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Math Help - homotopy

  1. #1
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    homotopy

    If dimM=m<p, show that every map(maybe continuous) M^m \to S^p 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.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Stiger View Post
    If dimM=m<p, show that every map(maybe continuous) M^m \to S^p 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.
    Is this topological dimension or dimensionality as a manifold? Is S^p=\mathbb{S}^p=\left\{\bold{x}\in\mathbb{R}^{p-1}:\|\bold{x}\|=1\right\}?
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  3. #3
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    sorry I was wrong
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    Is this topological dimension or dimensionality as a manifold? Is S^p=\mathbb{S}^p=\left\{\bold{x}\in\mathbb{R}^{p-1}:\|\bold{x}\|=1\right\}?
    Yes! You're right!!
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