[topological space]]. Write $Sing X \in$ [[sSet]] for its [[singular simplicial complex]]. \begin{defn} \label{}\hypertarget{}{} For $n \in \mathbb{N}$, a \textbf{singular $n$-chain} on $X$ is an element in the [[free abelian group]] $\mathbb{Z}[(Sing X)_n]$: a [[formal linear combinations]] of [[singular simplices]] in $X$. \end{defn} \begin{remark} \label{}\hypertarget{}{} These are the [[chains on a simplicial set]] on $Sing X$. \end{remark} The groups of singular chains combine to the [[simplicial abelian group]] $\mathbb{Z}[Sing X] \in Ab^{\Delta^{op}}$. \begin{defn} \label{SingularComplex}\hypertarget{SingularComplex}{} The [[alternating face map complex]] \begin{displaymath} C_\bullet(X) \coloneqq C_\bullet(\mathbb{Z}[Sing X]) \in Ch_\bullet \end{displaymath} is the \textbf{singular complex} of $X$. Its [[chain homology]] is the ordinary \textbf{singular homology} of $X$. \end{defn} One usually writes $H_n(X, \mathbb{Z})$ or just $H_n(X)$ for the singular homology of $X$ in degree $n$. See also at \emph{[[ordinary homology]]}. \begin{remark} \label{}\hypertarget{}{} So we have \begin{displaymath} C_\bullet(X) = [ \cdots \stackrel{\partial_2}{\to} \mathbb{Z}[(Sing X)_2] \stackrel{\partial_1}{\to} \mathbb{Z}[(Sing X)_1] \stackrel{\partial_0}{\to} \mathbb{Z}[(Sing X)_0] ] \end{displaymath} where the differentials are defined on [[basis]] elements $\sigma \in (Sing X)_n$ by \begin{displaymath} \partial_n \sigma = - \sum_{i = 0}^n (-1) d_i \sigma \end{displaymath} (with $d_i$ the $i$ [[simplicial set|simplicial face map]]) and then extended linearly. (One may change the global signs and obtain a [[quasi-isomorphism|quasi-isomorphic]] complex, in particular with the same homology groups.) \end{remark} \begin{remark} \label{}\hypertarget{}{} This means that a [[singular chain]] is a [[cycle]] if the formal linear combination of the oriented [[boundaries]] of all its constituent [[singular simplices]] sums to 0. See the \emph{\hyperlink{BasicExamples}{basic examples}} below \end{remark} More generally, for $R$ any unital [[ring]] one can form the degreewise [[free module]] $R[Sing X]$ over $R$. The corresponding homology is the \emph{singular homology with coefficients in $R$, denoted $H_n(X,R)$.} \begin{defn} \label{}\hypertarget{}{} Given a [[continuous map]] $f : X \to Y$ between topological spaces, and given $n \in \mathbb{N}$, every singular $n$-simplex $\sigma : \Delta^n \to X$ in $X$ is sent to a singular $n$-simplex \begin{displaymath} f_* \sigma : \Delta^n \stackrel{\sigma}{\to} X \stackrel{f}{\to} Y \end{displaymath} in $Y$. This is called the \textbf{push-forward} of $\sigma$ along $f$. Accordingly there is a push-forward map on groups of singular chains \begin{displaymath} (f_*)_n : C_n(X) \to C_n(Y) \,. \end{displaymath} \end{defn} \begin{prop} \label{PushForwardChainMap}\hypertarget{PushForwardChainMap}{} These push-forward maps make all diagrams of the form \begin{displaymath} \itexarray{ C_{n+1}(X) &\stackrel{(f_*)_{n+1}}{\to}& C_{n+1}(Y) \\ \downarrow^{\mathrlap{\partial^X_n}} && \downarrow^{\mathrlap{\partial^Y_n}} \\ C_n(X) &\stackrel{(f_*)_n}{\to}& C_n(Y) } \end{displaymath} commute. In other words, push-forward along $f$ constitutes a [[chain map]] \begin{displaymath} f_* : C_\bullet(X) \to C_\bullet(Y) \,. \end{displaymath} \end{prop} \begin{proof} It is in fact evident that push-forward yields a functor of [[singular simplicial complexes]] \begin{displaymath} f_* : Sing X \to Sing Y \,. \end{displaymath} From this the statement follows since $\mathbb{Z}[-] : sSet \to sAb$ is a functor. \end{proof} Accordingly we have: \begin{prop} \label{SingularHomologyAsAFunctor}\hypertarget{SingularHomologyAsAFunctor}{} Sending a topological space to its singular chain complex $C_\bullet(X)$, def. \ref{SingularComplex}, and a continuous map to its push-forward