Euler characteristic


The term Euler characteristicMathworldPlanetmath is defined for several objects.

If K is a finite simplicial complexMathworldPlanetmath of dimensionMathworldPlanetmath m, let αi be the number of simplexes of dimension i. The Euler characteristic of K is defined to be

χ(K)=i=0m(-1)iαi.

Next, if K is a finite CW complex, let αi be the number of i-cells in K. The Euler characteristic of K is defined to be

χ(K)=i0(-1)iαi.

If X is a finite polyhedron, with triangulation K, a simplicial complex, then the Euler characteristic of X is χ(K). It can be shown that all triangulations of X have the same value for χ(K) so that this is well-defined.

Finally, if C={Cq} is a finitely generatedMathworldPlanetmath graded group, then the Euler characteristic of C is defined to be

χ(C)=q0(-1)qrank(Cq).
Title Euler characteristic
Canonical name EulerCharacteristic
Date of creation 2013-03-22 16:12:51
Last modified on 2013-03-22 16:12:51
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 13
Author Mathprof (13753)
Entry type Definition
Classification msc 55N99