Euler characteristic
Noun
Meaning
Euler characteristic (plural Euler characteristics)
(topology, of a topological space) The sum of even-dimensional Betti numbers minus the sum of odd-dimensional ones.
A polygon's or polyhedron's Euler characteristic is just the number of corners minus the number of edges plus the number of faces.
Examples
Coxeter (1973) Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the development of topology and the treatment of a decomposition or CW-complex as analogous to a polytope.
His more significant results include: * The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one).
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology.
For a complete list of the Greek numeral prefixes see Numeral prefixes>Table of number prefixes in English>Greek>Quantitative Topological characteristics The topological class of a polyhedron is defined by its Euler characteristic and orientability.
This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.