MA4J7 Cohomology and Poincare Duality
(Term 1 Fall 2020)
Live lectures: Tuesdays 3:00-4:00
Live discussion of recorded lectures: Thursdays 4:00-5:00
Moodle course page, with links to recorded lectures:
https://moodle.warwick.ac.uk/course/view.php?id=41180
Topics to be covered:
- Cochain complexes and cohomology
- Relation between cohomology and homology
- Functoriality, relative cohomology, long exact sequence of a pair, excision, Mayer-Vietoris
- Ring structure of cohomology
- Cohomology of a product space
- Cap product
- Manifolds, fundamental class and orientation
- Cohomology with compact supports
- Poincare duality
Throughout all ideas will be illustrated with concrete examples.
Additional topics (some of which may be covered if time permits)
- H-spaces and Hopf algebras
- Cohomology of SO(n)
- Bockstein homomorphisms
- Transfer homomorphisms
- Cohomology with local coefficients
- Cohomology and homotopy theory
Assessment
There will be one 3-hour exam, counting for 85% of the grade. The remaining 15% will be based on exercises and quizzes during the course of term.