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 Poincaré 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 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