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Math Help - Hopf Invariant (Differential Topology)

  1. #1
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    Hopf Invariant (Differential Topology)

    Hello everyone, this is a problem out of Milnor (Topology from a Differentiable viewpoint) I have been assigned to solve. First, definition of the "Linking number"

    l(M,N)=deg(\lambda)\mbox{ where }\lambda :M\times N \to \mathbb{S}^{m+n} \mbox{ by }\lambda (x,y)=\frac{x-y}{|x-y|}

    Still with me? Ok, you can define the "Hopf Invariant" H(f) in the following way:

     \mbox{For }y\ne z \mbox{ regular values for } f:\mathbb{S}^{2p-1}\to\mathbb{S}^p \mbox{ then }H(f)=l(f^{-1}(y),f^{-1}(z))

    So I need to show three things (and any help on ANY them is appreciated :-) )

    a) The Hopf Invariant is locally constant as a function of y.

    b) If y and z and regular for g also and |f(x)-g(x)|<|y-z| then

    l(f^{-1}(y),f^{-1}(z))=l(g^{-1}(y),f^{-1}(z))=l(g^{-1}(y),g^{-1}(z))

    c) Prove H(f) depends only on the homotopy class of f and does not depend on the choice of y and z.

    Basically, prove H(f) is a well-defined invariant. Any thoughts?
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  2. #2
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    nobody? nothing?
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