Let be a finite CW complex with vertices, edges, and in general different -cells. The Euler characteristic

is a fundamental invariant of , and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups , but actually one can work on a more intuitive level characterized by the following slogan:

The Euler characteristic is a homotopy-invariant generalization of cardinality.

More precisely, the above expression for Euler characteristic can be deduced from three simple axioms:

**Cardinality:**.**Homotopy invariance:**If , then .**Inclusion-exclusion:**Suppose is the union of two subcomplexes whose intersection is a subcomplex of both and . Then .

Of course, this isn’t enough to conclude that there actually exists an invariant with these properties. Nevertheless, it’s enough to motivate the search for a proof that such an invariant exists.