640:541
Introduction to Algebraic Topology II
Christopher Woodward
MTh210:20 AM - 11:40 AMHLL-425BUS
Text: Allen Hatcher "Algebraic Topology," available from Cambridge University Press, as well as online at hatcher's site.

The alternate texts, Spanier's "Algebraic Topology" (which is somewhat encyclopedic) and Dold's "Lectures on algebraic topology" (which is great but doesn't cover very much) are available on reserve at the math library.

Prerequisites: Math 540 or knowledge of its contents.

Requirements: Homework problems from Hatcher and one or two presentations.

Presentation Schedule:

Andrew Oostergaard: Tor functors and universal coefficient theorem for homology
Jaret Flores : Kunneth theorem for chain complexes
David Duncan : Eilenberg-Zilber theorem and Kunneth for topological spaces
John Miller: Cross, (& Cup, and Cap products)
Chris Woodward : Cohomology of Fiber bundles: Leray-Hirsch theorem and Thom isomorphisms. Application to cohomology of projective spaces.
Susovan Pal: Orientations and existence of fundamental class
Ali Maalaoui: Poincare duality (relationship between compactly support cohomology and homology)
Chris Woodward: Introduction to higher homotopy groups
Priyam Patel : Whitehead theorem on homotopy type
Knight Fu:
Moulik Kallupalam Balasubra:
Description:
This course will be a sequel to Math 540, but can also be viewed as a mostly independent course on cohomology and homotopy theory for students who already have had an introduction to homology.

We will start with cohomology in Chapter 3 (the extent of the coverage depending on how far Math 540 gets into this chapter). We will then cover basic results on homotopy groups in Chapter 4, such as the long exact sequences for pairs of spaces and fiber bundles, and will take up a number for further topics that relate homotopy groups to homology and cohomology.

Depending on available time, the course may end with an introduction to vector bundles and characteristic classes, following a further book in progress by Hatcher on his web site, or Milnor and Stasheff's book Characteristic Classes.