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 boundary. The 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.)

Definition

Let $X \in$ Top be topological space. Write $Sing X \in$ 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 $\mathbb{Z}[(Sing X)_n]$:

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

Remark

These are the chains on a simplicial set on $Sing X$.

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

Definition

The alternating face map complex

$$C_\bullet(X) \coloneqq C_\bullet(\mathbb{Z}[Sing X]) \in Ch_\bullet$$

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

$$C_\bullet(X) = [ \cdots \stackrel{\partial_2}{\to} \mathbb{Z}[(Sing X)_2] \stackrel{\partial_1}{\to} \mathbb{Z}[(Sing X)_1] \stackrel{\partial_0}{\to} \mathbb{Z}[(Sing X)_0] ]$$

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

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

(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 $R[Sing X]$ 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

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

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

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

Proposition

These push-forward maps make all diagrams of the form

$$\array{ C_{n+1}(X) &\stackrel{(f_*)_{n+1}}{\to}& C_{n+1}(Y) \\ \downarrow^{\mathrlap{\partial^X_n}} && \downarrow^{\mathrlap{\partial^Y_n}} \\ C_n(X) &\stackrel{(f_*)_n}{\to}& C_n(Y) }$$

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

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

Proof

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

$$f_* : Sing X \to Sing Y \,.$$

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

Accordingly we have:

Proposition

Sending a topological space to its singular chain complex $C_\bullet(X)$, def. , and a continuous map to its push-forward chain map, prop. , constitutes a functor

$$C_\bullet(-,R) : Top \to Ch_\bullet(R Mod)$$

from the category Top to the category of chain complexes.

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

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

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

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

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

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

or graphically (using notation as for orientals)

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

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

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

Hence the boundary of the boundary is

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

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

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

Proof

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

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

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

$$\cdots \to \tilde H_{k+1}(S^n) \to \tilde H_k(S^{n-1}) \to \tilde H_k(D^n) \to \tilde H_k(S^n) \to \tilde H_{k-1}(S^{n-1}) \to \cdots \,.$$

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

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

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

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

(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

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

is an isomorphism.

In other words: the singular chain functor of prop. 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

$$\Phi : \pi_k(X,x) \to H_k(X)$$

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

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

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:

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

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

$$\Phi : \pi_1(X,x) \to H_1(X)$$

is surjective. Its kernel is the commutator subgroup of $\pi_1(X,x)$. Therefore it induces an isomorphism from the abelianization $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:

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

For higher connected $X$ we have the

Theorem

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

$$\Phi : \pi_n(X,x) \to H_n(X)$$

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

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

in Top.

Proposition

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

$$\cdots \to \tilde H_n(A) \to \tilde H_n(X) \to \tilde H_n(X/A) \to \tilde H_{n-1}(A) \to \tilde H_{n-1}(X) \to \tilde H_{n-1}(X/A) \to \cdots \,.$$

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 $H R$.

Related concepts

• Čech homology

• The dual notion is that of singular cohomology.

• The analogous notion in algebraic geometry is given by Chow groups.

• Kan-Thurston theorem

• Borel-Moore homology

References

General

Original references on chain homology/cochain cohomology and singular homology/singular cohomology:

• Henri Poincaré, Analysis Situs, Journal de l’École Polytechnique. (2). 1: 1–123 (1895) (Orig.: gallica:12148/bpt6k4337198/f7, Engl.: John Stillwell: pdf, pdf)

• Andrei Kolmogoroff, Über die Dualität im Aufbau der kombinatorischen Topologie, Recueil Mathématique 1(43) (1936), 97–102. (mathnet)

• Andrei Kolmogoroff, Homologiering des Komplexes und des lokal-bicompakten Raumes, Recueil Mathématique 1(43) (1936), 701–705. mathnet.

• J. W. Alexander, On the chains of a complex and their duals, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 (doi:10.1073/pnas.21.8.50)

• J. W. Alexander, On the ring of a compact metric space, Proc. Nat. Acad. Sci. USA, 21(1935), 511–512 (doi:10.1073/pnas.21.8.511)

• J. W. Alexander, On the connectivity ring of an abstract space, Ann. of Math., 37 (1936), 698–708 (doi:10.2307/1968484, pdf)

Lecture notes:

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

• David H. Fremlin, Singular homology for amateurs (2016) $[$pdf, 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

• Allen 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)