Families of relatively exact Lagrangians, free loop spaces and generalised homology
Noah Porcelli
备注：
Accepted version. Appendix B added containing a new example; another appendix removed; an error in Section 3 is fixed. More small changes made suggestions of referees and thesis examiners
本文研究 Hamilton 同痕对相对精确 Lagrangian 子流形的广义同调及自由环路空间的平凡作用，并据此推出多类情形中限制映射同伦于恒等，以及相关纤维丛的 Z/2 上同调分裂。
摘要：
我们证明，在依赖于 R 的适当定向条件下，辛流形 (M, ω) 的一个 Hamilton 同痕 ψ^1 若在集合意义下固定一个相对精确的 Lagrangian 子流形 L，则它必在 R_*(L) 上平凡作用，其中 R_* 是某种广义同调理论。我们采用的策略受 Hu、Lalonde 和 Leclercq（\cite{Hu-Lalonde-Leclercq}）启发；他们在 Z/2 上，以及在更强的定向假设下于 Z 上证明了类似结果。然而，方法上的差异使我们能够推出：若 L 是同伦球面，则 ψ^1|_L 同伦于恒等映射。我们的技术设置既不同于他们的设置，也不同于 Cohen、Jones 和 Segal（\cite{Cohen-Jones-Segal, Cohen}）的设置。我们还证明（在类似条件下）ψ^1|_L 在 R_*(\mathcal{L}L) 上平凡作用，其中 \mathcal{L}L 是 L 的自由环路空间。由此我们推出，当 L 是曲面或 K(π,1) 时，ψ^1|_L 同伦于恒等映射。利用 \cite{Lalonde-McDuff} 的方法，我们还表明：给定一族在球面或环面上均与 L Hamilton 同痕的 Lagrangian 子流形，相应的纤维丛在 Z/2 上发生上同调分裂。
英文摘要：
We prove that (under appropriate orientation conditions, depending on $R$) a Hamiltonian isotopy $ψ^1$ of a symplectic manifold $(M, ω)$ 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, Lalonde and Leclercq (\cite{Hu-Lalonde-Leclercq}), 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, $ψ^1|_L$ is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen, Jones and Segal (\cite{Cohen-Jones-Segal, Cohen}). We also prove (under similar conditions) that $ψ^1|_L$ 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 $K(π, 1)$, $ψ^1|_L$ is homotopic to the identity. Using methods of \cite{Lalonde-McDuff}, 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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