SoSe 24 Seminar: de Rham cohomology
Time and place: Wednesday 10-12 in M009
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 |
|---|---|---|
| 17.04 | The de Rham complex of Euclidean space | |
| 24.04 | The de Rham complex of a smooth manifold | |
| 01.05 | — | — |
| 08.05 | The Mayer–Vietoris sequence | |
| 15.05 | Orientation and integration | |
| 22.05 | Stokes’ theorem | |
| 29.05 | The Poincaré lemma | |
| 05.06 | The Poincaré lemma with compact supports | |
| 12.06 | Finiteness of de Rham cohomology | |
| 19.06 | Poincaré duality for oriented manifolds | |
| 26.06 | The degree of a proper map | |
| 03.07 | The Künneth formula and the Leray–Hirsch theorem | |
| 10.07 | The orientation bundle and the twisted de Rham complex | |
| 17.07 | Densities and Poincaré duality for nonorientable manifolds |