- Euler characteristic
In

mathematics , and more specifically inalgebraic topology andpolyhedral combinatorics , the**Euler characteristic**is atopological invariant , a number that describes one aspect of atopological 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 solid s.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 includesEuclidean 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 theantipodal 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"thsheaf cohomology group of $scriptstylemathcal\{F\}$.Another generalization of the concept of Euler characteristic on manifolds comes from

orbifold s. 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 theMöbius function in that poset'sincidence 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

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