Lectures on Topological K-theory

ITP Small Seminar Room
Fridays at 3:30 pm, Spring 2000

Friday, April 7, Xianzhe Dai, Introduction [Audio]
Friday, April 14, Siye Wu, The Grothendieck construction [Audio]
Friday, April 21, Bill Jacob, Algebraic K-theory (first steps) [Audio]
Friday, April 28, Doug Moore, K-theory and cohomology
Friday, May 5, Rick Ye, The Bott periodicity theorem [Audio]
Friday, May 12, Joe Polchinski, D-branes and K-theory [Audio]
Friday, May 19, Simeon Hellerman, D-branes and K-theory II [Audio]
Friday, May 26, Morten Krogh, Ramond-Ramond fields and K-theory I [Audio]
Friday, June 2, Morten Krogh, Ramond-Ramond fields and K-theory II [Audio]
Friday, June 9, Siye Wu, K-theory, T-duality and D-brane anomalies [Audio]
This will be an expository seminar on the elements of topological K-theory at a level suitable for graduate students in mathematics and physics.

Just like the fundamental group and the de Rham cohomology groups, K-theory provides topological invariants of smooth manifolds. These topological invariants are constructed from isomorphism classes of vector bundles over manifolds. Among its early applications to topology was a simple proof of the fact that the only spheres which possess trivial tangent bundles are those of dimensions 1,3 and 7. It was also one of the key tools used by Atiyah and Singer in their index theorem for systems of elliptic partial differential equations on smooth manifolds. More recently K-theory has become an important ingredient in the theory of D-branes from theoretical physics.

Here is a tentative list of topics we intend to discuss:

1. Vector bundles and their classification
2. The Grothendieck construction
3. Introduction to algebraic K-theory
4. Bott periodicity
5. Examples of calculations of K(X)
6. The Chern character
7. The Thom isomorphism
8. Clifford algebras and the Atiyah-Bott-Shapiro construction

Additional topics may be announced later, depending on the interests of participants. Suggestions are welcome.

Suggested references:

1. M. F. Atiyah, K-theory, W. A. Benjamin, New York, 1967.
2. H. B. Lawson and M-L Michelsohn, Spin geometry, Princeton Univ. Press, Princeton, NJ, 1989.
3. K. Olsen and R. J. Szabo, Constructing D-branes from K-theory, physics preprint, hep-th/9907140, 1999.

--Doug Moore (moore@math.ucsb.edu)