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Show that every map(maybe continuous)

  1. Jun 9, 2010 #1
    If dimM=m<p, show that every map(maybe continuous) Mm -> Sp is homotopic to a constant.

    This is the problem 5 in chap8. of 'topology from the differentiable viewpoint(Milnor)'.

    I proved it when the map is not onto. But I think it can be onto.
    Please help me.
  2. jcsd
  3. Jun 9, 2010 #2


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    Re: homotopy

    It can indeed be onto. For instance, for M=S^1 and S^p=S², take f:S^1-->[0,1]² one of the infamous space-filling curve (loop) (http://en.wikipedia.org/wiki/Space-filling_curve). Then make [0,1]² into S² by identifying all the edges together. Then p o f:S^1-->S² is a continuous surjection, where p:[0,1]²-->[0,1]²/~=S² is the quotient map.
  4. Jun 9, 2010 #3


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    Re: homotopy

    Is the map maybe supposed to be differentiable?
  5. Jun 11, 2010 #4
    Re: homotopy

    Maybe all you need is that it's homotopic to a smooth map :)

    (Then Sard's Theorem)
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