Cyclotomic Structures in Symplectic Topology

TL;DR

This paper develops a new equivariant Floer theory framework using stable homotopy types, enabling the construction of cyclotomic structures on symplectic cohomology for certain symplectic manifolds, linking Floer theory and p-adic Hodge theory.

Contribution

It introduces a general method for equivariant Floer theory avoiding transversality issues and constructs cyclotomic structures on symplectic cohomology in a broad setting.

Findings

Constructs equivariant orthogonal spectra from flow categories.

Establishes cyclotomic structures on symplectic cohomology.

Links Floer homology with p-adic Hodge theory.

Abstract

We extend the Cohen-Jones-Segal construction of stable homotopy types associated to flow categories of Morse-Smale functions $f$ to the setting where $f$ is equivariant under a finite group action and is Morse but no longer Morse-Smale. This setting occurs universally, as equivariant Morse functions can rarely be perturbed to nearby equivariant Morse-Smale functions. The method is very general, and allows one to do equivariant Floer theory while avoiding all the complications typically caused by issues of equivariant transversality. The construction assigns a (genuine) equivariant orthogonal spectrum to an equivariant framed virtually smooth flow category. Using this method, we construct, for a compact symplectic manifold $M$, which is symplectically atoroidal with contact boundary, and is equipped with an equivariant trivialization of its polarization class, a cyclotomic structure on the spectral lift of the symplectic cohomology $SH^*(M)$. This generalizes a variant of the map which sends loops to their $p$-fold covers on free loop spaces to the setting of general Liouville domains, and suggests a systematic connection between Floer homology and $p$-adic Hodge theory.