WiSe 20/21 Seminar: de Rham cohomology

Time and place: Monday 08-10 online


The goal of this seminar is to introduce and study de Rham cohomology, which is an invariant of smooth manifolds defined using differential forms. It is related both to analysis (via the fundamental theorem of calculus and Stokes’ theorem) and algebraic topology (as de Rham cohomology turns out to be isomorphic to singular cohomology). Moreover, the ideas underlying the definition of de Rham cohomology are quite versatile and can be applied in many other geometric contexts, for example in algebraic geometry.

The seminar will closely follow Chapter I of the book Differential Forms in Algebraic Topology by R. Bott and L. W. Tu and cover most of the results therein. The main theorem is Poincaré duality, which is a surprising symmetry in the cohomology of smooth compact manifolds. We will discuss it for both oriented and nonorientable manifolds.

Date Speaker Topic
02.11 Yaxi Zhu The de Rham complex of Euclidean space
09.11 Michael Friesen The de Rham complex of a smooth manifold
16.11 Michael Friesen The Mayer–Vietoris sequence
23.11 Arshay Sheth Orientation and integration
30.11 Arshay Sheth Stokes’ theorem
07.12 Matthias Uschold The Poincaré lemma
14.12 Hua Jing The Poincaré lemma with compact supports
21.12 Lukas Schweiger Finiteness of de Rham cohomology
11.01 Zhelun Chen Poincaré duality for oriented manifolds
18.01 Hua Jing The degree of a proper map
25.01 Hua Jing The Künneth formula and the Leray–Hirsch theorem
01.02 Marc Hoyois The orientation bundle and the twisted de Rham complex
08.02 Zhelun Chen Densities and Poincaré duality for nonorientable manifolds