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Hamiltonian no-torsion

Marcelo S Atallah and Egor Shelukhin

Geometry & Topology 27 (2023) 2833–2897

In 2002, Polterovich established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, also called Hamiltonian torsion, must be trivial. We prove the first higher-dimensional Hamiltonian no-torsion theorems beyond that of Polterovich, by considering the dynamical aspects of the problem. Our results are threefold.

First, we show that closed symplectic Calabi–Yau and negative monotone symplectic manifolds admit no Hamiltonian torsion. A key role is played by a new notion of a Hamiltonian diffeomorphism with nonisolated fixed points.

Second, going beyond topological constraints by means of Smith theory in filtered Floer homology, barcodes and quantum Steenrod powers, we prove that every closed positive monotone symplectic manifold admitting Hamiltonian torsion is geometrically uniruled by pseudoholomorphic spheres. In fact, we produce nontrivial homological counts of such curves, answering a close variant of Problem 24 from the introductory monograph of McDuff and Salamon. This provides additional no-torsion results and obstructions to Hamiltonian actions of compact Lie groups, related to a celebrated result of McDuff from 2009, and lattices such as SL (k, ) for k 2. We also prove that there is no Hamiltonian torsion diffeomorphism with noncontractible orbits.

Third, by defining a new invariant of a Hamiltonian diffeomorphism, we prove a first nontrivial symplectic analogue of Newman’s 1931 theorem on finite groups of transformations. Namely, for each monotone symplectic manifold there exists a neighborhood of the identity in the Hamiltonian group endowed with Hofer’s metric or Viterbo’s spectral metric that contains no finite subgroups.

Hamiltonian diffeomorphisms, finite group actions, Floer homology, barcodes, Smith theory, spectral invariants, quantum Steenrod powers
Mathematical Subject Classification
Primary: 37J11, 53D22, 53D40
Secondary: 57R17, 57S17
Received: 20 February 2021
Revised: 14 September 2021
Accepted: 5 January 2022
Published: 19 September 2023
Proposed: Yakov Eliashberg
Seconded: Anna Wienhard, David Fisher
Marcelo S Atallah
Department of Mathematics and Statistics
University of Montreal
Centre-Ville Montreal, QC
Egor Shelukhin
Department of Mathematics and Statistics
University of Montreal
Centre-Ville Montreal, QC

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