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Loops in the fundamental group of ${\mbox{Symp}} ({\mathbb C}{\mathbb P}^2\# \mbox{5}\overline { \mathbb C\mathbb P}\,\!^2,\omega )$ which are not represented by circle actions

Part of: Symplectic geometry, contact geometry Homology and homotopy of topological groups and related structures Differential topology

Published online by Cambridge University Press:  30 June 2022

Sílvia Anjos Open the ORCID record for Sílvia Anjos [Opens in a new window] ,
Miguel Barata ,
Martin Pinsonnault  and
Ana Alexandra Reis
Sílvia Anjos*
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
Miguel Barata
Affiliation:
Utrecht Geometry Center, Utrecht University, Budapestlaan 6,3584 CD Utrecht, The Netherlands e-mail: m.lourencohenriquesbarata@uu.nl
Martin Pinsonnault
Affiliation:
Department of Mathematics, University of Western Ontario, London, ON, Canada e-mail: mpinson@uwo.ca
Ana Alexandra Reis
Affiliation:
Department of Mathematics, Instituto Superior Técnico, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal e-mail: ana.alexandra.reis@tecnico.ulisboa.pt
*
e-mail: sanjos@math.tecnico.ulisboa.pt
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Abstract

We study generators of the fundamental group of the group of symplectomorphisms $\operatorname {\mathrm{Symp}} (\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2, \omega )$ for some particular symplectic forms. It was observed by Kȩdra (2009, Archivum Mathematicum 45) that there are many symplectic 4-manifolds $(M, \omega )$, where M is neither rational nor ruled, that admit no circle action and $\pi _1 (\operatorname {\mathrm {Ham}} (M,\omega ))$ is nontrivial. On the other hand, it follows from Abreu and McDuff (2000, Journal of the American Mathematical Society 13, 971–1009), Anjos and Eden (2019, Michigan Mathematical Journal 68, 71–126), Anjos and Pinsonnault (2013, Mathematische Zeitschrift 275, 245–292), and Pinsonnault (2008, Compositio Mathematica 144, 787–810) that the fundamental group of the group $ \operatorname {\mathrm{Symp}}_h(\mathbb C\mathbb P^2\#\,k\overline { \mathbb C\mathbb P}\,\!^2,\omega )$, of symplectomorphisms that act trivially on homology, with $k \leq 4$, is generated by circle actions on the manifold. We show that, for some particular symplectic forms $\omega $, the set of all Hamiltonian circle actions generates a proper subgroup in $\pi _1(\operatorname {\mathrm{Symp}}_{h}(\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2,\omega )).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon’s classification of Hamiltonian $S^1$-spaces, and the computation of Seidel elements of some circle actions.

Keywords

symplectic geometrysymplectomorphism groupfundamental groupHamiltonian circle actions

MSC classification

Primary: 53D35: Global theory of symplectic and contact manifolds
Secondary: 57R17: Symplectic and contact topology 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms 57T20: Homotopy groups of topological groups and homogeneous spaces
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 4 , August 2023 , pp. 1226 - 1271
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The first author is partially supported by FCT/Portugal through projects UID/MAT/04459/2019 and PTDC/MAT-PUR/29447/2017. The third author is partially supported by NSERC Discovery Grant RGPIN-2020-06428. All authors except the third are supported by the Calouste Gulbenkian Foundation through the program “New Talents in Mathematics.”

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