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Simplicial homology
From Wikipedia, the free encyclopedia
In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory.
It concerns topological spaces that can be triangulated. This means that each such 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 a given space.
It has been shown that all manifolds up to 3 dimensions allow for a triangulation. This, together with the fact that it is now possible to resolve the simplicial homology of a simplicial complex automatically and efficiently, make this theory feasible for application to real life situations, such as image analysis, medical imaging, and data analysis in general.
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Let S be a simplicial complex. A simplicial k-chain is a formal sum of k-simplices
.
For example σ1 + 5σ2 - 2σ3, where σ1, σ2, σ3 are k-simplices, is a simplicial k-chain. The group of all possible k-chains on S (the free abelian group defined on the set of k-simplices belonging to S) is denoted Ck. Some of the chains in Ck sum up to 0. Such chains are referred to as cycles and they form a subgroup of Ck denoted Zk. For any k-simplex,
,
the boundary operator 
is defined as
,
where the simplex
is the ith face of σ obtained by deleting its ith vertex (denoted by 
).
The definition extends to chains as well, by
.
For example, the boundary of the triangle 
is
.
The boundary of this chain is actually a cycle:
.
This example reflects the essential property of the boundary operator namely 
. It therefore makes sense to define another subgroup of chains, called boundaries, that are k-chains which form the boundary of a k+1-chain (the oriented sum of the triangle's faces in our example) denoted Bk.
The ground is now set for the definition of the kth homology group Hk of S as the quotient
.
An equivalent arguably more illuminating definition is possible if we consider that is a homomorphism (by definition). Thus if we denote
we get
.
A Homology group is not trivial if the complex at hand contains cycles which are not boundaries. This indicates that there are holes in the complex. For example the complex comprised by two triangles without their facets (area) connected at the base shown in the image (which can be thought of as the triangulation of the figure eight) contains two holes, as the edges of each triangle can form a cycle which is by construction not a boundary. Holes in general are of different dimensions, for instance in our example these are 1-holes. The dimension of a hole is given by the degree of the "missing" chain minus one (that were it to exist in the complex would have rendered the cycle in question a boundary). As a substantial amount of the information afforded by this construction is found in the rank of the homology groups, the numbers
are referred to as the Betti numbers of the space S.
As implied by the definition the Homology of a simplex is a topological invariant (i.e. is preserved by homeomorphism), and thus of the space which is triangulated. Moreover the simplicial homology of such a space is isomorphic to the singular homology of that space.
Recently there have been significant advances in the realization of simplicial homology as a viable computational tool by the introduction of persistent betti numbers. A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) in which one wishes to find hidden structure. Homology can serve as a qualitative tool to search for such structure. However, the data points have to first be triangulated (that is made into a simplicial complex). Computation of persistent homology (Edelsbrunner et. al.2002 Robins, 1999) involves analysis of homology at different resolutions, registering features (e.g. holes) that persist as the resolution is changed. Such features can be used to detect structures of molecules, tumors in X-rays, and cluster structures in complex data. A Matlab toolbox for computing persistent homology, Plex (Vin de Silva, Gunnar Carlsson), is available at this site. It should be noted that an equivalent, though more image-oriented, formulation of simplicial homology, cubical homology, has also been recently implemented.
- Lee, J.M., Introduction to Topological Manifolds, Springer-Verlag, Graduate Texts in Mathematics, Vol. 202 (2000) ISBN 0-387-98759-2
- Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.