Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology

Abstract

We relate the quantum Steenrod square to Seidel's equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We proceed similarly for $\mathbb{Z}/2$-equivariant symplectic cohomology, using an equivariant version of the continuation and $c^{\ast }$-maps. We prove a symplectic Cartan relation, pointing out the difficulties in stating it. We give a nonvanishing result for the equivariant pair-of-pants product for some elements of $S{H}^{\ast }(T^{\ast }{S}^n)$. We finish by calculating the symplectic square for the negative line bundles $M=\text{Tot}(\mathcal{O}(-1)\to {\mathbb{CP}}^m)$, proving an equivariant version of a result due to Ritter.

46 pages, 12 figures. Contents identical to previous version. Preprint of an article to appear in Journal of Topology and Analysis, DOI:10.1142/S1793525321500357 ©[copyright World Scientific Publishing Company], https://www.worldscientific.com/worldscinet/jta. Post-peer-review version will follow after a 12 month embargo period, as per open-access policy

References in corpus (1)

• A construction of the quantum Steenrod squares and their algebraic relations

Cited by in corpus (1)

• Formal groups and quantum cohomology