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    (Original post by dave345)
    Has anyone got anywhere at all with quesion 4, sheet 4? Soooo hard!
    4a) Id = sts^(-1)t
    b) let g \in G and write g as its (unique?) product under the generators a,b. Associate s with a and t with b. Then for each element in G there is a unique function R^2 to R^2 which is defined as the composition of the functions s and t replacing a and b in the factorisation of g. i.e. ig g = aba^2b^(-1) then the function is sts^2t^(-1). Then the action on R^2 is defined for g\in G by applying the associated function to (x,y).

    This action is free and properly dis-continious (take B((x,y),1/4) as your open neighbourhood), so then the map R^2 to R^2/G is a covering map, and the orbit space R^2/G is homeomorphic to the klein bottle (I can only prove this by drawing a picture if you sketch out what the quotient space looks like in R^2 though I think it is clear).

    c) s^2(x,y) = (x, y+1). So then it should be clear that by defining the group action of H on R^2 as in the previous part is free and properly discontinuous . Then the map R^2 to R^2/H is a covering space, and further H is isomorphic to the fundamental group of R^2/H as R^2 is simply connected (result in lectures). But the identification space identifies (x,y) with (x+n,y+m) for n,m integers and so it is clear (draw a picture) that this is homeomorphic to the torus. So R^2/H = T, but the fundamental group of T is ZxZ the product of the infinite cyclic group.

    d) I think the map you want maps you want does this:
    a
    _____>_____
    | |
    | |
    b ^ ^ b
    | |
    |_____>____|
    a
    Covering map :

    a c
    ___>____>__
    | | |
    | | |
    b ^ \/b ^ b
    | | |
    |__>__|__>_|
    a c

    I hope that picture clears it up?

    Edit: Piece of crap doesn't let me draw
    Ok well if you have the torus as a (unit) square with opposite edges identified without a twist, the take the following identification for (x,y) with x<= 1/2 sticks to (x +1/2, 1-y). This gives you a two-sheeted cover.
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    (Original post by jimbob_barnes)
    ..
    For (b) and (c): that the action is properly discontinuous would follow from it being free and \textbf{R}^2 being Hausdorff. However, I don't know how to prove it is free.
    Also, you're basically showing the homeomorphisms in (b) and (c) by visualizing them, this probably works for the course, but doesn't seem very satisfying in general.

    For (d), the picture is this:


    You slice the id space of the torus vertically in half and map the left part via translation and the right part by flipping it upside down.
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    Yes that was what I was trying to draw, as I'm not really a forum grinder I don't know how to embed images.

    You can easily make the R^2 covering space and map to the torus explicit, (x,y) maps to (e^(i2\pi x), e^(i2\pi y)) works for instance.
 
 
 
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