Euler characteristic

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The surfaces of nonconvex polyhedra can have various Euler characteristics:

This version holds both for convex polyhedra (where the densities are all 1) and the non-convex Kepler–Poinsot polyhedra.

Via stereographic projection the plane maps to the 2-sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchy's proof of Euler's formula given below.

There are many proofs of Euler's formula. One was given by Cauchy in 1811, as follows. It applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks.

Apply repeatedly either of the following two transformations, maintaining the invariant that the exterior boundary is always a simple cycle:

These transformations eventually reduce the planar graph to a single triangle. (Without the simple-cycle invariant, removing a triangle might disconnect the remaining triangles, invalidating the rest of the argument. A valid removal order is an elementary example of a shelling.)

The Euler characteristic behaves well with respect to many basic operations on topological spaces, as follows.

The product property holds much more generally, for fibrations with certain conditions.

This includes product spaces and covering spaces as special cases, and can be proven by the Serre spectral sequence on homology of a fibration.

The Euler characteristic can be calculated easily for general surfaces by finding a polygonization of the surface (that is, a description as a CW-complex) and using the above definitions.