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MA4J7 Cohomology and Poincaré Duality

Lecturer: Saul Schleimer

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 one hour lectures

Assessment: 85% 3 hour examination in the summer, 15% by assignments

Formal registration prerequisites: None

Assumed knowledge:

Useful knowledge: A certain level of mathematical maturity (comfort with proofs and routine computations). Some category theory (categories, functors, natural transformations) will make learning the material much easier.

Synergies: This is a capstone of the undergraduate mathematics programme. It is a valuable course for anybody with an interest in "modern topology" (bringing us up to the 1960's). It is essential background for postgraduate study in geometry or topology, as well as many areas of algebra, number theory, and applied mathematics.

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content:

  • Cochain complexes and cohomology
  • The duality between homology and cohomology
  • Chain approximations to the diagonal and products in cohomology
  • The cohomology ring
  • The cohomology ring of a product of spaces and applications
  • The PoincarĂ© duality theorem
  • The cohomology ring of projective spaces and applications
  • The Hopf invariant and the Hopf maps
  • Spaces with polynomial cohomology
  • Further applications of cohomology

Aims:

  • To introduce cohomology and products as an important tool in topology
  • Give a proof of the PoincarĂ© duality theorem and go on to use this theorem to compute products
  • There will be many applications of products including using products to distinguish between spaces with isomorphic homology groups
  • To use products to study the classical Hopf maps

Objectives: By the end of the module the student should be able to:

  • Define cup and cap products
  • Use the PoincarĂ© duality theorem
  • Compute the cohomology ring of many spaces including product spaces and projective spaces
  • Apply the cohomology ring to get topological results
  • Define, calculate and apply the Hopf invariant

Books:
Algebraic Topology, Allen Hatcher, CUP 2002
Algebraic Topology A First Course, Greenberg and Harper, Addison-Wesley 1981

Additional Resources