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Commuting symplectomorphisms on a surface and the flux homomorphism

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Abstract

Let \((S,\omega )\) be a closed connected oriented surface whose genus l is at least two equipped with a symplectic form. Then we show the vanishing of the cup product of the fluxes of commuting symplectomorphisms. This result may be regarded as an obstruction for commuting symplectomorphisms. In particular, the image of an abelian subgroup of \(\textrm{Symp}^c_0(S, \omega )\) under the flux homomorphism is isotropic with respect to the natural intersection form on \(H^1(S;{\mathbb {R}})\). The key to the proof is a refinement of the non-extendability result, previously given by the first-named and second-named authors, for Py’s Calabi quasimorphism \(\mu _P\) on \(\textrm{Ham}^c(S, \omega )\).

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Acknowledgements

We truly wish to thank the referees for useful comments and suggestions. We also thank Professors Masayuki Asaoka and Kaoru Ono, who have drawn the authors’ attention to Theorem 1.1. The first author is supported in part by JSPS KAKENHI Grant Numbers JP18J00765 and 21K13790. The second author is supported by JSPS KAKENHI Grant Number JP20H00114 and JST-Mirai Program Grant Number JPMJMI22G1. The third author and the fourth author are supported in part by JSPS KAKENHI Grant Numbers 19K14536 and 17H04822, respectively.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa, 252-5258, Japan

    Morimichi Kawasaki

  2. Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Mitsuaki Kimura

  3. Department of Mathematical Sciences, University of the Ryukyus, Nishihara-cho, Okinawa, 903-0213, Japan

    Takahiro Matsushita

  4. Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba, Aoba-ku, Sendai, 980-8578, Japan

    Masato Mimura

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  1. Morimichi Kawasaki
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Correspondence to Morimichi Kawasaki.

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Kawasaki, M., Kimura, M., Matsushita, T. et al. Commuting symplectomorphisms on a surface and the flux homomorphism. Geom. Funct. Anal. 33, 1322–1353 (2023). https://doi.org/10.1007/s00039-023-00644-9

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