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MA4J7 Cohomology and Poincare Duality

Course Log and Suggested exercises

Topics to be covered:

  • Cochain complexes and cohomology
  • Relation between cohomology and homology
  • Parallels with homology for spaces: Functoriality, relative cohomology, long exact sequence of a pair, excision, Mayer-Vietoris
  • Ring structure of cohomology
  • Cohomology of a product space
  • Cap product
  • Orientation of manifolds
  • 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. Suggested exercises will not be graded, but some exam questions will be taken from the exercises.