Math 227B: Algebraic Topology
Characteristic classes and K-Theory
Winter 2012
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Time and Place: MWF 12-12:50 pm in MS 5148
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Instructor: Ciprian Manolescu
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E-mail: cm_at_math.ucla.edu
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Office: MS 6921
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Office Hours: W 10-11am, Th 11am-12pm
The goal of the course will be to study vector bundles and fiber
bundles. We will discuss:
- Stiefel-Whitney classes;
- Chern classes;
- Euler and Pontryagin classes;
- K-Theory;
- 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
n-dimensional Euclidean space. Whitney's Theorems say that emb(M) is at
most 2m, and imm(M) at most 2m-1. 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 7-sphere and an exotic 7-sphere.
Textbooks:
- J. Milnor and J. Stasheff, Characteristic classes,
Princeton University Press, 1974.
- A. Hatcher, Vector Bundles and K-Theory, available online.
Other recommended books:
- M. F. Atiyah, K-Theory, W. A. Benjamin, 1967.
- R. Bott, Lectures on K(X), W. A. Benjamin, 1969.
- D. Husemoller, Fibre Bundles, Springer Verlag, 1993.
- M. Karoubi, K-Theory: An Introduction, Springer-Verlag, 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: