nLab
singular homology

Idea

The singular homology of a topological space $X$ is the simplicial homology of its singular simplicial complex:

a 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 homologous if they differ by a boudary. The singular homology of $X$ in degree $n$ is the group of $n$-cycles modulo 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).

Definition

Let $X\in$ Top be topological space. Write $\mathrm{Sing}X\in$ sSet for its singular simplicial complex.

Definition

For $n\in \mathbb{N}$, a singular $n$-chain on $X$ is an element in the free abelian group $\mathbb{Z}[(\mathrm{Sing}X{)}_n]$:

a formal linear combinations of singular simplices in $X$.

Remark

These are the chains on a simplicial set on $\mathrm{Sing}X$.

The groups of singular chains combine to the simplicial abelian group $\mathbb{Z}[\mathrm{Sing}X]\in {\mathrm{Ab}}^{{\Delta }^{\mathrm{op}}}$.

Definition

The alternating face map complex

$$C_{•}(X)≔C_{•}(\mathbb{Z}[\mathrm{Sing}X])\in {\mathrm{Ch}}_{•}$$C_\bullet(X) \coloneqq C_\bullet(\mathbb{Z}[Sing X]) \in Ch_\bullet

is the singular complex of $X$.

Its chain homology is the ordinary singular homology of $X$.

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 ordinary homology.

Remark

So we have

$$C_{•}(X)=[\cdots \stackrel{{\partial }_2}{\to }\mathbb{Z}[(\mathrm{Sing}X{)}_2]\stackrel{{\partial }_1}{\to }\mathbb{Z}[(\mathrm{Sing}X{)}_1]\stackrel{{\partial }_0}{\to }\mathbb{Z}[(\mathrm{Sing}X{)}_0]]$$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] ]

where the differentials are defined on basis elements $\sigma \in (\mathrm{Sing}X{)}_n$ by

$${\partial }_n\sigma =-\sum _{i=0}^n(-1)d_i\sigma$$\partial_n \sigma = - \sum_{i = 0}^n (-1) d_i \sigma

(with $d_i$ the $i$ simplicial face map) and then extended linearly.

(One may change the global signs and obtain a quasi-isomorphic complex, in particular with the same homology groups.)

Remark

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 basic examples below

More generally, for $R$ any unital ring one can form the degreewise free module $R[\mathrm{Sing}X]$ over $R$. The corresponding homology is the singular homology with coefficients in $R$, denoted $H_n(X,R)$.

Definition

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

$$f_{*}\sigma :{\Delta }^n\stackrel{\sigma }{\to }X\stackrel{f}{\to }Y$$f_* \sigma : \Delta^n \stackrel{\sigma}{\to} X \stackrel{f}{\to} Y

in $Y$. This is called the push-forward of $\sigma$ along $f$. Accordingly there is a push-forward map on groups of singular chains

$$(f_{*}{)}_n:C_n(X)\to C_n(Y).$$(f_*)_n : C_n(X) \to C_n(Y) \,.

Proposition

These push-forward maps make all diagrams of the form

$$\begin{array}{ccc}{C}_{n+1}(X)& \stackrel{(f_{*}{)}_{n+1}}{\to }& C_{n+1}(Y)\\ {\downarrow }^{{\partial }_n^X}& & {\downarrow }^{{\partial }_n^Y}\\ C_n(X)& \stackrel{(f_{*}{)}_n}{\to }& C_n(Y)\end{array}$$\array{ 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) }

commute. In other words, push-forward along $f$ constitutes a chain map

$$f_{*}:C_{•}(X)\to C_{•}(Y).$$f_* : C_\bullet(X) \to C_\bullet(Y) \,.

Proof

It is in fact evident that push-forward yields a functor of singular simplicial complexes

$$f_{*}:\mathrm{Sing}X\to \mathrm{Sing}Y.$$f_* : Sing X \to Sing Y \,.

