Stabilisation, scanning, and handle cancellation

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 Emb (D^n-1, S^1 D^n-1) ^j Emb (D^n-1-j, S^1 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 ₀ Emb (S^n-1, S^n) is a group with respect to the connected-sum operation for all n 2. This last result is perhaps only interesting when n=4, as when n 4, it follows from the resolution of the various generalised Schönflies problems.