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Math Help - Algebraic Topology Help

  1. #1
    Junior Member
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    Algebraic Topology Help

    Any help with these would be great.

    1. (a) Describe all possible covering maps X -> RP^2 x RP^2
    (b) Describe all possible covering maps X -> RP^2

    2. Show that there is a covering map from the cylinder to the Mobius band

    3. Find a two sheeted covering map
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  2. #2
    Senior Member Tinyboss's Avatar
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    (1) Hint: the fundamental group of a covering space is isomorphic to a subgroup of the fundamental group of the space it covers.
    (2) The Mobius band is not orientable because of that half-twist. But if you go around twice, it adds up to a full twist...
    (3) See (2).
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  3. #3
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    The idea of Galois connection is helpful as mentioned above.

    Let Y=RP^2 X RP^2
    Since Y is semi-locally simply connected, you can apply Galois connection here.

    Note that pi_1(Y)=Z/2 X Z/2.

    Now we apply Galois connection,

    {e} : S^2 X S^2 ----> This corresponds to a universal cover.
    Z/2 X {e} : RP^2 X S^2
    Z/2 X Z/2 : RP^2 X RP^2
    ...
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