Math 603D: K-Theory

Course Information

Instructor

The instructor for the course is me, Ben Williams

Meeting Time, Place

Provisionally, the course meets MWF from 11-12 in MATX 1102.

References

There is no general textbook reference for this course, but references will be given over the course of the term for specific aspects. Notes may also be made available.

Description

This is a topics course in K-theory. The first part will deal with the elementary K-theory of rings, i.e., the classical theory of K0(R) and K1(R). We will prove the Serre–Swan theorem, relating this K0-theory with K-theory for topological spaces. Then K-theory as an extraordinary cohomology theory on topological spaces will be presented, and we will introduce spectra and stable homotopy theory motivated by this theory. The last part of the course will return to algebraic K-theory and discuss the problem of constructing and then calculating “higher” algebraic K-theory.

A first course in algebraic topology (such as Math 527) will be assumed. Some familiarity with homotopy theory will be helpful, but not assumed.

Prerequisites

A first course in algebraic topology (such as Math 527) will be assumed, as will undergraduate algebra. Some familiarity with (unstable) homotopy theory will be helpful, but not assumed. Some facts about unstable homotopy theory, regarding fibrations and homotopy groups and so on, will be asserted without proof. Proofs of all such results can be found in May's Concise course in algebraic topology. Some results regarding classifying spaces are also assumed: these can be found in May's Classifying spaces and fibrations. The source for classifying spaces of topological groups is Segal's superb paper Classifying spaces and spectral sequences.

Grade

The grade for this course is based solely on exercises. Exercises will be given infrequently.
A pdf syllabus is online on the math department website, but in the case of a disagreement between the pdf and this website, this website should be assumed to be correct.

Exercises