UCSD Topology of Classifying Spaces Seminars 2004

UCSD Topology of Classifying Spaces 2004

Thursdays 10:30-12:00 in room 7218.

Organisers: Justin Roberts, Nitu Kitchloo, Jim Lin

The idea of a classifying space plays a central role in algebraic topology. The basic idea is that lots of interesting contravariant functors (in particular, cohomology functors such as singular cohomology or K-theory) may be represented as sets of homotopy classes of maps into a certain space. For example K(X) = [X, BU] where BU is the classifying space for complex vector bundles. Properties of the classifying space (for example, its cohomology) reflect properties of the functor (in this example, H*(BU) is the ring of universal Chern classes). Classifying spaces are only defined up to homotopy equivalence, but we will study several ways to construct them which are algebraically or geometrically "nice".

Closely related to classifying spaces are loop spaces, which occur in very many areas in mathematics and in physics, as the basic objects of string theory. Bott periodicity is the statement that BU and the loop space of U are homotopy equivalent.

We intend to look at some of the basic tools in the theory of classifying spaces and loop spaces, some of their applications, and to learn a bit of algebraic topology along the way.

As usual for our Thursday seminar, this is a ``learning seminar'' and participants will be expected to give talks. In the first meeting on Thursday Sept. 23, we will give a brief introduction and then survey the contents of the various lectures, so that participants can pick their favorite topic.

Various useful references: MilnorConstruction of the universal bundle, Geometric realisation of a semi-simplicial complex
Segal Categories and cohomology theories, Classifying spaces and spectral sequences
Bott and Tu Differential forms in algebraic topology
Griffiths and MorganRational homotopy theory
Perhaps: Stasheff, Adams, Chen, Hatcher/Thurston, Gelfand/Manin, Brown, Mosher/Tangora...

Provisional outline topics:

Sept 23: Nitu Kitchloo Introduction

Sept 30: Justin Roberts Fibrations in homotopy theory

Oct 7: Li Yu Spectral sequence of a fibration

Oct 14: Mike Gurvich Classifying spaces of groups

Oct 21: Dave Clark Classifying spaces of vector bundles

Oct 28: Dave Clark Stiefel-Whitney classes

Nov 4: Maia Averett Cohomology of classifying spaces

Nov 18: Nitu Kitchloo Bott periodicity

Dec 2: Nitu Kitchloo Spectra

Dec 9: Jim Lin Cohomology of loop spaces



justin@math.ucsd.edu