The $\mathbb{Z} /p \mathbb{Z}$-equivariant product-isomorphism in fixed point Floer cohomology

Let $p \geq 2$ be a prime, and $\mathbb{F}^p$ be the field with $p$ elements. We construct an isomorphism between the Floer cohomology of an exact or Hamiltonian symplectomorphism $\phi$, with $\mathbb{F}^p$ coefficients, and the $\mathbb{Z} /p \mathbb{Z}$‑equivariant Tate Floer cohomology of its $p$‑th power $\phi ^p$. This extends a result of Seidel for $p=2$. The construction involves a Kaledin-type quasi-Frobenius map, as well as a $\mathbb{Z} /p \mathbb{Z}$‑equivariant pants product: an equivariant operation with $p$ inputs and $1$ output. Our method of proof involves a spectral sequence for the action filtration, and introduces a new key component: a local $\mathbb{Z} /p \mathbb{Z}$‑equivariant coproduct providing an inverse on the $E^2$‑page. This strategy has the advantage of accurately describing the effect of the isomorphism on filtration levels. We describe applications to the symplectic mapping class group, as well as develop Smith theory for the persistence module of a Hamiltonian diffeomorphism $\phi$ on symplectically aspherical symplectic manifolds. We illustrate the latter by giving a new proof of the celebrated no-torsion theorem of Polterovich, and by relating the growth rate of the number of periodic points of the $p^k$‑th iteration of $\phi$ and its distance to the identity. Along the way, we prove a sharpening of the classical Smith inequality for actions of $\mathbb{Z} /p \mathbb{Z}$.

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Received 11 August 2020

Accepted 20 December 2020

Published 27 May 2022