I'd like to prove the following properties Hopf Invariant, which are Exercises at pag $428$ on Hatcher but I can't find any reference or proof.
I think I have to use directly the eaxct long sequence used to define the Hopf Invariant $H$, but I don't undertstand how the degree comes in play
Any help or reference would be appreciated.
Recall the definition of the Hopf invariant: given a map $f: S^{2n-1} \to S^n$, we form the mapping cone $C_f$, whose cohomology is given by $\mathbb{Z}$ in degrees $n$ and $2n$. Taking $\alpha \in H^n(C_f)$ and $\beta \in H^{2n}(C_f)$ to be generators, the Hopf invariant is the integer $H(f)$ satisfying $\alpha^2 = H(f) \beta$.
Now, given $S^{2n-1} \xrightarrow{f} S^{2n-1} \xrightarrow{g} S^n$, consider the map of cofiber sequences $$\require{AMScd} \begin{CD} S^n @>>> C_{gf} @>>> S^{2n}\\ @VVV @VVV @VV{-\Sigma f}V \\ S^n @>>> C_g @>>> S^{2n}. \end{CD}$$ Writing $\alpha, \beta$ (resp. $\alpha', \beta'$) for the generators in cohomology for the spheres in the top (resp. bottom) row, we have $\alpha'^2 = H(g) \beta'$. Pulling this equation back to the top row, we see that $\alpha^2 = H(g) (\deg f \cdot \beta)$. Since $H(gf)$ is defined so that $\alpha^2 = H(gf) \beta$, we deduce that $H(gf) = (\deg f) \cdot H(g)$.
For $S^{2n-1} \xrightarrow{f} S^n \xrightarrow{g} S^n$, we have $$\require{AMScd} \begin{CD} S^n @>>> C_f @>>> S^{2n}\\ @V{g}VV @VVV @VVV \\ S^n @>>> C_{gf} @>>> S^{2n}. \end{CD}$$
Writing $\alpha, \beta$ (resp. $\alpha', \beta'$) for the generators in cohomology for the spheres in the bottom (resp. top) row, we have $\alpha^2 = H(gf) \beta$. Pulling this equation back to the top row, we see that $(\deg g \cdot \alpha')^2 = H(gf) \beta'$. Substituting $\alpha'^2 = H(f) \beta'$, we deduce that $H(gf) = (\deg g)^2 \cdot H(f)$.