稳定化、扫描与把柄消除
Stabilisation, scanning and handle cancellation
AI总结：
本文通过关联圆盘微分同胚群、球面余维1嵌入球面空间与余维2平凡纽结空间的同伦型，结合消去把柄、扫描映射及嵌入演算框架，证明Schoenflies球面幺半群关于连通和构成群，该结果在四维情形意义独特。
摘要：
在这篇短文中，我们描述了一族论证，它们将以下三者的同伦型关联起来：a）圆盘$D^n$的微分同胚群，b）球面中余维为1的嵌入球面空间，以及c）球面中余维为2的平凡纽结空间的同伦型。我们还描述了这些论证的一些自然推广。\n我们从Cerf对Smale定理的“升级版”证明展开，该定理指出2维球面的微分同胚群具有等距群的同伦型。这一证明涉及一种消去把柄构造，与Budney-Gabai的“扫描”映射相关。我们进一步给出了Cerf构造的一种Bott式变体，以及适用于这些构造的一个相关的嵌入演算（Embedding Calculus）框架。我们利用这些论证证明了Schoenflies球面的幺半群关于连通和运算构成一个群。最后这一结果或许仅在四维情形下有意义，因为在其他维数下，它可由各类广义Schoenflies问题的解答直接推出。
英文摘要：
In this note we describe a family of arguments that link the homotopy-type of a) the diffeomorphism group of the disc $D^n$, b) the space of co-dimension one embedded spheres in a sphere and c) the homotopy-type of the space of co-dimension two trivial knots in a sphere. We also describe some natural extensions to these arguments. We begin with Cerf's `upgraded' proof of Smale's theorem, that the diffeomorphism group of the 2-sphere has the homotopy-type of the isometry group. This entails a canceling-handle construction, related to the `scanning' maps of Budney-Gabai. We further give a Bott-style variation on Cerf's construction, and a related Embedding Calculus framework for these constructions. We use these arguments to prove that the monoid of Schoenflies spheres is a group with respect to the connect-sum operation. This last result is perhaps only interesting when in dimension four, as in other dimensions it follows from the resolution of the various generalized Schoenflies problems.
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