- Simplicial homology
In

mathematics , in the area ofalgebraic topology ,**simplicial homology**is a theory with afinitary definition, and is probably the most tangible variant ofhomology theory .Simplicial homology concerns

topological spaces whose building blocks are "n"-simplex es, the "n"-dimensional analogs of triangles. By definition, such a space ishomeomorphic to asimplicial complex (more precisely, thegeometric realization of anabstract 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 givessingular homology . The simplicial homology of a simplicial complex is naturally isomorphic to thesingular 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

manifold s 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 asimage analysis ,medical imaging , anddata analysis in general.**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 "C_{k}".Consider a basis element of "C

_{k}", 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 "C

_{k}", 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 "B

_{k}" lies in "Z_{k}". 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 "H_{k}" of "S" is defined to be the quotient:$H\_k(S)\; =\; Z\_k/B\_k,\; .$

A homology group "H

_{k}" 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 ofpersistent homology ( [*http://graphics.stanford.edu/projects/lgl/paper.php?id=elz-tps-02 Edelsbrunner et. al.2002*] [*http://at.yorku.ca/b/a/a/k/28.htm 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. AMatlab toolbox for computing persistent homology, Plex (Vin de Silva ,Gunnar Carlsson ), is available at [*http://math.stanford.edu/comptop/programs/ this site*] . It should be noted that an equivalent, though more image-oriented, formulation of simplicial homology,cubical homology , has also been recently implemented.**References***Lee, J.M., "Introduction to Topological Manifolds",

Springer-Verlag , Graduate Texts in Mathematics, Vol. 202 (2000) ISBN 0-387-98759-2

*Hatcher, A., " [*http://www.math.cornell.edu/~hatcher/AT/ATchapters.html 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*** [

*http://math.stanford.edu/comptop/ Topological methods in scientific computing*]

* [*http://www.math.gatech.edu/~chomp/ Computational homology (also cubical homology)*]**ee also***

Homology theory

*Singular homology

*Cellular homology

*Wikimedia Foundation.
2010.*

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