Diffeomorphism groups of small high-dimensional manifolds

Ryan Budney (University of Victoria)

Monday 15th November, 2021 16:00-17:00 Online


This talk will outline an argument that the (n-4)-th homotopy group of the diffeomorphism group of S^1 x D^{n-1} is not finitely generated for all n>=4.   There are two key steps in this argument.  (1) We have to generate the diffeomorphisms. This is done via a method that is analogous to the manner in which Dehn constructs Dehn twists, and links our work to the study of spaces of string links.  Namely, we construct families of diffeomorphisms of elementary manifolds we call "barbell manifolds".  These are the boundary connect-sums of (two) trivial disc bundles over spheres. (2) We embed our barbell manifolds in S^1 x D^{n-1} and extend the diffeomorphisms.  We argue these extensions are non-trivial. To do this, we use a scanning construction of Cerf.  By thinking of {1} x D^{n-1} as 'fibered' over D^{n-2} by intervals, we construct a map from Diff(S^1 x D^{n-1}) to the (n-2)-nd iterated loop space of the space of embeddings of an interval into S^1 x D^{n-1}.   The low-dimensional homotopy groups of this space are computable via Embedding Calculus techniques of Goodwillie, Weiss and Klein.

The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/98078798957 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).

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