Pontryagin classes of fiber bundles with fiber R^n
Michael Weiss (University of Aberdeen)
Monday 25th October, 2010 16:00-17:00 204
Thom and Novikov proved long ago that fibre bundles with fiber R^n have characteristic classes which extend the rational Pontryagin classes for vector bundles. It is not clear that these extended rational Pontryagin classes satisfy all the familiar relations that Pontryagin classes of vector bundles satisfy. In particular, there is the relation "square of Euler class is a Pontryagin class" for orientable vector bundles of even fiber dimension. Does this also hold for orientable fibre bundles with fibre R^n when n is even? Is it true that the extended Pontryagin classes in cohomological dimension >2n vanish for a fibre bundle with fiber R^n ? I will explain how these questions translate into questions about certain spaces of regular (=nonsingular) smooth maps to the plane, and into a functor calculus question. I have been working on this jointly with R Reis (Abdn) for some years. We have been looking particularly at the "regular maps" side of things.