自同构如何作用于嵌入
How do autodiffeomorphisms act on embeddings
摘要：
我们的工作是平稳的。E. Rees在2002年：描述了S^p x S^q的自同构在嵌入S^p x S^q -> R^m的合痕类集合上的预复合作用。设g：S^p x S^q -> R^m为嵌入，使得g|_{axS ^q}：axS ^q -> R^m-g（bxS ^q）对于S^p中的某对不同点a，b是零同伦的.如果h是S^p x S^q在a x S^q的邻域上全同的自同构，对某个a \in S^p且p<q且2 m <3 p +3q+5，则g h与g同位素。设N是定向（p+q）-流形，f：N -> R^m，g：S^px S^q -> R^m是嵌入的合痕类.作为推论，我们得到在一定条件下，对于保向嵌入s：S^p x D^q -> N，S^p-参数嵌入连通和f#sg只依赖于f，g和s的同调类|_{S^p x 0}。
英文摘要：
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}.
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