Noun
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.
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. Source: Internet
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). Source: Internet
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. Source: Internet
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. Source: Internet
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. Source: Internet