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 Sp × Sq on the set of isotopy classes of embeddings Sp × Sq → ℝm.Let G: Sp × Sq → ℝm be an embedding such thatis null-homotopic for some pair of different points a, b ∈ Sp. We prove the following statement: if ψ is an autodiffeomorphism of Sp × Sq identical on a neighbourhood of a × Sq for some a ∈ Sp and p ⩽ q and 2m ⩾ 3p +3q + 4, then G◦ ψ is isotopic to G.Let N be an oriented (p + q)-manifold and let f, g be isotopy classes of embeddings N → ℝm, Sp × Sq → ℝm, respectively. As a corollary we obtain that under certain conditions for orientation-preserving embeddings s: Sp × Dq → N the Sp-parametric embedded connected sum f#sg depends only on f, g and the homology class of s|Sp × 0.