Problem 14 and 15 in Milnor is essentially to show the Hopf invariant with only tools from differential topology. Suppose we consider $f:S^3\to S^2$, and define $H(f)=\ell(f^{-1}(x),f^{-1}(y))$ to be the degree of the linking map $L:f^{-1}(x)\times f^{-1}(y)\to S^2$ where $L(x,y)=\frac{x-y}{|x-y|}$. Note that $x\neq y$ are regular values of $f$. We first need to show that $H$ is well defined, by Sard's we know there will be 3 regular values of $f$, namely $x,y,z$ so we must show that $\ell(f^{-1}(x),f^{-1}(y))=\ell(f^{-1}(x),f^{-1}(z))$.
My thoughts were that we have a diffeomorphism $h:S^2\to S^2$ such that $h(y)=z$ and is isotopic to the identity but I'm not really sure where to go from here, I was trying to follow the proof of the integer degree theorem.
My next question is why is it homotopy invariant, by Sard's we can pick regular values $x,y$ for $f,g:S^3\to S^2$ where $f,g$ are homotopic. How do we show that $\ell(f^{-1}(x),f^{-1}(y))=\ell(g^{-1}(x),g^{-1}(y))$.
To answer both of these problems, I figured we could define a well defined map $F:S^2\to S^2$ where $\deg(F)=\ell(f^{-1}(x),f^{-1}(y))$ for an arbitrary $f:S^3\to S^2$. Then $F=L\circ\phi\circ f^{-1}$ where $\phi$ is a chart from $S^3\to \mathbb{R}^3$ but this doesn't exactly make sense since $L$ isn't defined on $\mathbb{R}^3$.
Once we know it's well defined and Homotopy invariant, all thats left to do is compute $H(h)$ for the Hopf map which doesn't seem to tricky to do since we know the preimage of points are circles. Any help is appreciated.