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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).
The universal covering space of a finite connected simplicial complex X can also be described directly as a simplicial complex using edge-paths. Its vertices are pairs (w,γ) where w is a vertex of X and γ is an edge-equivalence class of paths from v to w.
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.
A simplicial proper action of a discrete group Γ on a simplicial complex X with finite quotient is said to be regular if it satisfies one of the following equivalent conditions: [9] X admits a finite subcomplex as fundamental domain; the quotient Y = X/Γ has a natural simplicial structure;
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.
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 ...
A d-dimensional simplicial complex is called pure if its maximal simplices all have dimension d.Let be a finite or countably infinite simplicial complex. An ordering ,, … of the maximal simplices of is a shelling if, for all =,, …, the complex
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.