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Math Help - Helicoid

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    MHF Contributor Bruno J.'s Avatar
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    Helicoid

    Prove that the helicoid is homeomorphic to the plane.
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    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Bruno J. View Post
    Prove that the helicoid is homeomorphic to the plane.
    We have  \begin{cases} x=r\cos(a\theta)\\y=r\sin(a\theta)\\z=\theta \end{cases} .

    Can't we just let  a\to0 (in a continuous manner to make it a homeomorphism)?
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by chiph588@ View Post
    We have  \begin{cases} x=r\cos(a\theta)\\y=r\sin(a\theta)\\z=\theta \end{cases} .

    Can't we just let  a\to0 (in a continuous manner to make it a homeomorphism)?
    So what is the image of (x,y,z) under this map? *

    *corrected
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    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Bruno J. View Post
    So what is the image of (x,y,z) under this map? *

    *corrected
     (x,y,z)\mapsto(r,0,\theta) where  r and  \theta are free variables, so this map is the x-z plane.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by chiph588@ View Post
     (x,y,z)\mapsto(r,0,\theta) where  r and  \theta are free variables, so this map is the x-z plane.
    Good!

    Here are a few abstract thoughts.

    We know the universal covering space of the punctured plane \mathbb{C}-\{0\} is the plane \mathbb{C}, with the projection \mbox{exp }: \mathbb{C} \to \mathbb{C}-\{0\}.

    On the other hand, the universal cover of \mathbb{C}-\{0\} is also the Riemann surface of the (multi-valued) inverse function \mbox{Log}, of which the helicoid is the most obvious model. By the universal property of the universal cover, there exists a homeomorphism between the helicoid and the plane.

    (Informally : the slit plane \mathbb{C}-\mathbb{R}_{\geq 0} is easily seen to be homeomorphic to a strip without its boundary, (any branch of the logarithm doing the job). Now lift the cut to the helicoid, and map each slice of the helicoid to a strip, identifying the appropriate boundaries of two strips if the corresponding boundaries of the slices are identified on the helicoid. In this way, the helicoid is mapped homeomorphically on the plane.)
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