Math 754 — Spring 2001 Algebraic Topology III: Spectral Sequences
This is a third-semester algebraic topology course. Some knowledge of cohomology and homotopy groups will be assumed, as covered in 753 for example, or Chapters 3 and 4 of my book. The main topic I have in mind for 754 is spectral sequences and their applications. The course would begin with an introduction to the Serre spectral sequence. This can be applied to compute homology and cohomology of various interesting spaces such as Eilenberg-MacLane spaces, and also to prove general theorems about finite generation of homology and homotopy groups, for example. Another application is localization of spaces at a set of primes. After the Serre spectral sequence, the next big topic would be the Adams spectral sequence. This can be used to compute some stable homotopy groups of spheres (see my webpage http://www.math.cornell.edu/~hatcher for pictures of the results). Possible further applications might include computing cobordism theories and constructing other cohomology theories such as BP cohomology.
After this there are several options for the later part of the course. There are other useful spectral sequences such as the Eilenberg-Moore spectral sequence. Or we could shift direction and talk about topological K-theory.