Conjugacy invariants for Brouwer mapping classes

TL;DR

This paper introduces new tools in homotopy Brouwer theory, including a canonical reducing set and a tangle, to classify Brouwer mapping classes relative to four orbits.

Contribution

It provides an explicit description of Brouwer mapping classes with four orbits by extending existing diagrams with a new tangle invariant.

Findings

Canonical reducing set of walls for plane decomposition

Explicit classification of Brouwer classes with four orbits

Introduction of a tangle invariant in the classification

Abstract

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set (the set of "walls") which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relatively to four orbits and describe them explicitly by adding to Handel's diagram and to the set of walls a "tangle", which is essentially an isotopy class of simple closed curves in the cylinder minus two points.