# Math Help - Mobius Band and its Boundary

1. ## Mobius Band and its Boundary

Hi all,
I need someone to explain me this one:

Prove that the the mobius band does not retract to its boundary.

It should be a well-known property from what I understand, but I cannot find the proof anywhere. Thanks in advance for your help.

2. Originally Posted by DJDorianGray
Hi all,
I need someone to explain me this one:

Prove that the the mobius band does not retract to its boundary.
A Mobius band deformation retracts to its middle circle. Thus, $\pi_1(M)=\pi_1(S)=\mathbb{Z}$, where M is a Mobius band.

Let B be a boundary circle of a Mobius band. Then $f:\pi_1(S) \rightarrow \pi_1(B)$ is induced by a degree 2 map of its central circle to itself. Thus $\pi_1(B) = 2\mathbb{Z}$. We conclude that B cannot be a retract of a Mobius band whose fundamental group is $\pi_1(M)=\pi_1(S)=\mathbb{Z}$.