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## Five proofs that the Euler characteristic of a closed orientable surface is even

Let $\Sigma_g$ be a closed orientable surface of genus $g$. (Below we will occasionally write $\Sigma$, omitting the genus.) Then its Euler characteristic $\chi(\Sigma_g) = 2 - 2g$ is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be $2 - 2g$, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.

## Euler characteristic as homotopy cardinality

Let $X$ be a finite CW complex with $c_0$ vertices, $c_1$ edges, and in general $c_i$ different $i$-cells. The Euler characteristic

$\displaystyle \chi(X) = \sum_{i \ge 0} (-1)^i c_i$

is a fundamental invariant of $X$, and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula $V - E + F = 2 = \chi(S^2)$ for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups $H_i(X, \mathbb{Q})$, 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:

1. Cardinality: $\chi(\text{pt}) = 1$.
2. Homotopy invariance: If $X \sim Y$, then $\chi(X) = \chi(Y)$.
3. Inclusion-exclusion: Suppose $X$ is the union of two subcomplexes $A, B$ whose intersection $A \cap B$ is a subcomplex of both $A$ and $B$. Then $\chi(X) = \chi(A) + \chi(B) - \chi(A \cap B)$.

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.