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Jan Steinebrunner: 3-manifolds and modular operads

Time: Tue 2023-04-04 10.15 - 12.00

Location: Cramer room, Albano

Participating: Jan Steinebrunner (Copenhagen)

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The goal of this talk is to explain how cyclic and modular (infinity-)operads can be used to study diffeomorphism groups of 3-manifolds and in particular their stable homology.

In the first part of the talk I will explain what cyclic and modular infinity-operads are and talk about joint work with Bregman and Boyd, in which we prove a 'derived version' of Milnor's prime decomposition theorem for 3-manifolds, and as a consequence compute the rational cohomology of B Diff( (S^1 x S^2) # (S^1 x S^2) ).

In the second part I will explain an ongoing project with Barkan, in which we study monoidal envelopes of modular infinity operads and their algebras. This allows us to establish universal properties for various bordism-style categories. As applications we obtain a new proof of Galatius' result about the stable homology of Aut(F_n), as well as a proof of Hatcher's conjecture about the stable homology of Diff( (S^1 x S^2)^{# g} rel D^3 ).