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Math Help - Torus T^2 homeomorphic to S^1 x S^1

  1. #1
    Super Member Bernhard's Avatar
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    Torus T^2 homeomorphic to S^1 x S^1

    I am reading Martin Crossley's book, Essential Topology.

    Example 5.43 on page 74 reads as follows:


    Torus T^2 homeomorphic to S^1 x S^1-crossley-example-5.43-page-74.png


    I am really struggling to get a good sense of why/how/wherefore Crossley came up with the maps f and g in EXAMPLE 5.43. How did he arrive at these maps?

    Why/how does f map  S^1 \times S^1 onto  T^2 and how does one check/prove that this is in fact a valid mapping between these topological spaces.

    Can anyone help in making the origins of these maps clear or perhaps just indicate the logic behind their design and construction? I am completely lacking a sense or intuition for this example at the moment ... ...

    Definitions for  T^2 and  S^1 are as follows:

    Torus T^2 homeomorphic to S^1 x S^1-equation-torus-crossley-page-25.png
    Torus T^2 homeomorphic to S^1 x S^1-s-1-crossley-page-32.png

    My ideas on how Crossley came up with f and g are totally bankrupt ... but to validate f (that is to check that it actually maps a point of  S^1 \times S^1 onto  T^2 - leaving out for the moment the concerns of showing that f is a continuous bijection ... ... I suppose one would take account of the fact that (x,y) and (x',y') are points of TEX] S^1 [/TEX] and so we have:

     x^2 + y^2 = 1 ... ... ... ... (1)

    and

     x'^2 + y'^2 = 1 ... ... ... ... (2)

    Then, keeping this in mind check that

     ((x' +2)x, (x' +2)y, y' is actually a point on the equation for  T^2 , namely:

     x^2 + y^2 + z^2 - 4 \sqrt{x^2 + y^2} = -3 ... ... ... (3)

    SO in (3) we must:

    - replace x by (x' +2)x
    - replace y by (x' +2)y
    - replace z by y'

    and then simplify and if necessary use (1) (2) to finally get -3.

    Is that correct? Or am I just totally confused ?

    Can someone please help?

    Peter


    NOTE: The above has also been posted on MHB
    Last edited by Bernhard; March 19th 2014 at 05:54 PM.
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  2. #2
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    Re: Torus T^2 homeomorphic to S^1 x S^1

    Bernhard,
    When you post the same question on another site, I will not answer in detail. Besides, this expressly violates a rule of MHF. However here's a hint:

    How do you get a torus? Answer, for one such torus, rotate the unit circle centered at (2,0) on the x-axis about the y-axis. Write down the parametric equations for the resulting surface and the homeomorphism should be clear.
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