# nLab singular homology

### Context

#### Topology

topology

algebraic topology

## Examples

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

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

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 $ℤ\left[\mathrm{Sing}X\right]\in {\mathrm{Ab}}^{{\Delta }^{\mathrm{op}}}$.

###### Definition
${C}_{•}\left(X\right)≔{C}_{•}\left(ℤ\left[\mathrm{Sing}X\right]\right)\in {\mathrm{Ch}}_{•}$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 ${H}_{n}\left(X,ℤ\right)$ or just ${H}_{n}\left(X\right)$ for the singular homology of $X$ in degree $n$. See also at ordinary homology.

###### Remark

So we have

${C}_{•}\left(X\right)=\left[\cdots \stackrel{{\partial }_{2}}{\to }ℤ\left[\left(\mathrm{Sing}X{\right)}_{2}\right]\stackrel{{\partial }_{1}}{\to }ℤ\left[\left(\mathrm{Sing}X{\right)}_{1}\right]\stackrel{{\partial }_{0}}{\to }ℤ\left[\left(\mathrm{Sing}X{\right)}_{0}\right]\right]$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 $\sigma \in \left(\mathrm{Sing}X{\right)}_{n}$ by

${\partial }_{n}\sigma =-\sum _{i=0}^{n}\left(-1\right){d}_{i}\sigma$\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 $R\left[\mathrm{Sing}X\right]$ over $R$. The corresponding homology is the singular homology with coefficients in $R$, denoted ${H}_{n}\left(X,R\right)$.

###### Definition

Given a continuous map $f:X\to Y$ between topological spaces, and given $n\in ℕ$, 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

$\left({f}_{*}{\right)}_{n}:{C}_{n}\left(X\right)\to {C}_{n}\left(Y\right)\phantom{\rule{thinmathspace}{0ex}}.$(f_*)_n : C_n(X) \to C_n(Y) \,.
###### Proposition

These push-forward maps make all diagrams of the form

$\begin{array}{ccc}{C}_{n+1}\left(X\right)& \stackrel{\left({f}_{*}{\right)}_{n+1}}{\to }& {C}_{n+1}\left(Y\right)\\ {↓}^{{\partial }_{n}^{X}}& & {↓}^{{\partial }_{n}^{Y}}\\ {C}_{n}\left(X\right)& \stackrel{\left({f}_{*}{\right)}_{n}}{\to }& {C}_{n}\left(Y\right)\end{array}$\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

${f}_{*}:{C}_{•}\left(X\right)\to {C}_{•}\left(Y\right)\phantom{\rule{thinmathspace}{0ex}}.$f_* : C_\bullet(X) \to C_\bullet(Y) \,.
###### Proof

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

${f}_{*}:\mathrm{Sing}X\to \mathrm{Sing}Y\phantom{\rule{thinmathspace}{0ex}}.$f_* : Sing X \to Sing Y \,.

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

Accordingly we have:

###### Proposition

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

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

from the category Top to the category of chain complexes.

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

${H}_{n}\left(-,R\right):\mathrm{Top}\to R\mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$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

${\sigma }^{1}\in {C}_{1}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\sigma^1 \in C_1(X) \,.

Then its boundary $\partial \sigma \in {H}_{0}\left(X\right)$ is

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

or graphically (using notation as for orientals)

$\partial \left(\sigma \left(0\right)\stackrel{\sigma }{\to }\sigma \left(1\right)\right)=\left(\sigma \left(0\right)\right)-\left(\sigma \left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\partial \left(\begin{array}{ccc}& & \sigma \left(1\right)\\ & {}^{\sigma \left(0,1\right)}↗& {⇓}^{\sigma }& {↘}^{{\sigma }^{1,2}}\\ \sigma \left(0\right)& & \underset{\sigma \left(0,2\right)}{\to }& & \sigma \left(2\right)\end{array}\right)=\left(\begin{array}{ccc}& & \sigma \left(1\right)\\ & {}^{\sigma \left(0,1\right)}↗& & \\ \sigma \left(0\right)\end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ \sigma \left(0\right)& \underset{\sigma \left(0,2\right)}{\to }& \sigma \left(2\right)\end{array}\right)+\left(\begin{array}{ccc}& & \sigma \left(1\right)\\ & & & {↘}^{{\sigma }^{1,2}}\\ & & & & \sigma \left(2\right)\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{rl}\partial \partial \sigma & =\partial \left(\left(\begin{array}{ccc}& & \sigma \left(1\right)\\ & {}^{\sigma \left(0,1\right)}↗& & \\ \sigma \left(0\right)\end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ \sigma \left(0\right)& \underset{\sigma \left(0,2\right)}{\to }& \sigma \left(2\right)\end{array}\right)+\left(\begin{array}{ccc}& & \sigma \left(1\right)\\ & & & {↘}^{{\sigma }^{1,2}}\\ & & & & \sigma \left(2\right)\end{array}\right)\right)\\ & =\left(\begin{array}{ccc}& & \\ & & & \\ \sigma \left(0\right)\end{array}\right)-\left(\begin{array}{ccc}& & \sigma \left(1\right)\\ & & & \\ \end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ \sigma \left(0\right)& & \end{array}\right)+\left(\begin{array}{ccc}& & \\ & & & \\ & & \sigma \left(2\right)\end{array}\right)+\left(\begin{array}{ccc}& & \sigma \left(1\right)\\ & & & \\ & & & & \end{array}\right)-\left(\begin{array}{ccc}& & \\ & & & \\ & & & & \sigma \left(2\right)\end{array}\right)\\ & =0\end{array}$\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 $n\in ℕ$ the reduced singular homology of the $n$-sphere ${S}^{n}$ is

