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How do autodiffeomorphisms act on embeddings?
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 148 / Issue 4 / August 2018
- Published online by Cambridge University Press:
- 28 December 2017, pp. 835-848
- Print publication:
- August 2018
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- Article
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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 that
is 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.
