# Euler characteristic

﻿
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological space's shape or structure. It is commonly denoted by $chi$ (Greek letter chi).

The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. Leonhard Euler, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic arises from homology and connects to many other invariants.

Polyhedra

The Euler characteristic $chi$ was classically defined for the surfaces of polyhedra, according to the formula

:$chi=V-E+F ,!$

where "V", "E", and "F" are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any "convex" polyhedron's surface has Euler characteristic

:$chi = V - E + F = 2. ,!$

This result is known as Euler's formula. A proof is given below.

Any contractible space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore its Euler characteristic is 1. This case includes Euclidean space $mathbb\left\{R\right\}^n$ of any dimension, as well as the solid unit ball in any Euclidean space &mdash; the one-dimensional interval, the two-dimensional disk, the three-dimensional ball, etc.

The "n"-dimensional sphere has Betti number 1 in dimensions 0 and "n", and all other Betti numbers 0. Hence its Euler characteristic is $1 + \left(-1\right)^n$ &mdash; that is, either 0 or 2.

The "n"-dimensional real projective space is the quotient of the "n"-sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere &mdash; either 0 or 1.

The "n"-dimensional torus is the product space of "n" circles. Its Euler characteristic is 0, by the product property.

Generalizations

More generally, one can define the Euler characteristic of any chain complex to be the alternating sum of the ranks of the homology groups of the chain complex.

A version used in algebraic geometry is as follows. For any sheaf $scriptstylemathcal\left\{F\right\}$ on a projective scheme "X", one defines its Euler characteristic:$scriptstylechi \left( mathcal\left\{F\right\}\right)= Sigma \left(-1\right)^i h^i\left(X,mathcal\left\{F\right\}\right)$,where $scriptstyle h^i\left(X, mathcal\left\{F\right\}\right)$ is the dimension of the "i"th sheaf cohomology group of $scriptstylemathcal\left\{F\right\}$.

Another generalization of the concept of Euler characteristic on manifolds comes from orbifolds. While every manifold has an integer Euler characteristic, an orbifold can have a fractional Euler characteristic. For example, the teardrop orbifold has Euler characteristic 1 + 1/"p", where "p" is a prime number corresponding to the cone angle 2"&pi;" / "p".

The concept of Euler characteristic of a bounded finite poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and largest elements; call them 0 and 1. The Euler characteristic of such a poset is defined as μ(0,1), where μ is the Möbius function in that poset's incidence algebra.

* List of uniform polyhedra
* List of topics named after Leonhard Euler

References

*Mathworld | urlname=EulerCharacteristic | title=Euler characteristic
*Mathworld | urlname=PolyhedralFormula | title=Polyhedral formula

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Euler characteristic — ▪ mathematics       in mathematics, a number, C, that is a topological characteristic of various classes of geometric figures based only on a relationship between the numbers of vertices (V), edges (E), and faces (F) of a geometric figure. This… …   Universalium

• Euler characteristic — noun The sum of even dimensional Betti numbers minus the sum of odd dimensional ones. A polygon or polyhedrons Euler characteristic is just the number of corners minus the number of edges plus the number of faces …   Wiktionary

• Characteristic — (from the Greek word for a property or attribute (= trait) of an entity) may refer to: In physics and engineering, any characteristic curve that shows the relationship between certain input and output parameters, for example: I V or current… …   Wikipedia

• Euler class — In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how quot;twisted quot; the vector bundle… …   Wikipedia

• Characteristic class — In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is twisted particularly, whether it possesses… …   Wikipedia

• Euler boolean operation — An Euler Boolean operation is a series of modifications to solid modelling which preserves the Euler characteristic in the boundary representation at every stage. One or more of these Euler Boolean operations is stored in a change state, so as to …   Wikipedia

• Euler's formula — This article is about Euler s formula in complex analysis. For Euler s formula in algebraic topology and polyhedral combinatorics see Euler characteristic.   Part of a series of articles on The mathematical constant e …   Wikipedia

• Euler-Charakteristik — Die Euler Charakteristik ist im mathematischen Teilgebiet der Topologie eine Kennzahl für geschlossene Flächen. Als Bezeichnung verwendet man üblicherweise χ. Benannt ist sie nach dem Mathematiker Leonard Euler, der 1758 bewies, dass für E die… …   Deutsch Wikipedia

• Euler-Poincare-Charakteristik — Die Euler Charakteristik ist in der Topologie (einem Teilgebiet der Mathematik) eine Kennzahl für geschlossene Flächen. Flächen, die unter topologischen Gesichtspunkten als gleich angesehen werden, haben dieselbe Euler Charakteristik. Sie ist… …   Deutsch Wikipedia

• Characteristic function — In mathematics, characteristic function can refer to any of several distinct concepts: The most common and universal usage is as a synonym for indicator function, that is the function which for every subset A of X, has value 1 at points of A and… …   Wikipedia