nLab singular homology




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

a singular nn-chain on XX is a formal linear combination of singular simplices σ:Δ nX\sigma : \Delta^n \to X, and a singular nn-cycle is such a chain such that its oriented boundary in XX vanishes. Two singular chains are homologous if they differ by a boundary. The singular homology of XX in degree nn is the group of nn-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 Δ topX\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.)


Let XX \in Top be topological space. Write SingXSing X \in sSet for its singular simplicial complex.


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

a formal linear combinations of singular simplices in XX.


These are the chains on a simplicial set on SingXSing X.

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


The alternating face map complex

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

is the singular complex of XX.

Its chain homology is the ordinary singular homology of XX.

One usually writes H n(X,)H_n(X, \mathbb{Z}) or just H n(X)H_n(X) for the singular homology of XX in degree nn. See also at ordinary homology.


So we have

C (X)=[ 2[(SingX) 2] 1[(SingX) 1] 0[(SingX) 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 σ(SingX) n\sigma \in (Sing X)_n by

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

(with d id_i the ii 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.)


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 RR any unital ring one can form the degreewise free module R[SingX]R[Sing X] over RR. The corresponding homology is the singular homology with coefficients in RR, denoted H n(X,R)H_n(X,R).


Given a continuous map f:XYf : X \to Y between topological spaces, and given nn \in \mathbb{N}, every singular nn-simplex σ:Δ nX\sigma : \Delta^n \to X in XX is sent to a singular nn-simplex

f *σ:Δ nσXfY f_* \sigma : \Delta^n \stackrel{\sigma}{\to} X \stackrel{f}{\to} Y

in YY. This is called the push-forward of σ\sigma along ff. Accordingly there is a push-forward map on groups of singular chains

(f *) n:C n(X)C n(Y). (f_*)_n : C_n(X) \to C_n(Y) \,.

These push-forward maps make all diagrams of the form

C n+1(X) (f *) n+1 C n+1(Y) n X n Y C n(X) (f *) n C n(Y) \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 ff constitutes a chain map

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

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

f *:SingXSingY. f_* : Sing X \to Sing Y \,.

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

Accordingly we have:


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

C (,R):TopCh (RMod) C_\bullet(-,R) : Top \to Ch_\bullet(R Mod)

from the category Top to the category of chain complexes.

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

H n(,R):TopRMod. H_n(-,R) : Top \to R Mod \,.


Basic examples


Let XX be a topological space. Let σ 1:Δ 1X\sigma^1 : \Delta^1 \to X be a singular 1-simplex, regarded as a 1-chain

σ 1C 1(X). \sigma^1 \in C_1(X) \,.

Then its boundary σH 0(X)\partial \sigma \in H_0(X) is

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

or graphically (using notation as for orientals)

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

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

( σ(1) σ(0,1) σ σ 1,2 σ(0) σ(0,2) σ(2))=( σ(1) σ(0,1) σ(0))( σ(0) σ(0,2) σ(2))+( σ(1) σ 1,2 σ(2)). \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

σ =(( σ(1) σ(0,1) σ(0))( σ(0) σ(0,2) σ(2))+( σ(1) σ 1,2 σ(2))) =( σ(0))( σ(1) )( σ(0) )+( σ(2))+( σ(1) )( σ(2)) =0 \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


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

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

The nn-sphere may be realized as the pushout

S nD n/S n1D n S n1* S^n \simeq D^n/S^{n-1} \coloneqq D^{n} \coprod_{S^{n-1}} *

which is the nn-ball with its boundary (n1)(n-1)-sphere identified with the point. The inclusion S n1D nS^{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

H˜ k+1(S n)H˜ k(S n1)H˜ k(D n)H˜ k(S n)H˜ k1(S n1). \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)0H_k(D^n) \simeq 0 for all k,nk,n by this example at reduced homology. This means that in the above long exact sequence all the morphisms

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

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

H˜ n(S 0){ ifn=0 0 otherwise \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 nn.


Homotopy invariance

Singular homology is homotopy invariant:


If f:XYf : 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)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. 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.


(Hurewicz homomorphism)

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

Φ:π k(X,x)H k(X) \Phi : \pi_k(X,x) \to H_k(X)

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

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

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


For XX a topological space the Hurewicz homomorphism in degree 0 exhibits an isomorphism between the free abelian group [π 0(X)]\mathbb{Z}[\pi_0(X)] on the set of connected components of XX and the degree-0 singular homlogy:

[π 0(X)]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 XX is connected.


For XX a connected topological space the Hurewicz homomorphism in degree 1

Φ:π 1(X,x)H 1(X) \Phi : \pi_1(X,x) \to H_1(X)

is surjective. Its kernel is the commutator subgroup of π 1(X,x)\pi_1(X,x). Therefore it induces an isomorphism from the abelianization π 1(X,x) abπ 1(X,x)/[π 1,π 1]\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]:

π 1(X,x) abH 1(X). \pi_1(X,x)^{ab} \stackrel{\simeq}{\to} H_1(X) \,.

For higher connected XX we have the


If XX is (n-1)-connected for n2n \geq 2 then

Φ:π n(X,x)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.


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

  1. AA is inhabited and closed in XX;

  2. AA has a neighbourhood in XX of which it is a deformation retract.

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

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

in Top.


If AXA \hookrightarrow X is a good pair, def. , then the singular homology of X/AX/A coincides with the relative homology of XX relative to AA. In particular, therefore, it fits into a long exact sequence of the form

H˜ n(A)H˜ n(X)H˜ n(X/A)H˜ n1(A)H˜ n1(X)H˜ n1(X/A). \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 HH \mathbb{Z} or HRH R.



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

Lecture notes:

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

See also

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)

Last revised on May 22, 2022 at 15:04:19. See the history of this page for a list of all contributions to it.