Bibliographic record
Abstract
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; Z/ 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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".