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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.
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).
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.
To do so with a simplicial complex, we need at least three vertices, and edges connecting them. But delta-sets allow for a simpler triangulation: thinking of S 1 {\displaystyle S^{1}} as the interval [0,1] with the two endpoints identified, we can define a triangulation with a single vertex 0, and a single edge looping between 0 and 0.
A triangulation of a set of points in the Euclidean space is a simplicial complex that covers the convex hull of , and whose vertices belong to . [1] In the plane (when P {\displaystyle {\mathcal {P}}} is a set of points in R 2 {\displaystyle \mathbb {R} ^{2}} ), triangulations are made up of triangles, together with their edges and vertices.
Let + be the (+)-simplex. + is a combinatorial n-sphere with its triangulation as the boundary of the n+1-simplex. Given a triangulated piecewise linear (PL) n-manifold , and a co-dimension 0 subcomplex together with a simplicial isomorphism : ′ +, the Pachner move on N associated to C is the triangulated manifold () (+ ′).
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Every flag complex is a clique complex: given a flag complex, define a graph G on the set of all vertices, where two vertices u,v are adjacent in G iff {u,v} is in the complex (this graph is called the 1-skeleton of the complex). By definition of a flag complex, every set of vertices that are pairwise-connected, is in the complex.