# Boundary of a Mobius band - I think S1 V S1, everyone else says S1?

1. Mar 5, 2013

### dumbQuestion

Boundary of a Mobius band - I think S1 V S1, everyone else says S1??

Hey I am having a huge problem! There are a few problems where I'm using Van Kampen's theorem and for one part of the problem I need to compute the fundamental group of the boundary of the Mobius band. Everyone keeps telling me this boundary is homotopic to S1 with fundamental group Z, but for the life of me I can't figure out how. Every way I do it I end up getting the wedge of two circles and so have a fundamental group of Z*Z. Can someone tell me who is right and who is wrong?

NEVERMIND! I figured it out. When I was drawing the boundary of the Mobius band, I was stupidly looking at the place where the lines "flip around" as a single point, and seeing that as the wedge point between two circles. Boy do I feel dumb!

2. Mar 5, 2013

### Bacle2

Something else you can do, other than SVK, is to use the fact that S1 is a retract of the Mobius Band, so that the two are homotopically equivalent.