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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:

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


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.