chain map, prop. \ref{PushForwardChainMap}, constitutes a [[functor]] \begin{displaymath} C_\bullet(-,R) : Top \to Ch_\bullet(R Mod) \end{displaymath} from the category [[Top]] to the [[category of chain complexes]]. In particular for each $n \in \mathbb{N}$ singular homology extends to a [[functor]] \begin{displaymath} H_n(-,R) : Top \to R Mod \,. \end{displaymath} \end{prop} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{BasicExamples}{}\subsubsection*{{Basic examples}}\label{BasicExamples} \begin{example} \label{}\hypertarget{}{} Let $X$ be a topological space. Let $\sigma^1 : \Delta^1 \to X$ be a singular 1-simplex, regarded as a 1-chain \begin{displaymath} \sigma^1 \in C_1(X) \,. \end{displaymath} Then its [[boundary]] $\partial \sigma \in H_0(X)$ is \begin{displaymath} \partial \sigma^1 = \sigma(0) -\sigma(1) \end{displaymath} or graphically (using notation as for [[orientals]]) \begin{displaymath} \partial \left( \sigma(0) \stackrel{\sigma}{\to} \sigma(1) \right) = (\sigma(0)) - (\sigma(1)) \,. \end{displaymath} Let $\sigma^2 : \Delta^2 \to X$ be a singular 2-chain. The boundary is \begin{displaymath} \partial \left( \itexarray{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & \Downarrow^{\mathrlap{\sigma}}& \searrow^{\mathrlap{\sigma^{1,2}}} \\ \sigma(0) &&\underset{\sigma(0,2)}{\to}&& \sigma(2) } \right) = \left( \itexarray{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & & \\ \sigma(0) } \right) - \left( \itexarray{ && \\ & & & \\ \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2) } \right) + \left( \itexarray{ && \sigma(1) \\ & & & \searrow^{\mathrlap{\sigma^{1,2}}} \\ && && \sigma(2) } \right) \,. \end{displaymath} Hence the boundary of the boundary is \begin{displaymath} \begin{aligned} \partial \partial \sigma &= \partial \left( \left( \itexarray{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & & \\ \sigma(0) } \right) - \left( \itexarray{ && \\ & & & \\ \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2) } \right) + \left( \itexarray{ && \sigma(1) \\ & & & \searrow^{\mathrlap{\sigma^{1,2}}} \\ && && \sigma(2) } \right) \right) \\ & = \left( \itexarray{ && \\ & & & \\ \sigma(0) } \right) - \left( \itexarray{ && \sigma(1) \\ & & & \\ } \right) - \left( \itexarray{ && \\ & & & \\ \sigma(0) && } \right) + \left( \itexarray{ && \\ & & & \\ && \sigma(2) } \right) + \left( \itexarray{ && \sigma(1) \\ & & & \\ && && } \right) - \left( \itexarray{ && \\ & & & \\ && && \sigma(2) } \right) \\ & = 0 \end{aligned} \end{displaymath} \end{example} For more illustrations see for instance (\hyperlink{Ghrist}{Ghrist, (4.5)}). \hypertarget{HomologyOfDisksAndSpheres}{}\subsubsection*{{Homology of cells: disks and spheres}}\label{HomologyOfDisksAndSpheres} \begin{prop} \label{}\hypertarget{}{} For all $n \in \mathbb{N}$ the [[reduced singular homology]] of the $n$-[[sphere]] $S^n$ is \begin{displaymath} \tilde H_k(S^n) = \left\{ \itexarray{ \mathbb{Z} & if\; k = n \\ 0 & otherwise } \right. \,. \end{displaymath} \end{prop} \begin{proof} The $n$-sphere may be realized as the [[pushout]] \begin{displaymath} S^n \simeq D^n/S^{n-1} \coloneqq D^{n} \coprod_{S^{n-1}} * \end{displaymath} which is the $n$-[[ball]] with its [[boundary]] $(n-1)$-sphere identified with the [[point]]. The inclusion $S^{n-1} \hookrightarrow D^n$ is a ``good pair'' in the sense of def. \ref{GoodPair}, and so the [[long exact sequence]] from prop. \ref{RelativeHomologyLongExactSequence} yields a long exact sequence \begin{displaymath} \cdots \to \tilde H_{k+1}(S^n) \to \tilde H_k(S^{n-1}) \to \tilde H_k(D^n) \to \tilde H_k(S^n) \to \tilde H_{k-1}(S^{n-1}) \to \cdots \,. \end{displaymath} Since the [[disks]] are all [[contractible topological spaces]] we have $H_k(D^n) \simeq 0$ for all $k,n$ by \href{reduced+homology#ReducedHomologyOfPoints}{this example} at \emph{[[reduced