In exercise 15 of Milnor's *Topology from a Differentiable Viewpoint*, one is asked to compute the Hopf invariant of the Hopf map. The way one is supposed to do this is to compute the linking number of two of the fibres, but Milnor doesn't define the linking number in terms of an integral. He says to compute it as the degree of the map $\frac{x-y}{||x-y||}$ from the product of two compact oriented boundaryless manifolds embedded in $\mathbf{R}^{k+1}$ to the sphere of dimension $k$ where the sum of the dimension of the manifolds is $k$.

I'm aware of other ways to compute the Hopf invariant by using deRham cohomology (see Bott and Tu, for instance), but I'm curious how one is actually supposed to do it by hand. Is there a particularly concrete way to compute the linking number without using this other machinery? Most of the other exercises in the book have cute little solutions, but is that true of this problem?

(Not homework!!)

isgiven by an integral. Think about $\int_M f^* \omega$, provided $\int_N \omega = 1$. – Ryan Budney Oct 17 '10 at 23:45