**Speaker**

Victor Turchin

**Abstract**

The smoothing theory provides delooping to the groups of relative to the boundary disk diffeomorphisms

Diff_\partial(D^m) = \Omega^{m+1} Top_m/O_m, m\neq 4;

Diff_\partial(D^m) = \Omega^{m+1} PL_m/O_m, 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 Emd_\partial(D^m,D^n), Emb^{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 Diff_\partial(D^m) and Emb^{fr}_\partial(D^m,D^n) with Hatcher’s O_{m+1} action on these spaces into an 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.

**Suggested Readings**

I will not be assuming that people are familiar with the smoothing theory. Below is a list of relevant references that a curious person can have a look at. But I do not encourage to spend too much time on them.

D. Burghelea and R. Lashof, The homotopy type of the space of diffeomorphisms I, Trans. Amer. Math. Soc. 196 (1974), 1–36.

Kirby, Robion C.; Siebenmann, Laurence C. Foundational essays on topological manifolds, smoothings, and triangulations. With notes by John Milnor and Michael Atiyah. Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. vii+355 pp. (Essay 5)

R. Lashof, Embedding spaces, Illinois J. Math. 20 (1976), 144–154.

R. Budney, Little cubes and long knots, Topology 46 (2007), 1–27.

A. Hatcher, “Topological moduli spaces of knots”. Unfinished preprint.

knotspaces.pdf (cornell.edu)

Sakai, Keiichi Deloopings of the spaces of long embeddings. Fund. Math. 227 (2014), no. 1, 27–34.