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 $f: |K| \rightarrow |L|$ 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

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

See also

• Simplicial approximation theorem