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A simplicial 3-complex. In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in the set (see illustration).
An abstract simplicial complex (ASC) is family of sets that is closed under taking subsets (the subset of a set in the family is also a set in the family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC.
Indeed it can be shown that for any subdivision ′ of a finite simplicial complex there is a unique sequence of maps between the homology groups : (′) such that for each in the maps fulfills () and such that the maps induces endomorphisms of chain complexes. Moreover, the induced map is an isomorphism: Subdivision does not change the ...
For the affine building, an apartment is a simplicial complex tessellating Euclidean space E n−1 by (n − 1)-dimensional simplices; while for a spherical building it is the finite simplicial complex formed by all (n − 1)! simplices with a given common vertex in the analogous tessellation in E n−2.
A d-dimensional simplicial complex is called pure if its maximal simplices all have dimension d. Let Δ {\displaystyle \Delta } be a finite or countably infinite simplicial complex. An ordering C 1 , C 2 , … {\displaystyle C_{1},C_{2},\ldots } of the maximal simplices of Δ {\displaystyle \Delta } is a shelling if, for all k = 2 , 3 ...
Via triangulation, one can assign a chain complex to topological spaces that arise from its simplicial complex and compute its simplicial homology. Compact spaces always admit finite triangulations and therefore their homology groups are finitely generated and only finitely many of them do not vanish.
Determining whether two finite simplicial complexes are homeomorphic. Determining whether a finite simplicial complex is (homeomorphic to) a manifold. Determining whether the fundamental group of a finite simplicial complex is trivial. Determining whether two non-simply connected 5-manifolds are homeomorphic, or if a 5-manifold is homeomorphic ...
An example is the chain complex defining the simplicial homology of a finite simplicial complex. A chain complex is bounded above if all modules above some fixed degree N are 0, and is bounded below if all modules below some fixed degree are 0. Clearly, a complex is bounded both above and below if and only if the complex is bounded.