Back Simpliziale Homologie German Homología simplicial Spanish 単体的ホモロジー Japanese 단체 호몰로지 Korean Симплициальные гомологии Russian Симпліційна гомологія Ukrainian
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).[1]: sec.5.3.2
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.[2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.
- ^ Prasolov, V. V. (2006), Elements of combinatorial and differential topology, American Mathematical Society, ISBN 0-8218-3809-1, MR 2233951
- ^ Armstrong, M. A. (1983), Basic topology, Springer-Verlag, ISBN 0-387-90839-0, MR 0705632