Reduced symplectic homology and string topology

TL;DR

This paper introduces a unified framework for loop product and coproduct in reduced loop homology, establishing a bialgebra structure and confirming a Sullivan conjecture with an additional term, within symplectic homology of Weinstein manifolds.

Contribution

It develops a common domain for loop operations, proves a bialgebra structure, and extends string topology results to reduced symplectic homology in Weinstein manifolds.

Findings

Unified structure for loop product and coproduct in reduced homology

Confirmation of Sullivan's conjecture with an extra term

Extension of string topology to symplectic homology of Weinstein manifolds

Abstract

We introduce a common domain of definition for the loop product and the loop coproduct, reduced loop homology, on which they combine to a unital infinitesimal anti-symmetric bialgebra structure. In particular, a relation conjectured by Sullivan holds with an extra term. The structure depends on choices governed by secondary continuation maps. These results on string topology are proved in the more general context of reduced symplectic homology for a suitable class of Weinstein manifolds.