A Classification of Fundamental Group Elements Representing simple closed curves on the punctured Klein Bottle

TL;DR

This paper classifies fundamental group elements representing simple closed curves on the punctured Klein bottle, describes the mapping class group explicitly, and provides a counterexample to the simple loop conjecture for certain representations.

Contribution

It offers a new classification of simple closed curves on the punctured Klein bottle and explicitly describes its mapping class group, with implications for the simple loop conjecture.

Findings

Classification of fundamental group elements for the punctured Klein bottle

Explicit description of the mapping class group

Counterexample to the simple loop conjecture

Abstract

In this paper we provide a classification of fundamental group elements representing simple closed curves on the punctured Klein bottle, Similar to the Birman-Series classification of curves on the punctured torus[1]. In the process, an explicit description of the mapping class group is given. We then apply this to give a counterexample the simple loop conjecture for representations from the Klein bottle group to PGL(2,R).