Smale收缩的一个接触几何变体

中文摘要 AI 辅助

受Smale关于二维圆盘紧支撑微分同胚群收缩的启发，我们研究开旋转对称Darboux球上的接触形式，这些形式在紧集外与标准接触形式一致，且其Reeb流没有俘获轨道。这样的形式称为垂直凸形式。在三维和五维中，我们将其空间在紧支撑微分同胚作用下的商空间与可缩的单值空间等同，并证明了一个等变乘积分裂。因此，标准接触形式的轨道是一个强形变收缩核。由此得出，在三维中垂直凸接触形式的空间是可缩的，在五维中它与紧支撑微分同胚群同伦等价，从而是连通的。定义标准接触结构的类似子空间在五维中具有相应紧支撑接触同胚群的同伦型，在三维中是可缩的。

英文摘要

Motivated by Smale's contraction of the compactly supported diffeomorphism group of the two-disc, we study contact forms on an open rotationally symmetric Darboux ball that agree with the standard contact form outside a compact set and whose Reeb flows have no trapped orbits. Such forms are called vertically convex. In dimensions three and five, we identify the quotient of their space by compactly supported diffeomorphisms with a contractible monodromy space and prove an equivariant product splitting. Consequently, the orbit of the standard contact form is a strong deformation retract. It follows that the space of vertically convex contact forms is contractible in dimension three and homotopy equivalent to the compactly supported diffeomorphism group, hence connected, in dimension five. The analogous subspace of forms defining the standard contact structure has the homotopy type of the corresponding compactly supported contactomorphism group in dimension five and is contractible in dimension three.

发表机构

• Ruhr-Universität Bochum（波鸿鲁尔大学）

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