homology]]}. This means that in the above long exact sequence all the morphisms \begin{displaymath} \tilde H_{k+1}(S^{n+1}) \to \tilde H_k(S^n) \end{displaymath} are [[isomorphisms]], for all $k \in \mathbb{N}$. Since \begin{displaymath} \tilde H_n(S^0) \simeq \left\{ \itexarray{ \mathbb{Z} & if \; n = 0 \\ 0 & otherwise } \right. \end{displaymath} (by \href{reduced+homology#ReducedHomologyOf0Sphere}{this example} at \emph{[[reduced homology]]}) the statement follows by [[induction]] on $n$. \end{proof} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{HomotopyInvariant}{}\subsubsection*{{Homotopy invariance}}\label{HomotopyInvariant} Singular homology is \emph{homotopy invariant}: \begin{prop} \label{}\hypertarget{}{} If $f : X \to Y$ is a [[continuous map]] between [[topological spaces]] which is a [[homotopy equivalence]], then the induced morphism on singular homology groups \begin{displaymath} H_n(f) : H_n(X) \to H_n(Y) \end{displaymath} is an [[isomorphism]]. In other words: the singular chain functor of prop. \ref{SingularHomologyAsAFunctor} sends [[weak homotopy equivalences]] to [[quasi-isomorphisms]]. \end{prop} A proof (via [[CW approximations]]) is spelled out for instance in (\hyperlink{Hatcher}{Hatcher, prop. 4.21}). \hypertarget{RelationToHomotopyGroups}{}\subsubsection*{{Relation to homotopy groups}}\label{RelationToHomotopyGroups} The singular [[homology groups]] of a topologial space serve to some extent as an approximation to the [[homotopy groups]] of that space. \begin{defn} \label{HurewiczHomomorphism}\hypertarget{HurewiczHomomorphism}{} \textbf{(Hurewicz homomorphism)} For $(X,x)$ a [[pointed object|pointed]] [[topological space]], the \textbf{Hurewicz homomorphism} is the [[function]] \begin{displaymath} \Phi : \pi_k(X,x) \to H_k(X) \end{displaymath} from the $k$th [[homotopy group]] of $(X,x)$ to the $k$th [[singular homology]] group defined by sending \begin{displaymath} \Phi : (f : S^k \to X)_{\sim} \mapsto f_*[S_k] \end{displaymath} a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the [[fundamental class]] $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$. \end{defn} \begin{prop} \label{}\hypertarget{}{} For $X$ a topological space the Hurewicz homomorphism in degree 0 exhibits an [[isomorphism]] between the [[free abelian group]] $\mathbb{Z}[\pi_0(X)]$ on the set of [[connected components]] of $X$ and the degree-0 singular homlogy: \begin{displaymath} \mathbb{Z}[\pi_0(X)] \simeq H_0(X) \,. \end{displaymath} \end{prop} Since a [[homotopy group]] in positive degree depends on the [[homotopy type]] of the [[connected component]] of the base point, while the singular homology does not depend on a basepoint, it is interesting to compare these groups only for the case that $X$ is connected. \begin{prop} \label{}\hypertarget{}{} For $X$ a [[connected topological space]] the [[Hurewicz homomorphism]] in degree 1 \begin{displaymath} \Phi : \pi_1(X,x) \to H_1(X) \end{displaymath} is [[surjection|surjective]]. Its [[kernel]] is the [[commutator subgroup]] of $\pi_1(X,x)$. Therefore it induces an [[isomorphism]] from the [[abelianization]] $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$: \begin{displaymath} \pi_1(X,x)^{ab} \stackrel{\simeq}{\to} H_1(X) \,. \end{displaymath} \end{prop} For higher connected $X$ we have the \begin{theorem} \label{}\hypertarget{}{} If $X$ is [[n-connected object in an (infinity,1)-topos|(n-1)-connected]] for $n \geq 2$ then \begin{displaymath} \Phi : \pi_n(X,x) \to H_n(X) \end{displaymath} is an [[isomorphism]]. \end{theorem} This is known as the \emph{[[Hurewicz theorem]]}. \hypertarget{RelationToRelativeHomology}{}\subsubsection*{{Relation to relative homology}}\label{RelationToRelativeHomology} For the present purpose one makes the following definition. \begin{defn} \label{GoodPair}\hypertarget{GoodPair}{} A [[topological subspace]] inclusion $A \hookrightarrow X$ in [[Top]] is called a \textbf{good pair} if \begin{enumerate}% \item $A$ is [[inhabited set|inhabited]] and [[closed subset|closed]] in $X$; \item $A$ has a [[neighbourhood]] in $X$ of which it is a [[deformation retract]]. \end{enumerate} \end{defn} Write $X/A$ for the [[cokernel]] of the inclusion, hence for the [[pushout]] \begin{displaymath} \itexarray{ A &\hookrightarrow& X \\ \downarrow && \downarrow \\ * &\to& X/A } \end{displaymath} in [[Top]]. \begin{prop} \label{RelativeHomologyLongExactSequence}\hypertarget{RelativeHomologyLongExactSequence}{} If $A \hookrightarrow X$ is a good pair, def. \ref{GoodPair}, then the singular homology of $X/A$ coincides with the [[relative homology]] of $X$ relative to $A$. In particular, therefore, it fits into a [[long exact sequence]] of the form \begin{displaymath} \cdots \to \tilde H_n(A) \to \tilde H_n(X) \to \tilde H_n(X/A) \to \tilde H_{n-1}(A) \to \tilde H_{n-1}(X) \to \tilde H_{n-1}(X/A) \to \cdots \,. \end{displaymath} \end{prop} For instance (\hyperlink{Hatcher}{Hatcher, theorem 2.13}). \hypertarget{relation_to_generalized_homology}{}\subsubsection*{{Relation to generalized homology}}\label{relation_to_generalized_homology} Singular homology computes the [[generalized homology]] with coefficients in the [[Eilenberg-MacLane spectrum]] $H \mathbb{Z}$ or $H R$. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item The [[duality|dual]] notion is that of \emph{[[singular cohomology]]}. \item The analogous notion in [[algebraic geometry]] is given by [[Chow groups]]. \item [[Kan-Thurston theorem]] \item [[Borel-Moore homology]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Lecture notes include \begin{itemize}% \item Rob Thompson, Len Evens , \emph{\href{http://math.hunter.cuny.edu/~rthompso/topology_notes/}{Topology notes}} \emph{Chapter 6, Singular homology} (\href{http://math.hunter.cuny.edu/~rthompso/topology_notes/chapter%20six.pdf}{pdf}) \end{itemize} Textbook discussion in the context of [[homological algebra]] is around Application 1.1.4 of \begin{itemize}% \item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]} \end{itemize} and in the context of [[algebraic topology]] in chapter 2.1 of \begin{itemize}% \item [[Allen Hatcher]], \emph{Algebraic Topology} (\href{http://www.math.cornell.edu/~hatcher/AT/ATpage.html}{web}) \end{itemize} and \href{http://www.math.upenn.edu/~ghrist/EAT/EATchapter4.pdf}{chapter 4} of \begin{itemize}% \item [[Robert Ghrist]], \emph{Elementary applied topology} (\href{http://www.math.upenn.edu/~ghrist/notes.html}{web}) \end{itemize} Discussion in the context of computing [[homotopy groups]] is in \begin{itemize}% \item [[Michael Hutchings]], \emph{Introduction to higher homotopy groups and obstruction theory} (\href{http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf}{pdf}) \end{itemize} Lecture notes include \begin{itemize}% \item [[Marco Gualtieri]], \emph{Homology} (\href{http://www.math.toronto.edu/mgualt/MAT1300/Week%206-9%20Term%202.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Singular_homology}{Singular homology}} \end{itemize} \hypertarget{examples_and_applications}{}\subsubsection*{{Examples and applications}}\label{examples_and_applications} \begin{itemize}% \item Michael Barratt, [[John Milnor]], \emph{An example of anomalous singular homology}, Proceedings of the American Mathematical Society Vol. 13, No. 2 (Apr., 1962), pp. 293-297 (\href{http://www.jstor.org/stable/2034486}{JSTOR}) \end{itemize} [[!redirects singular chain]] [[!redirects singular chains]] [[!redirects singular simplicial chain]] [[!redirects singular simplicial chains]] [[!redirects chain complex of singular simplices]] [[!redirects chain complexes of singular simplices]] [[!redirects singular chain homology]] \end{document}