[Paper Review] Hamiltonian no-torsion

Marcelo S Atallah, Egor Shelukhin|arXiv (Cornell University)|Aug 26, 2020

Geometric and Algebraic Topology73 references4 citations

TL;DR

This paper establishes higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case, proving that closed symplectic Calabi-Yau and negative monotone manifolds admit no nontrivial Hamiltonian torsion. Using generalized Morse-Bott methods, quantum Steenrod powers, and filtered Floer homology, it shows that positive monotone manifolds with Hamiltonian torsion must be geometrically uniruled, and that such subgroups cannot lie in arbitrarily small neighborhoods of the identity under natural norms on the Hamiltonian group.

ABSTRACT

In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case. We start by showing that closed symplectic Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian torsion. Going still beyond topological constraints, we prove that every closed positive monotone symplectic manifold $(M,ω)$ admitting Hamiltonian torsion is geometrically uniruled by holomorphic spheres for every $ω$-compatible almost complex structure, partially answering a question of McDuff-Salamon. This provides many additional no-torsion results, and as a corollary yields the geometric uniruledness of monotone Hamiltonian $S^1$-manifolds, a fact closely related to a celebrated result of McDuff from 2009. Moreover, the non-existence of Hamiltonian torsion implies the triviality of Hamiltonian actions of lattices like $SL(k,\mathbb{Z})$ for $k \geq 2,$ as well as those of compact Lie groups. Finally, for monotone symplectic manifolds admitting Hamiltonian torsion, we prove an analogue of Newman's theorem on finite transformation groups for several natural norms on the Hamiltonian group: such subgroups cannot be contained in arbitrarily small neighborhoods of the identity. Our arguments rely on generalized Morse-Bott methods, as well as on quantum Steenrod powers and Smith theory in filtered Floer homology.

Motivation & Objective

• To extend Polterovich's 2002 Hamiltonian no-torsion theorem beyond symplectically aspherical manifolds.
• To establish the nonexistence of Hamiltonian torsion in closed symplectic Calabi-Yau and negative monotone manifolds.
• To investigate the geometric and topological consequences of Hamiltonian torsion in positive monotone symplectic manifolds.
• To prove that Hamiltonian torsion implies ${\mathbb{F}}_p$-Steenrod uniruledness and geometric uniruledness in the positive monotone case.
• To show that Hamiltonian torsion subgroups cannot be contained in arbitrarily small neighborhoods of the identity under natural norms on the Hamiltonian group.

Proposed method

• Employ generalized Morse-Bott methods to analyze the structure of periodic orbits and fixed points of Hamiltonian diffeomorphisms.
• Use filtered Floer homology with ${\mathbb{F}}_p$ coefficients to study the local homology of generalized fixed points under finite group actions.
• Apply quantum Steenrod powers in the context of filtered Floer homology to detect uniruledness conditions.
• Utilize Smith theory in local Floer homology to derive inequalities on the dimensions of homology groups of fixed point sets.
• Leverage spectral invariants and the PSS isomorphism to relate action spectra to cohomological data.
• Establish the constancy of local Floer homology dimensions under iteration via Smith inequality and finite order constraints.

Experimental results

Research questions

• RQ1 Do closed symplectic Calabi-Yau and negative monotone manifolds admit nontrivial Hamiltonian torsion?
• RQ2 What geometric or cohomological constraints arise when a positive monotone symplectic manifold admits Hamiltonian torsion?
• RQ3 Can Hamiltonian torsion subgroups be arbitrarily small in neighborhoods of the identity under natural norms on the Hamiltonian group?
• RQ4 Does the existence of Hamiltonian torsion imply ${\mathbb{F}}_p$-Steenrod uniruledness or geometric uniruledness?
• RQ5 How do quantum Steenrod powers and filtered Floer homology detect the absence of torsion in symplectic manifolds beyond the aspherical case?

Key findings

• Closed symplectic Calabi-Yau and negative monotone manifolds do not admit Hamiltonian torsion.
• Every closed positive monotone symplectic manifold admitting Hamiltonian torsion is geometrically uniruled by $J$-holomorphic spheres for every $\omega$-compatible almost complex structure $J$.
• The existence of Hamiltonian torsion implies ${\mathbb{F}}_p$-Steenrod uniruledness for certain primes $p$, which in turn implies geometric uniruledness.
• Hamiltonian actions of lattices such as $SL(k,\mathbb{Z})$ for $k \geq 2$, and of compact Lie groups, must be trivial if the manifold admits no Hamiltonian torsion.
• For monotone symplectic manifolds with Hamiltonian torsion, the subgroup generated by such diffeomorphisms cannot be contained in arbitrarily small neighborhoods of the identity under any of several natural norms on the Hamiltonian group.
• The local Floer homology dimensions of iterated maps remain constant under iteration, due to the finite order of the diffeomorphism and the Smith inequality in filtered Floer homology.