Families of relatively exact Lagrangians, free loop spaces and generalised homology
Families of relatively exact Lagrangians, free loop spaces and generalised homology
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DOI：
10.1007/s00029-023-00910-6
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发表时间：
2024-04-01
影响因子：
1.4
通讯作者：
Porcelli，Noah W.
中科院分区：
文献类型：
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作者：
Porcelli，Noah W.
我们证明了（在适当的取向条件下，取决于r）固定相对精确拉格朗日的辛流形的哈密顿同位素必须平凡地作用于，其中有一些广义的同调理论。我们使用的策略受到Hu等人（Geom Topol 15:1617-1650, 2011）的启发，他们在更强的定向假设下反复证明了类似的结果。然而，由于我们在方法上的不同，我们可以推论出，如果它是同伦球，它就同伦于恒等球。我们的技术设置不同于他们和Cohen等人（见：algeaic topology，施普林格，Berlin, 2019）和Cohen（见：The flower memorial volume， Birkhäuser, Basel）。我们也证明了（在类似的条件下）作用于，其中是l的自由循环空间。由此我们推导出，当i是曲面或a时，与恒等式同伦。利用Lalonde和McDuff （Topology 42:309-347, 2003）的方法，我们还证明了给定一组拉格朗日量，它们都是哈密顿同位素在一个球体或环面上，相关的纤维束上同源地分裂。
We prove that (under appropriate orientation conditions, depending onR) a Hamiltonian isotopyof a symplectic manifoldfixing a relatively exact LagrangianLsetwise must act trivially on, whereis 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 overand overunder stronger orientation assumptions. However the differences in our approaches let us deduce that ifLis a homotopy sphere,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) thatacts trivially on, whereis the free loop space ofL. From this we deduce that whenLis a surface or a,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 toLover a sphere or a torus, the associated fibre bundle cohomologically splits over.