Watanabe's exotic families of diffeomorphisms

Speaker(s)

Mateusz Kujawski

Affiliation

UKEN Kraków

Language of the talk

English

Date

Oct. 15, 2025, 10:30 a.m.

Room

room 4070

Seminar

Seminar Algebraic Topology

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The study of diffeomorphism groups of manifolds has been an active area of research since the 1960's. One of the most important questions is understanding the homotopy type of Diff(D⁴, ∂D⁴) -- the group of diffeomorphisms of the n-dimensional disk that fix the boundary pointwise. In dimensions 3 and >4 a lot of the work has been done, while dimension 4 awaited a breakthrough until Watanabe's 2018 paper. In this article Watanabe shows the previously unknown existence of a nontrivial element of π₁ Diff(D⁴, ∂D⁴). Moreover, this element is trivial in π₁ Homeo(D⁴, ∂D⁴). In my talk I will present Watanabe's work focusing on the invariant Z₂ that detects the nontrivial element. In detail, I will review Kontsevich configuration space integrals: graph cohomology, configuration spaces and their applications to Diff(D⁴, ∂D⁴) by Watanabe. If time permits, I will also sketch some of my ideas on how to extend Watanabe's methods to other manifolds, and mention connections with the Poincaré Conjecture.

Major part of the talk is an overview of my Master's thesis prepared under supervision of Maciej Borodzik at UW. The rest is work in progress as part of my doctoral studies under  supervision of Maciej Borodzik and Piotr Pokora at UKEN, Kraków.

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