Pseudo-rotations and Steenrod squares

Abstract

In this note we prove that if a closed monotone symplectic manifold M of dimension 2n, satisfying a homological condition, that holds in particular when the minimal Chern number is N > n, admits a Hamiltonian pseudorotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudorotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.

Covariant constancy of quantum Steenrod operations

“…Quantum Steenrod operations, originally introduced by Fukaya [7], have recently appeared in a variety of contexts: their properties have been explored in [18] (which also contains the first nontrivial computations); they can be used to study arithmetic aspects of mirror symmetry [14]; and in Hamiltonian dynamics, they are relevant for the existence of pseudo-rotations [16,2,17]. Nevertheless, computing quantum Steenrod operations remains a challenging problem in all but the simplest cases.…”