Stabilisation, scanning, and handle cancellation

Abstract

In this note, we describe a family of arguments that link the homotopy type of (a) the diffeomorphism group of the disc $D^n$, (b) the space of co-dimension one embedded spheres in $S^n$, and (c) the homotopy type of the space of co-dimension two trivial knots in $S^n$. 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 $S^2$ has the homotopy type of the isometry group. This entails a cancelling-handle construction, related to recently studied “scanning” maps of spaces of embeddings $\mathrm{Emb}(D^{n-1},S^1\times D^{n-1})\to {\Omega }^j\mathrm{Emb}(D^{n-1-j},S^1\times D^{n-1})$. 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 ${\pi }_0\mathrm{Emb}(S^{n-1},S^n)$ is a group with respect to the connected-sum operation for all $n\ge 2$. This last result is perhaps only interesting when $n=4$, as when $n\ne 4$, it follows from the resolution of the various generalised Schönflies problems.