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