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- Q7520902 subject Q11711429.
- Q7520902 subject Q8521233.
- Q7520902 subject Q8800142.
- Q7520902 abstract "In algebraic topology, simplicial homology formalizes the idea of the number of holes of a given dimension in a simplicial complex. This generalizes the number of connected components (the case of dimension 0).Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-dimensional simplex), a line segment (1-dimensional simplex), a triangle (2-dimensional simplex) and a tetrahedron (3-dimensional simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). 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, by Cairns and Whitehead.Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another.Singular homology is a related theory which is more commonly used by mathematicians today. Singular homology is defined for all topological spaces, and it agrees with simplicial homology for spaces which can be triangulated. Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, and data analysis in general.".
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- Q7520902 comment "In algebraic topology, simplicial homology formalizes the idea of the number of holes of a given dimension in a simplicial complex. This generalizes the number of connected components (the case of dimension 0).Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles.".
- Q7520902 label "Simplicial homology".
- Q7520902 depiction Simplicial_homology_-_exactness_of_boundary_maps.svg.