Dozent: Fabian Hebestreit
Office: 3.023
Office hours: by appointment
eMail: f.hebestreit at math.uni-bonn.de
Seminar slot 1: mondays, 18:15 - 19:45 pm, 1.007
Seminar slot 2: tuesdays, 12:15 - 1:45 pm, 1.007

Characteristic Classes

I have updated the schedule. Furthermore, here you can find a script of mine, which covers the material of the first three talks, roughly from page 32. Beware though, it contains several (glaring) typos! For the second part we shall follow Whitehead's treatment in 'Elements of homotopy theory', Section 6, another reference being Hatcher's books 'Algebraic Topology', Section 3.H, and 'K-Theory and Vectorbundles', Section 3.3, available on his webpage. For the last part the standard reference is Milnor & Stasheff 'Characteristic Classes'.

Overview

The goal of the seminar is to study fibre bundles using cohomological methods. To this end, we will first go through the basics of fibre bundle theory, such as principal bundles and structure groups. We will then study the all important problem of constructing sections in fibre bundles using obstruction theory. As a byproduct we will obtain both the representability of cohomology by Eilenberg-Mac Lane spaces and the theory of universal fibre bundles. The characteristic classes of the title arise as the primary obstruction classes to finding certain types of section in a given bundle. We shall develop machinery to compute these classes and then consider several applications:
The relation between vector fields and the Euler characteristic, complex structures on spheres, dimensions of division algebras over the real numbers, estimates of immersion codimension, cobordism invariants and with a bit of luck we can even say a little bit about exotic spheres.


Schedule

Bundle Yoga
Talk 1 (Jan, 8.11.): Fibre bundles
Talk 2 (Didac, 13.11.): Principal and associated bundles
Talk 3 (Tikhon, 15.11): Structure groups and their reductions
Obstruction theory:
Talk 4 (Max, 19.11.): Homology with local coefficients
Talk 5 (Leo, 20.11.): Eilenberg's theorem
Talk 6 (Fabian, 26.11.): Universal fibre bundles
Characteristic classes:
Talk 7 (Xiaowen, 27.11.): Homotopy theory of Stiefel manifolds
Talk 8 (Domenico, 17.12.): The Thom isomorphism theorem
Talk 9 (Pedro, 18.12.): The projective bundle formula
Talk 10 (Pier, 21.1.): The cohomology of Grassmann manifolds
Talk 11 (Marco, 22.1.): Applications I
Talk 12 (Sami, 28.1.): Applications II
Talk 13 (Peng, 29.1.): Applications III