Fall 2003 MATH 541:Topics in Topology

TTh 1pm HB 423 Prof. Tim Cochran

This class will essentially be a standard second year graduate algebraic topology class taught primarily from the book by Davis and Kirk. This will involve a substantial amount of algebra, especially at the start. It is a “must” class for topologists of any sort and also will be extremely useful for anyone interested in working in algebraic geometry (say with Dr. Hassett). Here is an approximate syllabus. Prerequisites are Math 445 and 463. There WILL be some assigned homework but not as much as in a first year class (maybe half as much).

1. Review of modules (including over noncommutative rings)
2. Modules associated to topological spaces (homotopy groups, homology groups of covering spaces)

Tensor product of modules, Hom, adjoint functors

3. Homological algebra- free, projective, injective, flat modules; Universal coefficient theorems, classification theorems for modules
4. Cap products, cross products, Kunneth theorem ?
5. more on Homology with Local coefficients
6. Homotopy theory- fibrations, cofibrations, homotopy groups, Hurewicz and Whitehead theorems
7. Obstruction theory
8. Bordism
9. spectral sequences
10. group homology
11. Intersection theory