nLab
simplicial homology

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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.

Definition

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

Definition

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 this 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)) \,.
Remark

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.

Examples

Explicit

Example

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 (reflectind the fact that it is simply-connected);

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

General

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

Homology for spaces is discussed in chapter 2 of

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

Revised on November 27, 2012 09:29:09 by Ivanych? (178.46.126.141)