nLab simplicial homology

Idea

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

Definition

Let $S$ be a simplicial set and $A$ an abelian group.

Definition

For $n \in \mathbb{N}$ write

$$S_n \cdot A \coloneqq \mathbb{Z}[S_n] \otimes A$$

for the free abelian group on the set $S_n$ of $n$-simplices tensored with $A$: the group of formal linear combinations of $n$-simplices with coefficients in $A$.

These abelian groups arrange to a simplicial abelian group

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

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

$$C_\bullet(S \cdot A) = C_\bullet(S,A) \,.$$

The simplicial homology of $S$ is the chain homology of the complex of simplicial chains:

$$H_\bullet(S, A) \coloneqq H_\bullet(C_\bullet(S,A)) \,.$$

Remark

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

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

where $d_k : S_n \to S_{n-1}$ are the face maps of $S$.

Examples

Explicit

Example

Let $S = \partial \Delta^3$ 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

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

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

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

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

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

General

• For $X$ a topological space and $S = Sing X$ the singular simplicial complex of $X$, the simplicial homology of $Sing X$ is called the singular homology of $X$, denoted $H_\bullet(X,A)$.

Related concepts

• cochain on a simplicial set

• singular homology

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.

References

A basic discussion is for instance around application 1.1.3 of

• Charles Weibel. An Introduction to Homological Algebra

Homology for spaces is discussed in chapter 2 of

• Alan Hatcher, Algebraic topology, Cambridge Univ. Press 2002, starting p. 106

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