Rational Pontryagin classes of Euclidean bundles

Rational Pontryagin classes of Euclidean bundles

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Michael Weiss, University of Münster

Zoom link: https://princeton.zoom.us/j/96282936122

The Pontryagin classes are popular characteristic classes for real vector bundles. Since they are essentially the Chern classes of the complexified vector bundles, they satisfy certain vanishing relations: the Pontryagin classes of an n-dimensional vector bundle vanish in cohomological dimension greater than 2n. Thom and Novikov showed (late 1950s and early 1960s) that the *rationalized* Pontryagin classes can be defined for fiber bundles with fiber a euclidean space, i.e., in the absence of vector space structures on the fibers. More precisely, Thom showed it for PL bundles, using transversality arguments, and Novikov showed it in full generality. I ask(ed) whether these rationalized Pontryagin classes for fiber bundles with fiber a Euclidean space still satisfy the above-mentioned vanishing relations. It turns out that they do not. The proof is a combination of parametrized surgery and functor calculus methods.