From this the statement follows since $\mathbb{Z}[-]:\mathrm{sSet}\to \mathrm{sAb}$ is a functor.

Accordingly we have:

Proposition

Sending a topological space to its singular chain complex $C_{•}(X)$, def. 2, and a continuous map to its push-forward chain map, prop. 1, constitutes a functor

$$C_{•}(-,R):\mathrm{Top}\to {\mathrm{Ch}}_{•}(R\mathrm{Mod})$$C_\bullet(-,R) : Top \to Ch_\bullet(R Mod)

from the category Top to the category of chain complexes.

In particular for each $n\in \mathbb{N}$ singular homology extends to a functor

$$H_n(-,R):\mathrm{Top}\to R\mathrm{Mod}.$$H_n(-,R) : Top \to R Mod \,.

Examples

Basic examples

Example

Let $X$ be a topological space. Let ${\sigma }^1:{\Delta }^1\to X$ be a singular 1-simplex, regarded as a 1-chain

$${\sigma }^1\in C_1(X).$$\sigma^1 \in C_1(X) \,.

Then its boundary $\partial \sigma \in H_0(X)$ is

$$\partial {\sigma }^1=\sigma (0)-\sigma (1)$$\partial \sigma^1 = \sigma(0) -\sigma(1)

or graphically (using notation as for orientals)

$$\partial (\sigma (0)\stackrel{\sigma }{\to }\sigma (1))=(\sigma (0))-(\sigma (1)).$$\partial \left( \sigma(0) \stackrel{\sigma}{\to} \sigma(1) \right) = (\sigma(0)) - (\sigma(1)) \,.

Let ${\sigma }^2:{\Delta }^2\to X$ be a singular 2-chain. The boundary is

$$\partial \left(\begin{array}{ccc}& & \sigma (1)\\ & {}^{\sigma (0,1)}\nearrow & {\Downarrow }^{\sigma }& {\searrow }^{{\sigma }^{1,2}}\\ \sigma (0)& & \underset{\sigma (0,2)}{\to }& & \sigma (2)\end{array}\right)=\left(\begin{array}{ccc}& & \sigma (1)\\ & {}^{\sigma (0,1)}\nearrow & & \\ \sigma (0)\end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ \sigma (0)& \underset{\sigma (0,2)}{\to }& \sigma (2)\end{array}\right)+\left(\begin{array}{ccc}& & \sigma (1)\\ & & & {\searrow }^{{\sigma }^{1,2}}\\ & & & & \sigma (2)\end{array}\right).$$\partial \left( \array{ && \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( \array{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & & \\ \sigma(0) } \right) - \left( \array{ && \\ & & & \\ \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2) } \right) + \left( \array{ && \sigma(1) \\ & & & \searrow^{\mathrlap{\sigma^{1,2}}} \\ && && \sigma(2) } \right) \,.

Hence the boundary of the boundary is

$$\begin{array}{rl}\partial \partial \sigma & =\partial (\left(\begin{array}{ccc}& & \sigma (1)\\ & {}^{\sigma (0,1)}\nearrow & & \\ \sigma (0)\end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ \sigma (0)& \underset{\sigma (0,2)}{\to }& \sigma (2)\end{array}\right)+\left(\begin{array}{ccc}& & \sigma (1)\\ & & & {\searrow }^{{\sigma }^{1,2}}\\ & & & & \sigma (2)\end{array}\right))\\ & =\left(\begin{array}{ccc}& & \\ & & & \\ \sigma (0)\end{array}\right)-\left(\begin{array}{ccc}& & \sigma (1)\\ & & & \\ \end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ \sigma (0)& & \end{array}\right)+\left(\begin{array}{ccc}& & \\ & & & \\ & & \sigma (2)\end{array}\right)+\left(\begin{array}{ccc}& & \sigma (1)\\ & & & \\ & & & & \end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ & & & & \sigma (2)\end{array}\right)\\ & =0\end{array}$$\begin{aligned} \partial \partial \sigma &= \partial \left( \left( \array{ && \sigma(1) \\ & {}^{\mathllap{\sigma(0,1)}}\nearrow & & \\ \sigma(0) } \right) - \left( \array{ && \\ & & & \\ \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2) } \right) + \left( \array{ && \sigma(1) \\ & & & \searrow^{\mathrlap{\sigma^{1,2}}} \\ && && \sigma(2) } \right) \right) \\ & = \left( \array{ && \\ & & & \\ \sigma(0) } \right) - \left( \array{ && \sigma(1) \\ & & & \\ } \right) - \left( \array{ && \\ & & & \\ \sigma(0) && } \right) + \left( \array{ && \\ & & & \\ && \sigma(2) } \right) + \left( \array{ && \sigma(1) \\ & & & \\ && && } \right) - \left( \array{ && \\ & & & \\ && && \sigma(2) } \right) \\ & = 0 \end{aligned}

