Simplicial homology

Simplicial homology

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.


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 ight angle.

The boundary operator

:partial_k: C_k ightarrow 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 ight angle ,

where the simplex :left langle v^0 , ... , hat{v}^i , ... ,v^k ight angle

is the "i"th face of "σ" obtained by deleting its "i"th 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.

The "k"th 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

:eta_k = { m 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".

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.


*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.

External links

* [ Topological methods in scientific computing]
* [ Computational homology (also cubical homology)]

ee also

*Homology theory
*Singular homology
*Cellular homology

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