Abstracts and lecture notes

Localization of spaces: old and new: The modern theory of localization of spaces is due to Dror-Farjoun, with a large assist from Bousfield. It includes the original theories of inverting primes and of completion, but it goes beyond that to the inverting of any map of spaces. This has surprising applications which go back to the beginnings of homotopy theory and to the nature of the homotopy groups of finite complexes. Nontrivial homotopy theory may be said to begin with Hopf’s discovery that homotopy groups can be nonzero above the dimension of a space. He showed that the third homotopy group of the 2-dimensional sphere is nonzero. In the beginning of the golden age of homotopy theory, Serre generalized this to prove that simply connected noncontractible finite complexes have infinitely many nonzero homotopy groups. Using modern ideas like localization and Miller’s solution to the Sullivan conjecture, we give new proofs of Serre’s results. We also prove Serre’s conjecture that there is infinitely much torsion in these homotopy groups.

Notes from first hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8, page 9, page 10

Notes from second hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8, page 9

Notes from third hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8

Notes from fourth hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7

Chain models for loop spaces: We will discuss chain models for loop spaces and their multiplicative structures. These ideas go back to Adams and Hilton. We provide a transparent answer to the following question: If we start with a space with no multiplication and if we build from its chains a chain model for the loop space, then why does this model have a multiplication which corresponds to the multiplication of loops? To this end, we will discuss a generalization of the Eilenberg-Moore model to multiple homotopy pullbacks of fibrations.

Notes from first hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8, page 9, page 10, page 11, page 12

Notes from second hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8, page 9, page 10

Notes from third hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8, page 9, page 10

Notes from fourth hour: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8

Last updated: 24.04.07