For more illustrations see for instance (Ghrist, (4.5)).

Homology of cells: disks and spheres

Proposition

For all $n\in \mathbb{N}$ the reduced singular homology of the $n$-sphere $S^n$ is

$${\tilde{H}}_k(S^n)=\{\begin{array}{cc}\mathbb{Z}& \mathrm{if}k=n\\ 0& \mathrm{otherwise}\end{array}.$$\tilde H_k(S^n) = \left\{ \array{ \mathbb{Z} & if\; k = n \\ 0 & otherwise } \right. \,.

Proof

The $n$-sphere may be realized as the pushout

$$S^n\simeq D^n/S^{n-1}≔D^n\coprod _{S^{n-1}}*$$S^n \simeq D^n/S^{n-1} \coloneqq D^{n} \coprod_{S^{n-1}} *

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. 5, and so the long exact sequence from prop. 7 yields a long exact sequence

$$\cdots \to {\tilde{H}}_{k+1}(S^n)\to {\tilde{H}}_k(S^{n-1})\to {\tilde{H}}_k(D^n)\to {\tilde{H}}_k(S^n)\to {\tilde{H}}_{k-1}(S^{n-1})\to \cdots .$$\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 \,.

Since the disks are all contractible topological spaces we have $H_k(D^n)\simeq 0$ for all $k,n$ by this example at reduced homology. This means that in the above long exact sequence all the morphisms

$${\tilde{H}}_{k+1}(S^{n+1})\to {\tilde{H}}_k(S^n)$$\tilde H_{k+1}(S^{n+1}) \to \tilde H_k(S^n)

are isomorphisms, for all $k\in \mathbb{N}$. Since

$${\tilde{H}}_n(S^0)\simeq \{\begin{array}{cc}\mathbb{Z}& \mathrm{if}n=0\\ 0& \mathrm{otherwise}\end{array}$$\tilde H_n(S^0) \simeq \left\{ \array{ \mathbb{Z} & if \; n = 0 \\ 0 & otherwise } \right.

(by this example at reduced homology) the statement follows by induction on $n$.

Properties

Homotopy invariance

Singular homology is homotopy invariant:

Proposition

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

$$H_n(f):H_n(X)\to H_n(Y)$$H_n(f) : H_n(X) \to H_n(Y)

is an isomorphism.

In other words: the singular chain functor of prop. 2 sends weak homotopy equivalences to quasi-isomorphisms.

A proof (via CW approximations) is spelled out for instance in (Hatcher, prop. 4.21).

Relation to homotopy groups

The singular homology groups of a topologial space serve to some extent as an approximation to the homotopy groups of that space.

