# Victor Turchin: Smoothing theory deloopings of disk embedding and diffeomorphism spaces

**Time: **
Thu 2022-01-13 14.15 - 16.00

**Location: **
Zoom, meeting ID: 921 756 1880

**Video link: **
https://kva-se.zoom.us/j/9217561880

**Participating: **
Victor Turchin (Kansas State University)

**Abstract:** The smoothing theory provides delooping to the groups of relative to the boundary disk diffeomorphisms \(\mathrm{Diff}_\partial(D^m) = \Omega^{m+1} \mathrm{Top}_m/\mathrm{O}_m, m\neq 4\); \(\mathrm{Diff}_\partial(D^m) = \Omega^{m+1} \mathrm{PL}_m/\mathrm{O}_m, \textrm{any } m\). This result was established in the 70s and is due to contributions of several people: Cerf, Morlet, Burghelea, Lashof, Kirby, Siebenmann, Rourke, etc. In the talk I will briefly explain how this result is obtained. Less known is a similar statement for the spaces \(\mathrm{Emd}_\partial(D^m,D^n), \mathrm{Emd}^{fr}_\partial(D^m,D^n)\) of relative to the boundary (framed) disk embedding spaces. This latter result was hidden in a work of Lashof from 70s and was stated explicitly by Sakai nine years ago. The range stated by Sakai is \(n>4, n-m>2\). However, after a careful reading of the literature and with the help of Sander Kupers we got convinced that the delooping in question holds for any codimension \(n-m\) and any \(n\) (except \(n=4\) in the topological version of delooping). Of particular interest is the case \(m=2, n=4\). In the talk I will also explain how the smoothing theory techniques can be used to show that the delooping is compatible with the Budney \(E_{m+1}\)-action. The starting point in this project was the question whether it is possible to combine the Budney \(E_{m+1}\) action on \(\mathrm{Diff}_\partial(D^m)\) and \(\mathrm{Emb}^{fr}_\partial(D^m,D^n)\) with Hatcher's \(\mathrm{O}_{m+1}\) action on these spaces into an \(\mathrm{E}_{m+1}^{fr}\)-action. The answer is yes, it can be done by means of the smoothing theory delooping. Joint project in progress with Paolo Salvatore.