Introductory description

N/A

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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 Poincare 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

Learning outcomes

By the end of the module, students should be able to:

• Define cup and cap products.
• Use the Poincare 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.

Indicative reading list

Algebraic Topology, Allen Hatcher, CUP 2002
Algebraic Topology a first course, Greenberg and Harper, Addison-Wesley 1981

Subject specific skills

See learning outcomes

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.