Special Topics Course

Math 583 GA: Spectral Sequences

John Palmieri

Spring 2002, MWF 2:30-3:20

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Spectral sequences are a powerful computational and theoretical tool in algebra, topology, and category theory.

In this course I will discuss the basics of spectral sequences--how they arise and how they work--and I'll discuss many examples. The (essentially useless) definition of a spectral sequence is: a spectral sequence is a sequence of chain complexes E_r, for r > 0, so that

• the homology of E_r is E_r+1, and
• some underlying structure induces the differential on the chain complex E_r+1.

It is more helpful to think of a spectral sequence as a sort of black box, relating two algebraic objects (vector spaces or algebras, typically) to each other--one object is the input, the other is the output, and the spectral sequence is the complicated relationship between the two. The input is the E₁-term, and the E_r-terms for r > 1 interpolate between the input and the output, converging to the output as r approaches infinity.

Topics will include:

• the spectral sequence of a filtered complex
• the spectral sequence of a double complex
• the Serre spectral sequence
• the Bockstein spectral sequence
• exact couples and spectral sequences

Depending on the interests and background of the audience (and the lecturer), I can discuss other examples of spectral sequences from algebraic topology (e.g., the Eilenberg-Moore spectral sequence) or algebra/category theory (e.g., the composite functor spectral sequence). If you are interested in taking the course and would like to see specific spectral sequences, please let me know.

Prerequisites:

I will assume that you are comfortable with chain complexes, homology, and exact sequences; I will provide the topological or algebraic background for the various examples, as necessary.

Textbook:

To the extent that I use a book, it will probably be A User's Guide to Spectral Sequences by John M