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