algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
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.)
Let $X \in$ Top be topological space. Write $Sing X \in$ sSet for its singular simplicial complex.
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 $X$.
These are the chains on a simplicial set on $Sing X$.
The groups of singular chains combine to the simplicial abelian group $\mathbb{Z}[Sing X] \in Ab^{\Delta^{op}}$.
The alternating face map complex
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.
So we have
where the differentials are defined on basis elements $\sigma \in (Sing X)_n$ by
(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.)
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 $H_n(X,R)$.
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
in $Y$. This is called the push-forward of $\sigma$ along $f$. Accordingly there is a push-forward map on groups of singular chains
These push-forward maps make all diagrams of the form
commute. In other words, push-forward along $f$ constitutes a chain map
It is in fact evident that push-forward yields a functor of singular simplicial complexes
From this the statement follows since $\mathbb{Z}[-] : sSet \to sAb$ is a functor.
Accordingly we have:
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
from the category Top to the category of chain complexes.
In particular for each $n \in \mathbb{N}$ singular homology extends to a functor
Let $X$ be a topological space. Let $\sigma^1 : \Delta^1 \to X$ be a singular 1-simplex, regarded as a 1-chain
Then its boundary $\partial \sigma \in H_0(X)$ is
or graphically (using notation as for orientals)
Let $\sigma^2 : \Delta^2 \to X$ be a singular 2-chain. The boundary is
Hence the boundary of the boundary is
For more illustrations see for instance (Ghrist, (4.5)).
For all $n \in \mathbb{N}$ the reduced singular homology of the $n$-sphere $S^n$ is
The $n$-sphere may be realized as the pushout
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
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
are isomorphisms, for all $k \in \mathbb{N}$. Since
(by this example at reduced homology) the statement follows by induction on $n$.
Singular homology is homotopy invariant:
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
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).
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 $(X,x)$ a pointed topological space, the Hurewicz homomorphism is the function
from the $k$th homotopy group of $(X,x)$ to the $k$th singular homology group defined by sending
a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the fundamental class $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$.
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:
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.
For $X$ a connected topological space the Hurewicz homomorphism in degree 1
is surjective. Its kernel is the commutator subgroup of $\pi_1(X,x)$. Therefore it induces an isomorphism from the abelianization $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:
For higher connected $X$ we have the
If $X$ is (n-1)-connected for $n \geq 2$ then
is an isomorphism.
This is known as the Hurewicz theorem.
For the present purpose one makes the following definition.
A topological subspace inclusion $A \hookrightarrow X$ in Top is called a good pair if
$A$ 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
in Top.
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
For instance (Hatcher, theorem 2.13).
Singular homology computes the generalized homology with coefficients in the Eilenberg-MacLane spectrum $H \mathbb{Z}$ or $H R$.
The dual notion is that of singular cohomology.
The analogous notion in algebraic geometry is given by Chow groups.
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
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
Last revised on May 22, 2022 at 15:04:19. See the history of this page for a list of all contributions to it.