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Math Help - Klein Bottle Covering

  1. #1
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    Klein Bottle Covering

    Consider the equivalence relation ~ on \mathbb{R}^2.
    (x,y) \sim (x,y+1) and (x,y) \sim (x+1,-y)

    The quotient K = \mathbb{R}^2/ \sim is the Klein Bottle. Show that the quotient map p: \mathbb{R}^2\rightarrow K is a covering.

    Dont really know how to start I know I need to find an open cover of U_i wherethe preimage of each U_j under p is a disjoint union of open sets of which each is homeomorphic to U_j.
    Any hints are welcome. I basically dont see how this equivalence relation works. I know the Klein Bottle as quotient of I^2 with (0,y) ~ (1,y) and (x,0) ~ (1-x,1). Is this somehow related to this task?
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  2. #2
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    Re: Klein Bottle Covering

    choose any point, for example a=p(0,0), on K. choose a small open disk D centered at it with radius r<1/2,
    Then the pre-image of D is the union of disks centered at pre-images of a, which are all the points with integer coordinates.
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  3. #3
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    Re: Klein Bottle Covering

    Ok, you use the fact that all points equivalent to (0,0) are exactly the points (m,n) with m and n being integers. But what happens if you choose a non integer points, I mean where is the difference for example to the quotient map of the torus (i.e. where do we use the fact that the equivalence relation is not just (x,y) ~ (x,y+1) and (x,y) ~ (x+1,y) ? )
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  4. #4
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    Re: Klein Bottle Covering

    If you choose a non-integer point (x0,y0), then the equivalent points are (x0+m,y0+n) where m,n are integers.
    Locally there is no difference. But if you travel along a horizonal line in the plane, starting from (x,y), you jumps to different points after you proceed 1 unit of distance,
    That is, for a torus, from (x+1,y) to (x,y)
    for a Klein bottle, from (x+1,y) to (x,-y)
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  5. #5
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    Re: Klein Bottle Covering

    suppose we consider just the vertical strip IxR, and take a look at an open disk U of say, radius 1/8 on K, at the point p(1,1/2).

    then p^-1(U)∩(IxR) consists of a (left) half-circle of radius 1/8 centered at (1,1/2) and a (right) half-circle centered at (0,-1/2), plus vertical translates of these half-circles at unit intervals along IxR.

    if we extend our strip, to all of RxR, we see that we have alternating half-circles every unit (on the y-axis we have the "right half-circles", on x = 1, we have the "left haf-circles, etc.) horizontally, which also extend vertically.

    only disks which cross p({n}xR) display this unusual behavior (this is due to the "twist" of the klein bottle, whereas disks that intersect p(Rx{n}) just "unwrap" cylindrically, and their pre-images are just circles spaced at unit intervals).
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  6. #6
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    Re: Klein Bottle Covering

    So to prove that the quotient map p is a covering, I need to find an open cover of K, which open cover would be suitable?
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