Math 568  Spectral Sequences

 

The text:  User's Guide to Spectral Sequences

by   McCleary

is currently out of print. A new edition is in the works. We will use notes and xeroxed copies from various sources.

The following is a rough outline of the contents of the course. This can and will be modified according to the interests of the class. In particular, the depth into which we go in these various directions is not predetermined.

We will not give a proof of every result. In many cases we will outline the proof. The intention is to gain an understanding of the material and develop techniques for computation. Handouts containing complete proofs will be available
 

On Reserve:    User's Guide to Spectral Sequences by John McCleary
                                  Topology and Geometry by Glen Bredon
                                  Algebraic Topology: An Intuitive Approach by Hajime Sato
                                  Homotopy Theory: An Introduction to Algebraic Topology by Brayton Gray

 

TOPICS:    1.  Serre Spectral Sequence
 

This is by far the most important spectral sequence.

a. Fiber bundles and fibrations

This is an introduction to these topics which we will think of as generalizations of covering spaces. We will discuss numerous examples including those involving classical Lie groups, Grassmanians, projective spaces, and fibrations manufactured out of a continuous map such as loop spaces
 

b. The Spectral Sequence of a filtered differential  group

This is an algebraic machine made up out of many exact sequences. Most spectral sequences come from this situation

 c. Review of important facts from Homology and Cohomology

We will recall  the universal coefficient theorems, the    Kuenneth Theorem, cup products and Poincare duality.

d. The Serre Spectral Sequence

We begin with some examples where the algebra is quite simple and work our way toward more complicated examples. In particular we will calculate the cohomolgy of Grassmanians and characteristic classes and discuss the comparison theorem.

e. Homotopy groups and the Hurewicz theorem

We introduce homotopy groups and their relation to fiberings. Then we develop the Hurewicz theorem and the Whitehead theorem.

f. Steenrod operations and Eilenberg-MacLane spaces

These classical and useful topics will be developed.

g. Cohomology of Groups

Examples will be given, and techniques developed.

2. Other Spectral Sequences

The content here will depend on the interests of the class. Some possible examples are the Eilenberg-Moore spectral sequence, the Adams Spectral sequence, the Leray spectral sequence, the Bockstein spectral sequence, other spectral sequences from algebra, algebraic geometry, and algebraic K theory.