Emmy-Noether-Seminar: "Pontryagin classes and representations up to homotopy"

Jul 15
15-07-2022 14:30 Uhr bis 15:30 Uhr
04.363

Pontryagin classes and representations up to homotopy

Madeleine Jotz Lean (Würzburg)

Abstract: This talk elaborates on a Theorem by Bott stating that if M is a
smooth manifold and F is an involutive subbundle of TM of codimension
q, then the k-th Pontryagin classes of the conormal bundle of F have
to vanish for k>2q.

In fact, this theorem can be immediately generalised to a theorem on
the Pontryagin classes of a reducible vector bundle, i.e. a vector
bundle E over M with a flat F connection on E for F an involutive
subbundle of TM. In particular, if none of the Pontryagin classes of a
given vector bundle vanish, then it has no quotient to a vector bundle
of the same rank over a manifold of dimension smaller than dim M/2.

We will see the proof of this theorem in the general context of Lie
algebroid representations up to homotopy. If time permits, I will then explain
how this gives obstructions to the existence of ideals in Lie
algebroids.

Friedrich-Alexander-Universität Erlangen-Nürnberg