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In the mathematical discipline of [[simplicial homology]] theory, a '''simplicial map''' is a [[Map (mathematics)|map]] between [[simplicial complex]]es 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 verticesIn 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 coordinate system (mathematics)|barycentric coordinates]].
 
Simplicial maps which are [[bijection|bijective]] are called ''simplicial [[isomorphism]]s''.
 
==Simplicial approximation==
Let <math>f: |K| \rightarrow |L|</math> be a continuous map between the underlying polyhedra of simplicial complexes and let us write <math>\text{st}(v)</math> for the [[simplicial complex|star]] of a vertex. A simplicial map <math>f_\triangle :K \rightarrow L</math> such that <math>f(\text{st}(v)) \subseteq \text{st}(f_\triangle (v))</math>, is called a '''simplicial approximation''' to <math>f</math>.
 
A simplicial approximation is [[homotopy|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]]
 
[[Category:Algebraic topology]]

Latest revision as of 05:31, 10 January 2013

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 be a continuous map between the underlying polyhedra of simplicial complexes and let us write for the star of a vertex. A simplicial map such that , is called a simplicial approximation to .

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