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.

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