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Math Help - Mobius Band and its Boundary

  1. #1
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    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.
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  2. #2
    Senior Member
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    Quote Originally Posted by DJDorianGray View Post
    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}.
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