On the Filtered Symplectic Homology of Prequantization Bundles

TL;DR

This paper investigates the dynamics of Reeb flows on prequantization bundles and introduces new algebraic tools to analyze symplectic homology, providing bounds, stability results, and a novel proof of the contact Conley conjecture.

Contribution

It introduces the linking number filtration on symplectic homology and applies it to prove the non-degenerate case of the contact Conley conjecture without contact homology.

Findings

Established upper bounds on symplectic capacity growth.

Proved uniform instability of filtered symplectic homology.

Provided a new proof of the existence of infinitely many Reeb orbits.

Abstract

We study Reeb dynamics on prequantization circle bundles and the filtered (equivariant) symplectic homology of prequantization line bundles, aka negative line bundles, with symplectically aspherical base. We define (equivariant) symplectic capacities, obtain an upper bound on their growth, prove uniform instability of the filtered symplectic homology and touch upon the question of stable displacement. We also introduce a new algebraic structure on the positive (equivariant) symplectic homology capturing the free homotopy class of a closed Reeb orbit -- the linking number filtration -- and use it to give a new proof of the non-degenerate case of the contact Conley conjecture (i.e., the existence of infinitely many simple closed Reeb orbits), not relying on contact homology.