Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2311.12164 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910718406688768 |
|---|---|
| author | Humilière, Vincent Jannaud, Alexandre Leclercq, Rémi |
| author_facet | Humilière, Vincent Jannaud, Alexandre Leclercq, Rémi |
| contents | We initiate the study of the fundamental group of natural completions of the group of Hamiltonian diffeomorphisms, namely its $C^0$-closure $\overline{\mathrm{Ham}}(M,ω)$ and its completion with respect to the spectral norm $\widehat{\mathrm{Ham}}(M,ω)$. We prove that in some situations, namely complex projective spaces and rational Hirzebruch surfaces, certain Hamiltonian loops that were known to be non-trivial in $π_1\big(\mathrm{Ham}(M,ω)\big)$ remain non-trivial in $π_1\big(\widehat{\mathrm{Ham}}(M,ω)\big)$. This yields in particular cases, including $\mathbb C\mathrm P^2$ and the monotone $S^2\times S^2$, the injectivity of the map $π_1\big(\mathrm{Ham}(M,ω)\big)\toπ_1\big(\widehat{\mathrm{Ham}}(M,ω)\big)$ induced by the inclusion. The same results hold for the Hofer completion of $\mathrm{Ham}(M,ω)$. Moreover, whenever the spectral norm is known to be $C^0$-continuous, they also hold for $\overline{\mathrm{Ham}}(M,ω)$.
Our method relies on computations of the valuation of Seidel elements and hence of the spectral norm on $π_1\big(\mathrm{Ham}(M,ω)\big)$. Some of these computations were known before, but we also present new ones which might be of independent interest. For example, we show that the spectral pseudo-norm is degenerate when $(M,ω)$ is any non-monotone $S^{2}\times S^{2}$. At the contrary, it is a genuine norm when $M$ is the 1-point blow-up of $\mathbb C\mathrm P^{2}$; it is unbounded for small sizes of the blow-up and become bounded starting at the monotone one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12164 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Essential loops in completions of Hamiltonian groups Humilière, Vincent Jannaud, Alexandre Leclercq, Rémi Symplectic Geometry 53D05 We initiate the study of the fundamental group of natural completions of the group of Hamiltonian diffeomorphisms, namely its $C^0$-closure $\overline{\mathrm{Ham}}(M,ω)$ and its completion with respect to the spectral norm $\widehat{\mathrm{Ham}}(M,ω)$. We prove that in some situations, namely complex projective spaces and rational Hirzebruch surfaces, certain Hamiltonian loops that were known to be non-trivial in $π_1\big(\mathrm{Ham}(M,ω)\big)$ remain non-trivial in $π_1\big(\widehat{\mathrm{Ham}}(M,ω)\big)$. This yields in particular cases, including $\mathbb C\mathrm P^2$ and the monotone $S^2\times S^2$, the injectivity of the map $π_1\big(\mathrm{Ham}(M,ω)\big)\toπ_1\big(\widehat{\mathrm{Ham}}(M,ω)\big)$ induced by the inclusion. The same results hold for the Hofer completion of $\mathrm{Ham}(M,ω)$. Moreover, whenever the spectral norm is known to be $C^0$-continuous, they also hold for $\overline{\mathrm{Ham}}(M,ω)$. Our method relies on computations of the valuation of Seidel elements and hence of the spectral norm on $π_1\big(\mathrm{Ham}(M,ω)\big)$. Some of these computations were known before, but we also present new ones which might be of independent interest. For example, we show that the spectral pseudo-norm is degenerate when $(M,ω)$ is any non-monotone $S^{2}\times S^{2}$. At the contrary, it is a genuine norm when $M$ is the 1-point blow-up of $\mathbb C\mathrm P^{2}$; it is unbounded for small sizes of the blow-up and become bounded starting at the monotone one. |
| title | Essential loops in completions of Hamiltonian groups |
| topic | Symplectic Geometry 53D05 |
| url | https://arxiv.org/abs/2311.12164 |