(also nonabelian homological algebra)
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For a simplicial set and an abelian group, the simplicial homology of is the chain homology of the chain complex corresponding under the Dold-Kan correspondence to the simplicial abelian group of -chains on : formal linear combinations of simplices in with coefficients in .
Let be a simplicial set and an abelian group.
For write
for the free abelian group on the set of -simplices tensored with : the group of formal linear combinations of -simplices with coefficients in .
These abelian groups arrange to a simplicial abelian group
The alternating face map complex of these groups is called the complex of simplicial chains on
The simplicial homology of is the chain homology of the complex of simplicial chains:
This means that the differentials in are given on basis elements by the formal linear combination
where are the face maps of .
Let 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 is of the form
By writing out the two non-trivial differentials, one can deduce explicitly that
(reflecting the fact that the tetrahedron is connected);
(reflecting the fact that it is simply-connected);
(reflecting the fact, by the Hurewicz theorem, that the second homotopy group of the 2-sphere is );
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.