Sergei Yakovenko's blog: on Math and Teaching

Monday, January 2, 2017

Lecture 8, Dec 26, 2016

De Rham and Cech cohomology of smooth manifolds

Using the exterior differential d on smooth differential forms and the fact that d^2=0, we define the de Rham cohomology with real coefficients H^k_{\mathrm dR}(M,\mathbb R) as the quotient space of closed k-forms by exact k-forms. This is a global invariant of a manifold M (for non-compact manifolds we may also consider a version for compactly supported forms, which yields different results).

De Rham cohomology can be computed using the Poincare lemma. If \mathfrak U=\{U_i\} is an open covering of M such that all opens sets and all their non-empty finite intersections are topologically trivial (homeomorphic to open balls), then for any closed form \omega\in\Omega^k(M) one can construct its primitives \xi_i\in\Omega^{k-1}(U_i) such that \mathrm d\xi_i=\omega in U_i. The (k-1)th forms \xi_i may disagree on the intersections U_{ij}=U_i\cap U_j, but one can attempt to twist them by suitable closed forms \mathrm d\phi_i. The corresponding system of (k-2)-forms \{\phi_i\} satisfies certain linear conditions on pairwise intersections; to satisfy these conditions one has to look for forms on triple intersections etc.

This construction gives rise to the notion of the Cech cohomology defined via systems of linear algebraic equations and reduces computation of the de Rham cohomology to a problem from linear algebra, determined by the combinatorics of the pairwise and multiple intersections of the sets U_i. In particular, one can conclude that the de Rham cohomology of compact manifolds is finite-dimensional.

An ultra-concise set of notes is available here (I hope to return and expand this text). The notion of Cech cohomology is further elaborated here.

NB. The class was shorter than usual because of the Hanukka lighting ceremony.

חנוכה שמח and Happy New Year, С наступающим Новым годом!

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