Simplicial homology
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).
Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead).
It is a remarkable fact that simplicial homology only depends on the associated topological space.
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.
A simplicial k-chain is a finite formal sum where each ci is an integer and σi is an oriented k-simplex.
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.
Let σ = (v0,...,vk) be an oriented k-simplex, viewed as a basis element of Ck.
Equivalently, the abelian groups form a chain complex.
For example, consider the complex S obtained by gluing two triangles (with no interior) along one edge, shown in the image.
One can compute that the homology group H1(S) is isomorphic to Z2, with a basis given by the two cycles mentioned.
Indeed, all three vertices become equal in the quotient group; this expresses the fact that S is connected.
The construction of the homology groups of a tetrahedron is described in detail here.
In general, if S is a d-dimensional simplex, the following holds: Let S and T be simplicial complexes.
A simplicial map f from S to T is a function from the vertex set of S to the vertex set of T such that the image of each simplex in S (viewed as a set of vertices) is a simplex in T. A simplicial map f: S → T determines a homomorphism of homology groups Hk(S) → Hk(T) for each integer k. This is the homomorphism associated to a chain map from the chain complex of S to the chain complex of T. Explicitly, this chain map is given on k-chains by if f(v0), ..., f(vk) are all distinct, and otherwise f((v0, ..., vk)) = 0.
This is essential to applications of the theory, including the Brouwer fixed point theorem and the topological invariance of simplicial homology.
Singular homology is defined for all topological spaces and depends only on the topology, not any triangulation; and it agrees with simplicial homology for spaces which can be triangulated.
[4]: thm.2.27 Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, and data analysis in general.
A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.)
Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex.
More generally, simplicial homology plays a central role in topological data analysis, a technique in the field of data mining.
