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Simplicial Complex

SimplicialComplex

A simplicial complex is a space with a triangulation. Formally, a simplicial complex K in R^n is a collection of simplices in R^n such that

1. Every face of a simplex of K is in K, and

2. The intersection of any two simplices of K is a face of each of them

(Munkres 1993, p. 7).

Objects in the space made up of only the simplices in the triangulation of the space are called simplicial subcomplexes. When only simplicial complexes and simplicial subcomplexes are considered, defining homology is particularly easy (and, in fact, combinatorial because of its finite/counting nature). This kind of homology is called simplicial homology.

See also

Abstract Simplicial Complex, Boundary Complex, Homology, Induced Subcomplex, Neighborly Triangulation, Nerve, Simplex, Simplicial Homology, Simplicial Set, Simplicial Subcomplex, Space, Tight Triangulation, Triangulation

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References

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 7, 1994.Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.Knill, O. "The Amazing World of Finite Abstract Simplicial Complexes." Apr. 23, 2018. https://www.youtube.com/watch?v=W9CWvUiycoc.Munkres, J. R. "Simplicial Complexes and Simplicial Maps." §1.2 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 7-14, 1993.

Referenced on Wolfram|Alpha

Simplicial Complex

Cite this as:

Weisstein, Eric W. "Simplicial Complex." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SimplicialComplex.html

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