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, R).


Received: November 2, 2017


Daniel Gomez
Department of Mathematics
University of Pittsburgh


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