Euler characteristic Guide, Meaning , Facts, Information and Description

In algebraic topology, the Euler characteristic is a topological invariant (infact homotopy invariant) defined for broad class of topological spaces. It is usually denoted by .

In case of 2-dimensional topological polyhedron it can be calculated using the following formula: where F,E and V are the numbers of faces, edges and vertices correspondently. In particular for any polyhedron homeomorphic to a sphere we have

For instance, for a cube we have 6 − 12 + 8 = 2 and for a tetrahedron we have 4 − 6 + 4 = 2. The last formula is also called Euler's formula.

Table of contents

1 Definitions and properties 2 Partially ordered set 3 History

Definitions and properties

For finite CW-complex and in particular for finite simplicial complex the Euler characteristic can be defined as the alternating sum

where denotes the number of cells of dimension .

Then, one can define the Euler characteristic of a manifold as the Euler characteristic of a simplicial complex homeomorphic to it. For example, circle and torus have Euler characteristic 0 and solid balls have Euler characteristic 1.

The Euler characteristic of closed oriented surfaces can be calculated using its genus g

For closed Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature, see the Gauss-Bonnet theorem for two-dimensional case and generalized Gauss-Bonnet theorem for general case. A discrete analog of Gauss-Bonnet theorem is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry).

More generally still, for any topological space, we can define the n-th Betti number as the rank of the n-th homology group. The Euler characteristic can then be defined as the alternating sum

This definition has sense if the Betti numbers are all finite and zero beyond a certain index .

Two topological spaces which are homotopy equivalent have isomorphic homology groups and hence the same Euler characteristic.

From this definition and Poincaré duality, it follows that Euler characteristic of any closed odd-dimensional manifold is zero.

Partially ordered set

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

History

The ?first? rigorous proof of Euler formula was given by 20-year old Cauchy.

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