Families of relatively exact Lagrangians, free loop spaces and generalised homology

Selecta Mathematica Pub Date : 2024-02-03 DOI:10.1007/s00029-023-00910-6

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Abstract

We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy $\psi ^1$ of a symplectic manifold $(M, \omega )$ fixing a relatively exact Lagrangian L setwise must act trivially on $R_*(L)$ , where $R_*$ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over ${\mathbb {Z}}/2$ and over ${\mathbb {Z}}$ under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, $\psi ^1|_L$ is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that $\psi ^1|_L$ acts trivially on $R_*({\mathcal {L}}L)$ , where ${\mathcal {L}}L$ is the free loop space of L. From this we deduce that when L is a surface or a $K(\pi , 1)$ , $\psi ^1|_L$ is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over ${\mathbb {Z}}/2$ .

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相对精确拉格朗日族、自由环空间和广义同源性

摘要我们证明（在适当的取向条件下，取决于 R）交折流形 $(M, \omega )$ 的哈密顿等位（(\psi ^1\)固定一个相对精确的拉格朗日 L setwise）必须在 $R_*(L)$上起微不足道的作用，其中 $R_*(L)\是某种广义同调理论。其中 \(R_*$ 是某种广义同调理论。我们使用的策略受到了 Hu 等人的启发（Geom Topol 15:1617-1650, 2011），他们证明了在${\mathbb {Z}}/2$ 和${\mathbb {Z}}$上的类似结果。然而，我们方法的不同让我们推导出，如果 L 是一个同调球，那么 $\psi ^1|_L$ 与同一性是同调的。我们的技术设置既不同于他们的，也不同于科恩等人（收录于《代数拓扑学》，施普林格出版社，柏林，2019年）和科恩（收录于《弗洛尔纪念卷》，伯克豪泽出版社，巴塞尔）的。我们还证明（在类似条件下），$\psi ^1|_L$ 作用于 $R_*({\mathcal {L}}L)$ 是微不足道的。由此我们可以推导出，当 L 是一个曲面或一个 $K(\pi , 1)$ 时，$\psi ^1|L_)作用于\(R_*({\mathcal {L}}L)$。时，$\psi ^1|_L$与同一性同构。使用 Lalonde 和 McDuff 的方法（《拓扑学》42:309-347, 2003），我们还证明了给定一个拉格朗日族，所有这些拉格朗日族都是在球面或环面上与 L 同构的，相关的纤维束在 ${\mathbb {Z}}/2$ 上同调分裂。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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