Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism
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| Format: | Preprint |
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2025
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| _version_ | 1866913734659670016 |
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| author | Kawasaki, Morimichi Kimura, Mitsuaki Maruyama, Shuhei Matsushita, Takahiro Mimura, Masato |
| author_facet | Kawasaki, Morimichi Kimura, Mitsuaki Maruyama, Shuhei Matsushita, Takahiro Mimura, Masato |
| contents | Given a closed connected symplectic manifold $(M,ω)$, we construct an alternating $\mathbb{R}$-bilinear form $\mathfrak{b}=\mathfrak{b}_{μ_{\mathrm{Sh}}}$ on the real first cohomology of $M$ from Shelukhin's quasimorphism $μ_{\mathrm{Sh}}$. Here $μ_{\mathrm{Sh}}$ is defined on the universal cover of the group of Hamiltonian diffeomorphisms on $(M,ω)$. This bilinear form is invariant under the symplectic mapping class group action, and $\mathfrak{b}$ yields a constraint on the fluxes of commuting two elements in the group of symplectomorphisms on $(M,ω)$. These results might be seen as an analog of Rousseau's result for an open connected symplectic manifold, where he recovered the symplectic pairing from the Calabi homomorphism. Furthermore, $\mathfrak{b}$ controls the extendability of Shelukhin's quasimorphisms, as well as the triviality of a characteristic class of Reznikov. To construct $\mathfrak{b}$, we build general machinery for a group $G$ of producing a real-valued $\mathbb{Z}$-bilinear form $\mathfrak{b}_μ$ from a $G$-invariant quasimorphism $μ$ on the commutator subgroup of $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism Kawasaki, Morimichi Kimura, Mitsuaki Maruyama, Shuhei Matsushita, Takahiro Mimura, Masato Symplectic Geometry Differential Geometry Group Theory Geometric Topology 53D05, 53D22 (Primary) 20J05, 37E35, 55R40, 20F65 (Secondary) Given a closed connected symplectic manifold $(M,ω)$, we construct an alternating $\mathbb{R}$-bilinear form $\mathfrak{b}=\mathfrak{b}_{μ_{\mathrm{Sh}}}$ on the real first cohomology of $M$ from Shelukhin's quasimorphism $μ_{\mathrm{Sh}}$. Here $μ_{\mathrm{Sh}}$ is defined on the universal cover of the group of Hamiltonian diffeomorphisms on $(M,ω)$. This bilinear form is invariant under the symplectic mapping class group action, and $\mathfrak{b}$ yields a constraint on the fluxes of commuting two elements in the group of symplectomorphisms on $(M,ω)$. These results might be seen as an analog of Rousseau's result for an open connected symplectic manifold, where he recovered the symplectic pairing from the Calabi homomorphism. Furthermore, $\mathfrak{b}$ controls the extendability of Shelukhin's quasimorphisms, as well as the triviality of a characteristic class of Reznikov. To construct $\mathfrak{b}$, we build general machinery for a group $G$ of producing a real-valued $\mathbb{Z}$-bilinear form $\mathfrak{b}_μ$ from a $G$-invariant quasimorphism $μ$ on the commutator subgroup of $G$. |
| title | Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism |
| topic | Symplectic Geometry Differential Geometry Group Theory Geometric Topology 53D05, 53D22 (Primary) 20J05, 37E35, 55R40, 20F65 (Secondary) |
| url | https://arxiv.org/abs/2503.10283 |