I’ve been away from this blog for too long–partially it’s because most of my expository energy has gone into preparing a collection of notes on algebraic geometry (to help me learn the subject). Someday I’ll post them.

Today, however, I’d like to talk about a clever proof I learned recently.

The following result is neat:

Theorem: Let ${M}$ be a compact smooth manifold. Then the de Rham cohomology groups ${H^i_{DR}(X, \mathbb{R})}$ are finite-dimensional.

I’m pretty sure it follows from Hodge theory and the finite-dimensionality of the harmonic forms. However, I learned a neat elementary proof that I’d like to discuss.

By de Rham’s theorem, we can compute the de Rham cohomology groups as the sheaf cohomology groups ${H^i(X, \mathbb{R})}$ for ${\mathbb{R}}$ denoting the constant sheaf associated to the group ${\mathbb{R}}$. Now, pick a Riemannian metric on ${M}$. Each point has a neighborhood ${U}$ such that any two points in ${U}$ are joined by a unique geodesic contained in ${U}$—such a neighborhood is called geodesically convex. It is clear that a geodesically convex neighborhood is homeomorphic (via the exponential map) to a convex set in ${\mathbb{R}^n}$, which has trivial de Rham cohomology, and also that the intersection of two geodesically convex sets is geodesically convex.

So pick a finite cover of ${M}$ by geodesically convex sets ${U_1, \dots, U_k}$. Then on every intersection ${U_{i_1} \cap \dots \cap U_{i_n}}$, the sheaf ${\mathbb{R}}$ has trivial cohomology because this intersection is geodesically convex, hence diffeomorphic to a convex set in ${\mathbb{R}^n}$. In particular, the cover ${\{U_i\}}$ satisfies the hypotheses of Leray’s theorem. We can apply Cech cohomology with this cover to compute ${H^i(X, \mathbb{R})}$, or equivalently the de Rham cohomology.

But there are finitely many sets in this cover, and the sections of the sheaf ${\mathbb{R}}$ over each of these sets is just the abelian group ${\mathbb{R}}$ by connectedness of anything geodesically convex. So the Cech complex consists of finite-dimensional vector spaces; its cohomology thus consists of finite-dimensional vector spaces. $\Box$

I learned this from Bott and Tu’s Differential Forms in Algebraic Topology, which appears to be a really fun read.