# The fundamental group of the punctured Klein bottle and the simple loop conjecture

By Daniel Gomez – GJM, Volume 2, Issue 2 (2017), 59-66.

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 to the simple loop conjecture for representations from the Klein bottle group to $PGL(2, {\mathbb R})$.

Categories: Issue2, 2017