Extending the flux homomorphism to volume-preserving homeomorphisms
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910835987709952 |
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| author | Tchuiaga, Stéphane |
| author_facet | Tchuiaga, Stéphane |
| contents | This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, δ)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic homeomorphism of $(T^2, ω)$ with trivial flux, is a finite energy Hamiltonian homeomorphism of $(T^2, ω)$. We discuss the cohomology groups $H^\ast(Homeo_0(M,Ω), \mathcal{C}(M, \mathbb{R}) )$ of $ Homeo_0(M,Ω)$ with coefficients in $ \mathcal{C}(M, \mathbb{R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09647 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Extending the flux homomorphism to volume-preserving homeomorphisms Tchuiaga, Stéphane Symplectic Geometry Metric Geometry This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, δ)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic homeomorphism of $(T^2, ω)$ with trivial flux, is a finite energy Hamiltonian homeomorphism of $(T^2, ω)$. We discuss the cohomology groups $H^\ast(Homeo_0(M,Ω), \mathcal{C}(M, \mathbb{R}) )$ of $ Homeo_0(M,Ω)$ with coefficients in $ \mathcal{C}(M, \mathbb{R})$. |
| title | Extending the flux homomorphism to volume-preserving homeomorphisms |
| topic | Symplectic Geometry Metric Geometry |
| url | https://arxiv.org/abs/2206.09647 |