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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).
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 ...
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.
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.
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.
A set-family Δ is called an abstract simplicial complex if, for every set X in Δ, and every non-empty subset Y ⊆ X, the set Y also belongs to Δ. The finite sets that belong to Δ are called faces of the complex, and a face Y is said to belong to another face X if Y ⊆ X , so the definition of an abstract simplicial complex can be restated ...
A polyhedral space is a (usually finite) simplicial complex in which every simplex has a flat metric. (Other spaces of interest are spherical and hyperbolic polyhedral spaces, where every simplex has a metric of constant positive or negative curvature). In the sequel all polyhedral spaces are taken to be Euclidean polyhedral spaces.
In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces , such that