Seminar

Smoothing theory deloopings of disk embedding and diffeomorphism spaces

Higher algebraic structures in algebra, topology and geometry

13 January 14:15 - 16:00

Victor Turchin - Kansas State University

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.

 

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Organizers
Gregory Arone
Stockholm University
Tilman Bauer
KTH Royal Institute of Technology
Alexander Berglund
Stockholm University
Søren Galatius
University of Copenhagen
Jesper Grodal,
University of Copenhagen
Thomas Kragh
Uppsala University

Program
Contact

Alexander Berglund

alexb@math.su.se

Søren Galatius

galatius@math.ku.dk

Thomas Kragh

thomas.kragh@math.uu.se

Other
information

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