MA4J7 Cohomology and Poincaré Duality

Lecturer: Dr. Saul Schleimer

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 one hour lectures

Assessment: Three hour examination

Prerequisites: MA3F1 Introduction to Topology, MA3H6 Algebraic Topology

Content:
1. Cochain complexes and cohomology.
2. The duality between homology and cohomology.
3. Chain approximations to the diagonal and products in cohomology.
4. The cohomology ring.
5. The cohomology ring of a product of spaces and applications.
6. The Poincaré duality theorem.
7. The cohomology ring of projective spaces and applications.
8. The Hopf invariant and the Hopf maps.
9. Spaces with polynomial cohomology.
10. Further applications of cohomology.

Aims:
To introduce cohomology and products as an important tool in topology. Give a proof of the Poincaré 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 Poincaré 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