Hamiltonian no-torsion

Atallah, Marcelo S¹ ; Shelukhin, Egor¹

¹ Department of Mathematics and Statistics, University of Montreal, Centre-Ville Montreal, QC, Canada

Geometry & topology, Tome 27 (2023) no. 7, pp. 2833-2897.

Résumé

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 $\mathrm{SL}(k,\mathbb{Z})$ for $k\ge 2$. 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.

Détail

DOI : 10.2140/gt.2023.27.2833

Keywords: Hamiltonian diffeomorphisms, finite group actions, Floer homology, barcodes, Smith theory, spectral invariants, quantum Steenrod powers

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