Math 227B: Algebraic Topology
Characteristic classes and KTheory
Winter 2012

Time and Place: MWF 1212:50 pm in MS 5148

Instructor: Ciprian Manolescu

Email: cm_at_math.ucla.edu

Office: MS 6921

Office Hours: W 1011am, Th 11am12pm
The goal of the course will be to study vector bundles and fiber
bundles. We will discuss:
 StiefelWhitney classes;
 Chern classes;
 Euler and Pontryagin classes;
 KTheory;
 Bott periodicity;
 Spectra and generalized homology theories.
Here are a few motivating questions which can be answered with the
techniques from this course:
Given a smooth manifold M of dimension m, let emb(M) resp. imm(M)
be the
smallest values of n such that M can be embedded, resp. immersed, in the
ndimensional Euclidean space. Whitney's Theorems say that emb(M) is at
most 2m, and imm(M) at most 2m1. It is also clear that emb(M) and
imm(M) are at least m. What stronger lower bounds can we obtain, for
particular manifolds?
For what values of n is there a bilinear multiplication on R^n
without
zero divisors? One is familiar with the values n=1 (real numbers), n=2
(complex numbers), n=4 (quaternions) and n=8 (octonions). In fact these
are the only possible values.
How can one distinguish smooth manifolds that are homeomorphic
but not
diffeomorphic? Milnor (1956) gave the first example of such a pair,
consisting of the 7sphere and an exotic 7sphere.
Textbooks:
 J. Milnor and J. Stasheff, Characteristic classes,
Princeton University Press, 1974.
 A. Hatcher, Vector Bundles and KTheory, available online.
Other recommended books:
 M. F. Atiyah, KTheory, W. A. Benjamin, 1967.
 R. Bott, Lectures on K(X), W. A. Benjamin, 1969.
 D. Husemoller, Fibre Bundles, Springer Verlag, 1993.
 M. Karoubi, KTheory: An Introduction, SpringerVerlag, 1978.
 J. P. May, A Concise Course in Algebraic Topology, Chicago
Univ. Press, 1999.
 N. Steenrod, Topology of fiber bundles, Princeton Univ. Press,
1951.
Problem sets: