Sphere systems, standard form, and cores of products of trees

TL;DR

This paper introduces a standard form for pairs of maximal sphere systems in doubled handlebodies, establishing a bijective correspondence with certain square complexes, thus advancing the understanding of their structure.

Contribution

It defines a new standard form for sphere systems in doubled handlebodies and proves an existence and uniqueness theorem, linking these systems to specific square complexes.

Findings

Existence and uniqueness of the standard form for sphere systems

Bijection between sphere systems and square complexes

Extension of Hatcher's normal form

Abstract

We introduce the concept of a standard form for two embedded maximal sphere systems in the doubled handlebody, and we prove an existence and uniqueness result. In particular, we show that pairs of maximal sphere systems in the doubled handlebody (up to homeomorphism) bijectively correspond to square complexes satisfying a set of properties. This work is a variant on Hatcher's normal form.