Strings, fermions and the topology of curves on annuli

TL;DR

This paper explores the algebraic structure of string homology on annuli, revealing complex fermionic properties and higher-order structures, extending previous work on discs and connecting to Floer homology.

Contribution

It introduces the computation and analysis of string homology for annuli, uncovering rich algebraic and fermionic structures beyond the disc case.

Findings

String homology on annuli has a complex, fermionic algebraic structure.

Compared to discs, annuli exhibit additional higher-order structures in string homology.

The relationship between string homology and Floer homology is more intricate for annuli.

Abstract

In previous work with Schoenfeld, we considered a string-type chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs. In this paper we consider the corresponding "string homology" of annuli. We find this homology has a rich algebraic structure which can be described, in various senses, as fermionic. While for discs we found an isomorphism between string homology and the sutured Floer homology of a related 3-manifold, in the case of annuli we find the relationship is more complex, with string homology containing further higher-order structure.