How do autodiffeomorphisms act on embeddings

Abstract

We work in the smooth category. The following problem was suggested by E. Rees in 2002: describe the precomposition action of self-diffeomorphisms of S^p x S^q on the set of isotopy classes of embeddings S^p x S^q -> R^m. Let g : S^p x S^q -> R^m be an embedding such that g |_{a x S^q} : a x S^q -> R^m - g (b x S^q) is null-homotopic for some pair of different points a,b in S^p. Theorem. If h is an autodiffeomorphism of S^p x S^q identical on a neighborhood of a x S^q for some a\in S^p and p<q and 2m<3p+3q+5, then g h is isotopic to g. Let N be an oriented (p+q)-manifold and f : N -> R^m, g : S^p x S^q -> R^m isotopy classes of embeddings. As a corollary we obtain that under certain conditions for orientation-preserving embeddings s : S^p x D^q -> N the S^p-parametric embedded connected sum f#_sg depends only on f,g and the homology class of s|_{S^p x 0}.