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Homotopic maps on a sphere

  1. Jul 15, 2013 #1
    If I take two arbitrary continuous maps ##f,g:S^n\rightarrow S^n## such that ##f(x) \neq -g(x)## for any ##x\in S^n##, then ##f## and ##g## are homotopic.

    How do I show this result? I really don't see how to use the condition that ##f## and ##g## never occupy two antipodal points. Any hint would be appreciated.
     
  2. jcsd
  3. Jul 15, 2013 #2

    WannabeNewton

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    Consider the straight line homotopy ##H(x,t) = f(x) + t(g(x) - f(x))##. Try to force the straight line homotopy onto the sphere; you'll see how the constraint on ##f(x)## and ##g(x)## not occupying antipodal points comes into play.
     
  4. Jul 15, 2013 #3
    Thanks a lot, miss!

    So, my idea is to take

    [tex]H(x,t) = \frac{f(x) + t(g(x) - f(x))}{\|f(x) + t(g(x) - f(x))\|}[/tex]

    I guess the constraint on ##f## and ##g## comes into play because we don't want the denominator to vanish? But I have troubles proving this rigorously. Assume that the denominator is ##0##, then

    [tex](t-1)f(x) = tg(x)[/tex]

    I'm pretty stuck now!
     
  5. Jul 15, 2013 #4

    WannabeNewton

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    ##f(x)## and ##g(x)## are both elements of ##S^{n}##; take the norm of both sides.
     
  6. Jul 15, 2013 #5
    Wow, I didn't think of that! Why are you so smart?
     
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