Definition

(Hurewicz homomorphism)

For $(X,x)$ a pointed topological space, the Hurewicz homomorphism is the function

$$\Phi :{\pi }_k(X,x)\to H_k(X)$$\Phi : \pi_k(X,x) \to H_k(X)

from the $k$th homotopy group of $(X,x)$ to the $k$th singular homology group defined by sending

$$\Phi :(f:S^k\to X{)}_{\sim }\mapsto f_{*}[S_k]$$\Phi : (f : S^k \to X)_{\sim} \mapsto f_*[S_k]

a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the fundamental class $[S_k]\in H_k(S^k)\simeq \mathbb{Z}$.

Proposition

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:

$$\mathbb{Z}[{\pi }_0(X)]\simeq H_0(X).$$\mathbb{Z}[\pi_0(X)] \simeq H_0(X) \,.

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.

Proposition

For $X$ a connected topological space the Hurewicz homomorphism in degree 1

$$\Phi :{\pi }_1(X,x)\to H_1(X)$$\Phi : \pi_1(X,x) \to H_1(X)

is surjective. Its kernel is the commutator subgroup of ${\pi }_1(X,x)$. Therefore it induces an isomorphism from the abelianization ${\pi }_1(X,x{)}^{\mathrm{ab}}≔{\pi }_1(X,x)/[{\pi }_1,{\pi }_1]$:

$${\pi }_1(X,x{)}^{\mathrm{ab}}\stackrel{\simeq }{\to }{H}_1(X).$$\pi_1(X,x)^{ab} \stackrel{\simeq}{\to} H_1(X) \,.

For higher connected $X$ we have the

Theorem

If $X$ is (n-1)-connected for $n\ge 2$ then

$$\Phi :{\pi }_n(X,x)\to H_n(X)$$\Phi : \pi_n(X,x) \to H_n(X)

is an isomorphism.

This is known as the Hurewicz theorem.

Relation to relative homology

For the present purpose one makes the following definition.

Definition

A topological subspace inclusion $A\hookrightarrow X$ in Top is called a good pair if

1. $A$ is inhabited and closed in $X$;

2. $A$ has a neighbourhood in $X$ of which it is a deformation retract.

Write $X/A$ for the cokernel of the inclusion, hence for the pushout

$$\begin{array}{ccc}A& \hookrightarrow & X\\ \downarrow & & \downarrow \\ *& \to & X/A\end{array}$$\array{ A &\hookrightarrow& X \\ \downarrow && \downarrow \\ * &\to& X/A }

in Top.

Proposition

If $A\hookrightarrow X$ is a good pair, def. 5, 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

$$\cdots \to {\tilde{H}}_n(A)\to {\tilde{H}}_n(X)\to {\tilde{H}}_n(X/A)\to {\tilde{H}}_{n-1}(A)\to {\tilde{H}}_{n-1}(X)\to {\tilde{H}}_{n-1}(X/A)\to \cdots .$$\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 \,.

For instance (Hatcher, theorem 2.13).

Relation to generalized homology

Singular homology computes the generalized homology with coefficients in the Eilenberg-MacLane spectrum $H\mathbb{Z}$ or $HR$.

Related concepts

The dual notion is that of singular cohomology.

References

General

Lecture notes include

• Rob Thompson, Len Evens , Topology notes Chapter 6, Singular homology (pdf)

Textbook discussion in the context of homological algebra is around Application 1.1.4 of

• Charles Weibel, An Introduction to Homological Algebra

and in the context of algebraic topology in chapter 2.1 of

• Alan Hatcher, Algebraic Topology (web)

and chapter 4 of

• Robert Ghrist, Elementary applied topology (web)

Discussion in the context of computing homotopy groups is in

• Michael Hutchings, Introduction to higher homotopy groups and obstruction theory (pdf)

Lecture notes include

• Marco Gualtieri, Homology (pdf)

See also

• Wikipedia, Singular homology

Examples and applications

• Michael Barratt, John Milnor, An example of anomalous singular homology, Proceedings of the American Mathematical Society Vol. 13, No. 2 (Apr., 1962), pp. 293-297 (JSTOR)