Euler characteristic

The Euler characteristic of a polyhedron is V − E + F where V, E, and F are respectively the numbers of vertices, edges, and faces. (The name Euler is pronounced "oiler"; see Leonhard Euler).

The Euler characteristic of any polyhedron homeomorphic to a sphere is 2. For instance, for a cube we have 8 − 12 + 6 = 2 and for a tetrahedron we have 4 − 6 + 4 = 2.

In general, the Euler characteristic is a topological invariant, i.e., any two polyhedra that are homeomorphic to each other have the same Euler characteristic. One can therefore extend the definition to more general surfaces than polyhedra, and speak of the Euler characteristic of, for example, a torus, which would be the Euler characteristic of any polyhedron homeomorphic to a torus. In this sense, a torus has Euler characteristic 0.

One can also define the concept of Euler characteristic of manifolds of dimension other than 2. One approach is to define the Euler characteristic of any simplicial complex as the alternating sum

{number of points} − {number of 1-simplices} + {number of 2-simplices} − {number of 3-simplices} + ...

and then define the Euler characteristic of a manifold as the Euler characteristic of any simplicial complex homeomorphic to it. With this definition, circles and squares have Euler characteristic 0 and solid balls have Euler characteristic 1.

The Euler characteristic χ of a manifold is closely related to its genus g: if the manifold is orientable, we have

and if it is not orientable, we have

For two-dimensional orientable Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature, see the Gauss-Bonnet theorem.

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

b₀ − b₁ + b₂ − b₃ + ...

if these Betti numbers are all finite and zero beyond a certain index n₀. Two topological spaces which are homotopy equivalent have isomorphic homology groups and hence the same Euler characteristic. This notion generalizes all the definitions given above and can be extended to other homology theories outside of topology.

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.