In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead). Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another.
Definitions
Orientations A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v0, ..., vk), with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean.
Chains Let S be a simplicial complex. A simplicial k-chain is a finite formal sum
∑ i = 1 N c i σ i , {\displaystyle \sum _{i=1}^{N}c_{i}\sigma _{i},\,}
where each ci is an integer and σi is an oriented k-simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example,
( v 0 , v 1 ) = − ( v 1 , v 0 ) . {\displaystyle (v_{0},v_{1})=-(v_{1},v_{0}).}
The group of k-chains on S is written Ck. This is a free abelian group which has a basis in one-to-one correspondence with the set of k-simplices in S. To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.
Boundaries and cycles Let σ = (v0, ..., vk) be an oriented k-simplex, viewed as a basis element of Ck. The boundary operator
∂ k : C k → C k − 1 {\displaystyle \partial _{k}:C_{k}\rightarrow C_{k-1}}
is the homomorphism defined by:
∂ k ( σ ) = ∑ i = 0 k ( − 1 ) i ( v 0 , … , v i ^ , … , v k ) , {\displaystyle \partial _{k}(\sigma )=\sum _{i=0}^{k}(-1)^{i}(v_{0},\dots ,{\widehat {v_{i}}},\dots ,v_{k}),}
where the oriented simplex
( v 0 , … , v i ^ , … , v k ) {\displaystyle (v_{0},\dots ,{\widehat {v_{i}}},\dots ,v_{k})}
is the ith face of σ, obtained by deleting its ith vertex. In Ck, elements of the subgroup
Z k := ker ∂ k {\displaystyle Z_{k}:=\ker \partial _{k}}
are referred to as cycles, and the subgroup
B k := im ∂ k + 1 {\displaystyle B_{k}:=\operatorname {im} \partial _{k+1}}
is said to consist of boundaries.
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