Stabilisation, scanning, and handle cancellation

  • Ryan Budney

    University of Victoria, Victoria, BC, Canada
Stabilisation, scanning, and handle cancellation cover

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Abstract

In this note, we describe a family of arguments that link the homotopy type of (a) the diffeomorphism group of the disc , (b) the space of co-dimension one embedded spheres in , and (c) the homotopy type of the space of co-dimension two trivial knots in . We also describe some natural extensions to these arguments. We begin with Cerf’s “upgraded” proof of Smale’s theorem, showing that the diffeomorphism group of has the homotopy type of the isometry group. This entails a cancelling-handle construction, related to recently studied “scanning” maps of spaces of embeddings . We further give a Bott-style variation on Cerf’s construction and a related embedding calculus framework for these constructions. We use these arguments to prove that the monoid of Schönflies spheres is a group with respect to the connected-sum operation for all . This last result is perhaps only interesting when , as when , it follows from the resolution of the various generalised Schönflies problems.

Cite this article

Ryan Budney, Stabilisation, scanning, and handle cancellation. Enseign. Math. (2024), published online first

DOI 10.4171/LEM/1080