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.
It may be interesting to refer to Karoubi's chapter on Bott periodicity in Handbook of K-theory, but we will not follow this to prove Bott periodicity, although we possibly should. Instead, we will prove only complex Bott periodicity, following the paper of Harris: Bott periodicity via simplicial spaces. This has the advantage of being reasonably quick, and causes us to encounter the group-completion theorem. We prove the group completion theorem by following the paper of McDuff and Segal Homology fibrations and the 'group-completion' theorem
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.
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.