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Math Help - Retracts of the Mobius Band

  1. #1
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    Retracts of the Mobius Band

    Let M be the Mobius band. Prove there is no retract from M to its boundary.

    Can I use fundamental group stuff here?
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  2. #2
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    Quote Originally Posted by robeuler View Post
    Let M be the Mobius band. Prove there is no retract from M to its boundary.

    Can I use fundamental group stuff here?
    The below lemma can be found in the standard topology/algebraic topology books.

    ================================================== =======
    Lemma 1. If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion j:A \rightarrow X is injective.

    Proof: If  r:X \rightarrow A is a retraction, then the composite map r \cdot j equals the identity map of A. It follows that r_* \cdot j_* is the identity map of \pi_1(A,a), so that j_* must be injective, where r_* and j_* denote the induced homomorphism by topological continuous maps r and j.
    ================================================== =======

    A deformation retract of a Mobius band is its middle circle. A degree 1 map of a circle involves a degree 1 map of a middle circle of a Mobius band. Thus, the induced map j_*:\pi(S^1) \rightarrow \pi(M) in Lemma 1 is an isomorphism.

    However, a Mobius band is a non-orientable two-dimensional manifold (surface). A two dimensional figure in a Mobius band cannot be moved around the middle circle once and back to where it started. Actually, coming back to the initial point of a Mobius band involves two turns of its middle (or inner) circle. If the boundary were to be a retract of a Mobius band, it is homotopic to the degree 2 map of a middle circle to itself.

    A degree 1 map of a circle to a subset A of a Mobius band and degree 2 map of a circle of a subset B of a Mobius band cannot induce the same homomorphisms (isomorphisms) between their fundamental groups. Since the middle circle is a deformation retract of a Mobius band, the boundary of a Mobius band cannot be the deformation retract of a Mobius band. It follows that r_* \cdot j_* in lemma 1 is not the identity map if the boundary were to be a retract of a Mobius band.

    Thus, there is no retract from a Mobius band to its boundary.
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