Pseudo-rotations and Steenrod squares revisited

TL;DR

This paper demonstrates that the existence of Hamiltonian pseudo-rotations on closed monotone symplectic manifolds imposes restrictions on quantum Steenrod squares, revealing geometric constraints and uniruledness properties.

Contribution

It proves that pseudo-rotations deform the quantum Steenrod square of the point class, extending previous results with a new elementary calculation involving capped periodic orbits.

Findings

Quantum Steenrod squares are deformed by pseudo-rotations.

Pseudo-rotations imply uniruledness by pseudo-holomorphic spheres.

The proof introduces an elementary calculation with capped periodic orbits.

Abstract

In this note we prove that if a closed monotone symplectic manifold admits a Hamiltonian pseudo-rotation, which may be degenerate, then the quantum Steenrod square of the cohomology class Poincar\'{e} dual to the point must be deformed. This result gives restrictions on the existence of pseudo-rotations, implying a form of uniruledness by pseudo-holomorphic spheres, and generalizes a recent result of the author. The new component in the proof consists in an elementary calculation with capped periodic orbits.