I'm having some trouble proving the following statement: let $g:S^{2n-1} \to S^{2n-1}, f: S^{2n-1} \to S^n$. Then $H(f\circ g) = \deg g H(f)$ where $H(f)$ is the Hopf invariant. The definition I am using for Hopf invariant is as follows: let $C_f = D^{2n} \sqcup_f S^{n}$ where $D^{2n}$ is the $2n$ disk which we attach to $S^{n}$ via $f$. Let $\alpha, \beta$ be generators for $H^n(C_f)$ and $H^{2n}(C_f)$ respectively. Then $H(f)$ is defined by $\alpha \cup \alpha = H(f) \beta$. I can prove the statement using the integral formula for $H(f)$ but would like to prove it cohomologically.
I think I should consider the map $G: C_{f\circ g} \to C_f$ that takes $x\in \partial D^{2n}$ to $g(x)$ and acts as the identity on everything else. But then I don't know how to compute what $G^* \alpha$ is.
Thanks!