  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 a sphere and c) the homotopy-type of the space of
co-dimension two trivial knots in a sphere. We also describe some natural
extensions to these arguments.
  We begin with Cerf's `upgraded' proof of Smale's theorem, that the
diffeomorphism group of the 2-sphere has the homotopy-type of the isometry
group. This entails a canceling-handle construction, related to the `scanning'
maps of Budney-Gabai. 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 Schoenflies
spheres is a group with respect to the connect-sum operation. This last result
is perhaps only interesting when in dimension four, as in other dimensions it
follows from the resolution of the various generalized Schoenflies problems.
