A flexible construction of equivariant Floer homology and applications

TL;DR

This paper introduces a new flexible approach to equivariant Floer cohomology, establishing invariance properties, comparing spectral sequences, improving inequalities, and defining new invariants with applications in symplectic and knot theory.

Contribution

It provides a novel construction of equivariant Floer cohomology, proves invariance of spectral sequences, and introduces new invariants and conjecturally related spectral sequences.

Findings

Proved invariance properties of spectral sequences

Improved Smith-type inequalities

Defined new concordance invariants

Abstract

Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsv\'ath-Szab\'o, Lipshitz-Treumann, Szab\'o, Sarkar-Seed-Szab\'o, and others. In this paper we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith-type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar-Seed-Szab\'o's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.