Families of relatively exact Lagrangians, free loop spaces and generalised homology
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929216280330240 |
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| author | Porcelli, Noah |
| author_facet | Porcelli, Noah |
| contents | 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 $\mathbb{Z}/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_09677 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Families of relatively exact Lagrangians, free loop spaces and generalised homology Porcelli, Noah Symplectic Geometry 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 $\mathbb{Z}/2$. |
| title | Families of relatively exact Lagrangians, free loop spaces and generalised homology |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2202.09677 |