Divisibility results for signatures of bundles via Grothendieck—Witt theory

Hybrid talk.
For zoom details contact Barthel, Ozornova, Ray, Teichner.

The talk will begin with a recollection of the question: to what extent is signature multiplicative in fibre bundles of manifolds (and generalisations thereof) and summarise what is known about this question. I will then explain a divisibility result for (twisted) signatures of local systems of unimodular forms over stably framed base manifolds. For local systems coming from smooth manifold bundles, this recovers an old theorem of Atiyah and Mayer which is index theoretic in nature. In contrast to the manifold bundle case, the divisibilities obtained in my theorem are sharp. The proof of the theorem goes via a translation into a question about hermitian K-theory, so I will briefly recall what hermitian K-theory is, how it appears in this question and what recent developments in the theory are used in the proof of the theorem.