Durham Geometry and Topology Seminar - Mauricio Bustamante: Diffeomorphisms of solid tori

The homotopy groups of the diffeomorphism group of a high dimensional manifold with infinite fundamental group can be infinitely generated. The simplest example of this sort is the solid torus $T=S^1\times D^{d-1}$. In fact, using Hatcher, Igusa, and Waldhausen's approach to pseudoisotopy theory, it is possible to show that in the range of degrees up to (roughly) $d/3$, the homotopy groups of $Diff(T)$ contain infinitely generated torsion subgroups.

In this talk, I will discuss an alternative point of view to study $Diff(T)$ which does not invoke pseudoisotopy theory: when $d=2n$, we interpret $Diff(T)$ as the "difference" between diffeomorphisms and certain self-embeddings of the manifold $X_g$ which is the connected sum of $T$ with the g-fold connected sum of $S^n \times S^n$.

We will see how infinitely generated torsion subgroups appear from this perspective, and that they can be found even up to degrees $d/2$. This is ongoing joint work with O. Randal-Williams.

This talk was held in the Durham Geometry and Topology Seminar on 19 November 2020.