- 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 "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 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
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