Let $X$ be a compact, oriented, connected manifold. I recently learned about a cohomological version of Lefschetz' fixed point theorem, which states:
If $f: X \to X$ is a continuous self-map, then $$\langle [\Delta] \cup [\Gamma_f]\rangle_{X \times X} = \sum_i (-1)^i \operatorname{tr}(f^*: H^i(X, \mathbb R) \to H^i(X, \mathbb R)).$$
Here $\Delta \subset X \times X$ is the diagonal, and $\Gamma_f \subset X \times X$ is the graph of $f$. The classes $[\Delta]$ and $[\Gamma_f]$ are the (Poincaré duals of) fundamental classes considered in $H^*(X \times X)$, and $\langle - \rangle: H^{2n}(X \times X) \to \mathbb R$ is evaluation against the fundamental class of $X \times X$.
Now if one applies this theorem to the identity map $f = \operatorname{id}_X$ then $[\Gamma_f] = [\Delta]$, and the traces all become the Betti numbers. So one obtains $$[\Delta]^2 = \sum_i (-1)^i \dim H^i(X, \mathbb R).$$ So this provides proof that two ways to define the Euler characteristic agree. Is there a more down-to-earth proof of this fact, maybe using differential forms (deRham cohomology)?