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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.
The barycentric subdivision is an operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations: The substitution allows to assign combinatorial invariants as the Euler characteristic to the spaces.
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 Whitney triangulation or clean triangulation of a two-dimensional manifold is an embedding of a graph G onto the manifold in such a way that every face is a triangle and every triangle is a face. If a graph G has a Whitney triangulation, it must form a cell complex that is isomorphic to the Whitney complex of G.
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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.