Simplicial map

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

A simplicial approximation is homotopic to the map it approximates.

References

• Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2.

See also

• Simplicial approximation theorem
• Simplicial set