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 Dⁿ , (b) the space of co-dimension one embedded spheres in Sⁿ , and (c) the homotopy type of the space of co-dimension two trivial knots in Sⁿ . 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 S² 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(Dⁿ⁻¹, S¹× Dⁿ⁻¹) → Ωʲ Emb(Dⁿ⁻¹⁻ʲ, S¹ × Dⁿ⁻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 π₀ Emb(Sⁿ⁻¹, Sⁿ) 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.