Show that the Hopf invariant of a composition \(S^{2n-1} \xrightarrow{f} S^{2n-1} \xrightarrow{g} S^n\) is given by \(H(gf) = (\deg f) H(g)\), and for a composition \(S^{2n-1} \xrightarrow{f} S^n \xrightarrow{g} S^n\) the Hopf invariant satisfies \(H(gf) = (\deg g)^2 H(f)\).
Solution
1. The Hopf invariant \(H(g)\) is defined via the cohomology of the mapping cone \(C_g\): \(H(g)\) is the integer such that \(\alpha \smile \alpha = H(g)\beta\) in \(H^{2n}(C_g)\), where \(\alpha \in H^n(C_g)\) and \(\beta \in H^{2n}(C_g)\) are generators.
definition of the Hopf invariant.
2. First composition: \(S^{2n-1} \xrightarrow{f} S^{2n-1} \xrightarrow{g} S^n\).
2.1. The mapping cone \(C_{gf}\) is homotopy equivalent to \(C_g\) (since \(f\) is a self-map of the domain sphere, and the cone of \(gf\) is obtained from the cone of \(g\) by precomposing the attaching map with \(f\)).
\(C_{gf} = S^n \cup_{gf} e^{2n}\), and \(C_g = S^n \cup_g e^{2n}\).
2.2. The attaching map \(gf\) is homotopic to \((\deg f) g\) in \(\pi_{2n-1}(S^n)\).
precomposition by \(f\) multiplies the homotopy class by \(\deg f\) (the action of \(\pi_{2n-1}(S^{2n-1}) = \ZZ\) on \(\pi_{2n-1}(S^n)\)).
2.3. Hence \(H(gf) = (\deg f) H(g)\).
the Hopf invariant is linear in the attaching map, so scaling the attaching map by \(\deg f\) scales the Hopf invariant by \(\deg f\).
3. Second composition: \(S^{2n-1} \xrightarrow{f} S^n \xrightarrow{g} S^n\).
3.1. The mapping cone \(C_{gf}\) has attaching map \(gf\), and \(gf\) is homotopic to \((\deg g) f\) in \(\pi_{2n-1}(S^n)\).
postcomposition by \(g\) multiplies the homotopy class by \(\deg g\).
3.2. The Hopf invariant satisfies \(H(gf) = (\deg g)^2 H(f)\).
the Hopf invariant is a quadratic form on \(\pi_{2n-1}(S^n)\) (for \(n\) even it is a quadratic form; the general fact is that \(H\) is a homomorphism for \(n\) odd and a quadratic form for \(n\) even, and in both cases postcomposition by a degree-\(\deg g\) map multiplies \(H\) by \((\deg g)^2\)).
4. Q.E.D.
2 and 3.