So I understand that one can create a closed loop (inside/on) a mobius strip. And cutting if we cut along the loop what remains will still be a connected surface.
Meaning that the genus is $\geq 1$. However I don't understand what stops us from drawing another loop. Following this around the loop we get that it does not intersect our previous loop, or itself. And if we cut along this loop, we again get a connected surface. So I can't see why the genus of a mobius strip is 1.
What am I missing here?
Best Answer
The connected surface that results from cutting along a loop on a Mobius strip is homeomorphic to a cylinder. But cutting along a loop on a cylinder will always produce a disconnected space (either a disk and a cylinder minus a disk, or two cylinders).