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 $c^*$-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 $SH^*(T^* S^n)$. We finish by calculating the symplectic square for the negative line bundles $M = \text{Tot}(\mathcal{O}(-1) \rightarrow \mathbb{CP}^m)$, proving an equivariant version of a result due to Ritter.