nLab simplicial homology



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




For SS a simplicial set and AA an abelian group, the simplicial homology of SS is the chain homology of the chain complex corresponding under the Dold-Kan correspondence to the simplicial abelian group SAS \cdot A of AA-chains on SS: formal linear combinations of simplices in SS with coefficients in AA.


Let SS be a simplicial set and AA an abelian group.


For nn \in \mathbb{N} write

S nA[S n]A S_n \cdot A \coloneqq \mathbb{Z}[S_n] \otimes A

for the free abelian group on the set S nS_n of nn-simplices tensored with AA: the group of formal linear combinations of nn-simplices with coefficients in AA.

These abelian groups arrange to a simplicial abelian group

XAAb Δ op. X \cdot A \in Ab^{\Delta^{op}} \,.

The alternating face map complex of these groups is called the complex of simplicial chains on SS

C (SA)=C (S,A). C_\bullet(S \cdot A) = C_\bullet(S,A) \,.

The simplicial homology of SS is the chain homology of the complex of simplicial chains:

H (S,A)H (C (S,A)). H_\bullet(S, A) \coloneqq H_\bullet(C_\bullet(S,A)) \,.

This means that the differentials in C (S,A)C_\bullet(S,A) are given on basis elements σS n\sigma \in S_n by the formal linear combination

σ= k=0 n(1) kd kσ, \partial \sigma = \sum_{k = 0}^{n} (-1)^k d_k \sigma \,,

where d k:S nS n1d_k : S_n \to S_{n-1} are the face maps of SS.




Let S=Δ 3S = \partial \Delta^3 be the boundary of the simplicial 3-simplex, the (hollow) simplicial tetrahedron.

Since this has

  • 4 non-degenerate vertices

  • 6 non-degenerate edges

  • 4 non-degenerate faces

the normalized chain complex of \mathbb{Z} is of the form

00 4 6 40. \cdots \to 0 \to 0 \to \mathbb{Z}^4 \to \mathbb{Z}^6 \to \mathbb{Z}^4 \to 0 \,.

By writing out the two non-trivial differentials, one can deduce explicitly that

  • H 0(Δ 3)=H_0(\partial \Delta^3) = \mathbb{Z} (reflecting the fact that the tetrahedron is connected);

  • H 1(Δ 3)=0H_1(\partial \Delta^3) = 0 (reflecting the fact that it is simply-connected);

  • H 2(Δ 3)=H_2(\partial \Delta^3) = \mathbb{Z} (reflecting the fact, by the Hurewicz theorem, that the second homotopy group of the 2-sphere is \mathbb{Z} );


Terminology and a bit of history

The term simplicial homology is also used in the literature for the homology of polyhedral spaces, based on the theory of simplicial complexes. That homology is defined by first looking at a chain complex of simplicial chains on, say, a triangulation of a space, and then passing to the corresponding homology. The theory then proceeds by proving that the end result is independent of the triangulation used. The resulting homology theory is isomorphic to singular homology, but historically was the earlier theory.


A basic discussion is for instance around application 1.1.3 of

Homology for spaces is discussed in chapter 2 of

and this includes a discussion of the homology of simplicial complexes.

Last revised on June 7, 2019 at 19:03:46. See the history of this page for a list of all contributions to it.