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Math Help - Topology question

  1. #1
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Topology question

    I am looking for a surjective map

    g:\{S^{n-1}\times [-1,1] \} \to S^{n}

    Where S^n=\{ x \in \mathbb{R}^{n+1}:||x||=1 \}

    I have figured out the case for n=1(The easy one)

    It is basically converting to sphereical coordinates.

    I can't find a closed form for it(I'm not even sure if one exists) for general n.

    Just for context I'm trying to show that the suspension of S^{n-1} is homeomorphic to S^n

    So I need the function g above to be a quotient map.

    Thanks for any input
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  2. #2
    Behold, the power of SARDINES!
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    let \vec x \in S^{n-1} and t\in [-1,1]

    What I have been thinking is this (I know it is not a function)

    g(\vec x, t)=(h(\vec x),1-t^2)

    Where h( \vec x,t) =\{x_1^2+x_2^2+...x_{n-1}^2=1-t^2 \}

    The problem with this is it returns a set and is not well defined. I guess what I'm not sure about is it that if I require each x_i from the input to have the same sign in the output would this fix the defect?

    My intution says yes, but then again I have been working on this for a while and maybe I just want it to be right?

    Thanks again

    TES
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  3. #3
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    g(\vec x, t)=((1-t^2)^{1/2}\vec x,t)
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  4. #4
    Behold, the power of SARDINES!
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    Quote Originally Posted by Opalg View Post
    g(\vec x, t)=((1-t^2)^{1/2}\vec x,t)
    Thank you.
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