论文标题
相对精确的Lagrangians，免费循环空间和广义同源性的家族
Families of relatively exact Lagrangians, free loop spaces and generalised homology
论文作者
论文摘要
我们证明（在适当的定向条件下，取决于$ r $），哈密顿同位素$ψ^1 $的符号歧管$（m，ω）$固定相对精确的lagrangian $ l $ setwise必须在$ r _*（l）$上琐碎地行为，而$ r _*$ r _*$是某些常规人体学理论。我们使用了由Hu，Lalonde和Leclercq（\ cite {hu-lalonde-leclercq}）启发的策略，他们证明了与$ \ Mathbb {z}/2 $以及$ \ \ \ \ \ \ m athbb {z} $相似的结果。但是，我们的方法的差异使我们推断出，如果$ l $是同质球，则$ψ^1 | _l $对身份是同型。我们的技术设置不同于他们的技术和Cohen，Jones和Segal（\ Cite {Cohen-Jones-Segal，Cohen}）。我们还证明（在类似条件下）$ψ^1 | _l $在$ r _*（\ Mathcal {l} l）$上微不足道，其中$ \ Mathcal {l} l $是$ L $的免费循环空间。从中，我们推断出$ l $是表面或$ k（π，1）$，$ψ^1 | _l $与身份是同质的。使用\ cite {lalonde-mcduff}的方法，我们还表明，鉴于一个lagrangian家族都是hamiltonian同位素到$ l $ to phore或torus上的$ l $，相关的纤维捆绑包在$ \ mathbb {z}}/2 $上均超过$ \ mathbb {z}/2 $。
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 $\mathbb{Z}/2$.
