# Hopf Invariant (Differential Topology)

• Nov 15th 2008, 03:46 PM
cduston
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?
• Nov 19th 2008, 06:38 AM
cduston
nobody? nothing?