Given two functions $f: S^{2p-1} \to S^p$ and $g: S^{p} \to S^{p}$, I want to prove that
\begin{equation} H(f) = (\text{deg}(g))^2 H(f), \end{equation}
where $H(f)$ is the Hopf invariant of $f$. I first tried to analyze the problem by assuming that $g$ preserves orientations. That is if $y \neq z$ are two distinct regular values of $g \circ f$ and since $(g \circ f)^{-1}(w) = \coprod_{u \in g^{-1}(w)} f^{-1}(u)$, then
\begin{equation} \begin{split} H(g\circ f) & = l\left((g \circ f)^{-1}(y),(g \circ f)^{-1}(z)\right) \\ &= l\left(\coprod_{x \in g^{-1}(y)} f^{-1}(x), \coprod_{u \in g^{-1}(z)} f^{-1}(u)\right) \\ & = \sum_{x \in g^{-1}(y)} \sum_{u \in g^{-1}(z)} l\left( f^{-1}(x),f^{-1}(u)\right) \\ & = \sum_{x \in g^{-1}(y)} \sum_{u \in g^{-1}(z)} H(f) \\ & = H(f) \sum_{x \in g^{-1}(y)} \sum_{u \in g^{-1}(z)} 1 \\ & = H(f) \; (\text{deg}(g))^2. \end{split} \end{equation}
The sixth equality holds if $g$ preserves orientations. Here $l$ represents the linking number.
Is the above correct?
This was my idea to solve the problem in the case that g preserves orientations; however, in the general case I am stuck. This is problem 15 of Milnor's Topology from the Differentiable Point of View.
