Euler characteristic

Summary

The Euler characteristic $\chi(X)$ of a space is the alternating sum $b_0 - b_1 + b_2 - \cdots$ of its Betti numbers. For a finite CW complex it equals the alternating count of cells, which recovers $V - E + F$ for polyhedra. It is a homotopy invariant, and it is multiplicative for products and finite coverings. It links topology with geometry through results such as the Gauss–Bonnet theorem.