Euler Characteristic - Topological Definition

Topological Definition

The polyhedral surfaces discussed above are, in modern language, two-dimensional finite CW-complexes. (When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic can be defined as the alternating sum

where k_n denotes the number of cells of dimension n in the complex.

Similarly, for a simplicial complex, the Euler characteristic equals the alternating sum

where k_n denotes the number of n-simplexes in the complex.

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

This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index n₀. For simplicial complexes, this is not the same definition as in the previous paragraph but a homology computation shows that the two definitions will give the same value for .