MAGIC011: Manifolds and homology

Course details

Semester

Autumn 2009
Monday, October 5th to Friday, December 11th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
0

Timetable

Wednesdays
10:05 - 10:55
Thursdays
10:05 - 10:55

Description

The course will cover the cohomology of topological spaces, with a heavy emphasis on interesting examples, most of which are manifolds.

Prerequisites

Some knowledge of general topology and commutative rings.

Syllabus

  1. Topological manifolds: definition and examples.
  2. Cohomology rings: basic properties, without construction. Description (without proof) of the cohomology rings of many interesting manifolds.
  3. Cohomology of configuration spaces: partial proof of stated description.
  4. Geometry of balls and spheres.
  5. Geometry of Hermitian spaces.
  6. Cohomology of balls and spheres.
  7. Cohomology of unitary groups.
  8. Cohomology of projective spaces.
  9. Vector bundles.
  10. Smooth structures and the tangent bundle.
  11. The Thom isomorphism theorem.
  12. Homotopical classification of vector bundles and line bundles.
  13. Cohomology of projective bundles; Chern classes; cohomology of flag manifolds and Grassmannians.
  14. Normal bundles, tubular neighbourhoods, and the Pontrjagin-Thom construction.
  15. Poincaré duality.
  16. The universal coefficient theorem.
  17. Cohomology of complex hypersurfaces.

Lecturer

  • NS

    Prof Neil Strickland

    University
    University of Sheffield

Bibliography

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Assessment

Description

x6

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