An algebraic approach to the algebraic Weinstein conjecture

TL;DR

This paper explores the connection between Hochschild homology's behavior with colimits and the Weinstein conjecture in contact geometry, proposing an algebraic approach to a major open problem.

Contribution

It establishes an equivalence between the Weinstein conjecture for certain contact manifolds and the failure of Hochschild homology to commute with specific colimits in representation categories.

Findings

Hochschild homology's colimit behavior relates to Reeb orbit detection.

For polarizably Weinstein fillable manifolds, the conjecture aligns with algebraic colimit failure.

Provides an algebraic framework for understanding a geometric conjecture.

Abstract

How does one measure the failure of Hochschild homology to commute with colimits? Here I relate this question to a major open problem about dynamics in contact manifolds -- the assertion that Reeb orbits exist and are detected by symplectic homology. More precisely, I show that for polarizably Weinstein fillable contact manifolds, said property is equivalent to the failure of Hochschild homology to commute with certain colimits of representation categories of tree quivers. So as to be intelligible to algebraists, I try to include or black-box as much of the geometric background as possible.