Алгебралық топологияда симплекстік гомология

Concept in algebraic topology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n simplices, the n dimensional analogs of triangles. This includes a point (0 simplex), a line segment (1 simplex), a triangle (2 simplex) and a tetrahedron (3 simplex). 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. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead). Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another.

A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k simplex is given by an ordering of the vertices, written as (v0, ,vk), with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1 simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2 simplex amounts to choosing what "counterclockwise" should mean.

Let S be a simplicial complex. A simplicial k chain is a finite formal sum

where each ci is an integer and σi is an oriented k simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example,

The group of k chains on S is written Ck. This is a free abelian group which has a basis in one to one correspondence with the set of k simplices in S. To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.

Because , where is the second face removed, In geometric terms, this says that the boundary of a boundary of anything has no boundary. Equivalently, the abelian groups

form a chain complex. Another equivalent statement is that Bk is contained in Zk. As an example, consider a tetrahedron with vertices oriented as w,x,y,z. By definition, its boundary is given by: xyz wyz + wxz wxy. The boundary of the boundary is given by: 1=(yz xz+xy) (yz wz+wy)+(xz wz+wx) (xy wy+wx) = 0.

Let S and T be simplicial complexes. A simplicial map f from S to T is a function from the vertex set of S to the vertex set of T such that the image of each simplex in S (viewed as a set of vertices) is a simplex in T. A simplicial map f: S → T determines a homomorphism of homology groups Hk(S) → Hk(T) for each integer k. This is the homomorphism associated to a chain map from the chain complex of S to the chain complex of T. Explicitly, this chain map is given on k chains by

if f(v0), , f(vk) are all distinct, and otherwise 1=f((v0, , vk)) = 0. This construction makes simplicial homology a functor from simplicial complexes to abelian groups. This is essential to applications of the theory, including the Brouwer fixed point theorem and the topological invariance of simplicial homology.

Singular homology is a related theory that is better adapted to theory rather than computation. Singular homology is defined for all topological spaces and depends only on the topology, not any triangulation; and it agrees with simplicial homology for spaces which can be triangulated. Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real life situations, such as image analysis, medical imaging, and data analysis in general. Another related theory is Cellular homology.

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 a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex. However, the data points have to first be triangulated, meaning one replaces the data with a simplicial complex approximation. Computation of persistent homology involves analysis of homology at different resolutions, registering homology classes (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. More generally, simplicial homology plays a central role in topological data analysis, a technique in the field of data mining.

Exact, efficient, computation of simplicial homology of large simplical complexes can be computed using the GAP Simplicial Homology. A MATLAB toolbox for computing persistent homology, Plex (Vin de Silva, Gunnar Carlsson), is available at this site. Stand alone implementations in C++ are available as part of the Perseus, Dionysus and PHAT software projects. For Python, there are libraries such as scikit tda, Persim, giotto tda and GUDHI, the latter aimed at generating topological features for machine learning. These can be found at the PyPI repository.