Simplicial map

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In the mathematical discipline of simplicial homology theory, a simplicial map is a map between simplicial complexes with the property that the images of the vertices of a simplex always span a simplex. Note that this implies that vertices have vertices for images.

Simplicial maps are thus determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes.

Simplicial maps induce continuous maps between the underlying polyhedra of the simplicial complexes: one simply extends linearly using barycentric coordinates.

Simplicial maps which are bijective are called simplicial isomorphisms.

Simplicial approximation

Let f:|K|\rightarrow |L| be a continuous map between the underlying polyhedra of simplicial complexes and let us write {\text{st}}(v) for the star of a vertex. A simplicial map f_{\triangle }:K\rightarrow L such that f({\text{st}}(v))\subseteq {\text{st}}(f_{\triangle }(v)), is called a simplicial approximation to f.

A simplicial approximation is homotopic to the map it approximates.

References

See also

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