Localization of Floer homology for $\mathbb{Z}_p$-actions

Abstract

For an odd prime $p$, we prove a Smith inequality over $\mathbb{F}_p$ for exact Lagrangian Floer homology under a symplectic $\mathbb{Z}_p$-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a $p$-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free $\mathbb{Z}_p$-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.