${\stackrel{˜}{H}}_{k}\left({S}^{n}\right)=\left\{\begin{array}{cc}ℤ& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}k=n\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(n-1\right)$-sphere identified with the point. The inclusion ${S}^{n-1}↪{D}^{n}$ is a “good pair” in the sense of def. 5, and so the long exact sequence from prop. 7 yields a long exact sequence

$\cdots \to {\stackrel{˜}{H}}_{k+1}\left({S}^{n}\right)\to {\stackrel{˜}{H}}_{k}\left({S}^{n-1}\right)\to {\stackrel{˜}{H}}_{k}\left({D}^{n}\right)\to {\stackrel{˜}{H}}_{k}\left({S}^{n}\right)\to {\stackrel{˜}{H}}_{k-1}\left({S}^{n-1}\right)\to \cdots \phantom{\rule{thinmathspace}{0ex}}.$\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 ${H}_{k}\left({D}^{n}\right)\simeq 0$ for all $k,n$ by this example at reduced homology. This means that in the above long exact sequence all the morphisms

${\stackrel{˜}{H}}_{k+1}\left({S}^{n+1}\right)\to {\stackrel{˜}{H}}_{k}\left({S}^{n}\right)$\tilde H_{k+1}(S^{n+1}) \to \tilde H_k(S^n)

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

${\stackrel{˜}{H}}_{n}\left({S}^{0}\right)\simeq \left\{\begin{array}{cc}ℤ& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}n=0\\ 0& \mathrm{otherwise}\end{array}$\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

${H}_{n}\left(f\right):{H}_{n}\left(X\right)\to {H}_{n}\left(Y\right)$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 $\left(X,x\right)$ a pointed topological space, the Hurewicz homomorphism is the function

$\Phi :{\pi }_{k}\left(X,x\right)\to {H}_{k}\left(X\right)$\Phi : \pi_k(X,x) \to H_k(X)

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

$\Phi :\left(f:{S}^{k}\to X{\right)}_{\sim }↦{f}_{*}\left[{S}_{k}\right]$\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 $\left[{S}_{k}\right]\in {H}_{k}\left({S}^{k}\right)\simeq ℤ$.

###### Proposition

For $X$ a topological space the Hurewicz homomorphism in degree 0 exhibits an isomorphism between the free abelian group $ℤ\left[{\pi }_{0}\left(X\right)\right]$ on the set of connected components of $X$ and the degree-0 singular homlogy:

$ℤ\left[{\pi }_{0}\left(X\right)\right]\simeq {H}_{0}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\Phi :{\pi }_{1}\left(X,x\right)\to {H}_{1}\left(X\right)$\Phi : \pi_1(X,x) \to H_1(X)

is surjective. Its kernel is the commutator subgroup of ${\pi }_{1}\left(X,x\right)$. Therefore it induces an isomorphism from the abelianization ${\pi }_{1}\left(X,x{\right)}^{\mathrm{ab}}≔{\pi }_{1}\left(X,x\right)/\left[{\pi }_{1},{\pi }_{1}\right]$:

${\pi }_{1}\left(X,x{\right)}^{\mathrm{ab}}\stackrel{\simeq }{\to }{H}_{1}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\Phi :{\pi }_{n}\left(X,x\right)\to {H}_{n}\left(X\right)$\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 $A↪X$ 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

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

in Top.

###### Proposition

If $A↪X$ 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

$\cdots \to {\stackrel{˜}{H}}_{n}\left(A\right)\to {\stackrel{˜}{H}}_{n}\left(X\right)\to {\stackrel{˜}{H}}_{n}\left(X/A\right)\to {\stackrel{˜}{H}}_{n-1}\left(A\right)\to {\stackrel{˜}{H}}_{n-1}\left(X\right)\to {\stackrel{˜}{H}}_{n-1}\left(X/A\right)\to \cdots \phantom{\rule{thinmathspace}{0ex}}.$\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 $Hℤ$ or $HR$.

## References

### General

Lecture notes include

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

and in the context of algebraic topology in chapter 2.1 of

and chapter 4 of

Discussion in the context of computing homotopy groups is in

Lecture notes include