Simplicial map

Definition and observations

In 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 maps defined on a finite number of points, and there exists a finite amount of them which can be defined between any two given 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 $st(v)$ for the star of a vertex. A simplicial map $f_{\triangle }:K\rightarrow L$ such that $f(st(v))\subseteq st(f_{\triangle }(v))$, is called a simplicial approximation to $f$.

A simplicial approximation is homotopic to the map it approximates.

References

• Munkres, James R.: Elements of Algebraic Topology, Westview Press, 1995. ISBN 978-0201627282.

See also

• Simplicial approximation theorem