Simplicial homology

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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.

Simplicial homology concerns topological spaces whose building blocks are n-simplexes, the n-dimensional analogs of triangles. 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. Replacing n-simplexes by their continuous images in a given topological space gives singular homology. The simplicial homology of a simplicial complex is naturally isomorphic to the singular homology of its geometric realization. This implies, in particular, that the simplicial homology of a space does not depend on the triangulation chosen for the 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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[edit] Definition

Let S be a simplicial complex. A simplicial k-chain is a formal sum of k-simplices

\sum_{i=1}^N c_i \sigma^i \,.

The group of k-chains on S, the free abelian group defined on the set of k-simplices in S, is denoted Ck.

Consider a basis element of Ck, a k-simplex,

\sigma = \left \langle v^0 , v^1 , ... ,v^k\right \rangle.

The boundary operator

\partial_k: C_k \rightarrow C_{k-1}

is a homomorphism defined by:

\partial_k(\sigma)=\sum_{i=0}^K (-1)^i \left \langle v^0 , ... , \hat{v}^i , ... ,v^k\right \rangle ,

where the simplex

\left \langle v^0 , ... , \hat{v}^i , ... ,v^k\right \rangle

is the ith face of σ obtained by deleting its ith vertex.

In Ck, elements of the subgroup

Z_k = \ker \partial_k

are referred to as cycles, and the subgroup

B_k = \operatorname{im} \partial_{k+1}

is said to consist of boundaries.

Direct computation shows that Bk lies in Zk. The boundary of a boundary must be a cycle. In other words,

(C_k, \partial_k)

form a simplicial chain complex.

A simplicial complex with 2 1-holes
A simplicial complex with 2 1-holes

The kth homology group Hk of S is defined to be the quotient

H_k(S) = Z_k/B_k\, .

A homology group Hk is not trivial if the complex at hand contains k-cycles which are not boundaries. This indicates that there are k-dimensional holes in the complex. For example consider the complex obtained by glueing two triangles (with no interior) along one egde, shown in the image. This is a triangulation of the figure eight. The edges of each triangle form a cycle. These two cycles are by construction not boundaries (there are no 2-chains). Therefore the figure has two "1-holes".

Holes can be of different dimensions. The rank of the homology groups, the numbers

\beta_k = {\rm rank} (H_k(S))\,

are referred to as the Betti numbers of the space S, and gives a measure of the number of k-dimensional holes in S.

[edit] Numerical implementation and application

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.

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