Connectedness of the Cut System Complex on Nonorientable Surfaces

Fatema Ali, Ferihe Atalan

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. In this note, we show that the cut system complex of $N$ is connected for $g < 4$ and disconnected for $g \geq 4$. We then define a related complex and show that it is connected for $g \geq 4$.