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 $mathbb\{R\}^n$ of any dimension, as well as the solid unit ball in any Euclidean space — 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 $1\; +\; (-1)^n$ — 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 — 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 $scriptstylemathcal\{F\}$ on a projective scheme "X", one defines its Euler characteristic:$scriptstylechi\; (\; mathcal\{F\})=\; Sigma\; (-1)^i\; h^i(X,mathcal\{F\})$,where $scriptstyle\; h^i(X,\; mathcal\{F\})$ is the dimension of the "i"th sheaf cohomology group of $scriptstylemathcal\{F\}$.

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"π" / "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.

See also

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

References

External links

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