Adams' theorem on Hopf invariant one

Remarks

Conversely, maps of Hopf invariant one exist for $2n = 2, 4, 8$: they are the Hopf fibrations $S^3 \to S^2$, $S^7 \to S^4$ and $S^{15} \to S^8$, built from the complex numbers, the quaternions and the octonions.

Adams proved the theorem in 1960 with secondary cohomology operations. A map of Hopf invariant one gives a two-cell complex $X = S^{2n} \cup_f D^{4n}$ with $Sq^{2n}$ nonzero from $H^{2n}(X; \F_2)$ to $H^{4n}(X; \F_2)$. For $2n$ not a power of $2$, the Adem relations write $Sq^{2n}$ through lower squares, which vanish on $H^{2n}(X; \F_2)$ because they land in zero groups, so this is impossible. For $2n = 2^k$ with $k \ge 4$, Adams decomposed $Sq^{2^k}$ through secondary operations. A shorter proof by Adams and Atiyah uses the Adams operations in complex K-theory.

The theorem has several consequences. For $m \ge 1$, the sphere $S^{m}$ admits the structure of an H-space only for $m \in \set{1, 3, 7}$. The space $\R^m$ has the structure of a real division algebra, with bilinear multiplication and no zero divisors, only for $m \in \set{1, 2, 4, 8}$. For $m \ge 1$, the sphere $S^{m}$ is parallelizable only for $m \in \set{1, 3, 7}$.

Going deeper

For a student

The theorem settles the Hopf invariant one problem. In the Adams spectral sequence for the sphere at the prime $2$, it says that the classes $h_j$ in $\operatorname{Ext}^{1, 2^j}$ survive only for $j \le 3$. For $j \ge 4$ there is a nonzero differential $d_2(h_j) = h_0 h_{j-1}^2$. Through the stable homotopy groups of spheres, it is linked to the image of the J-homomorphism and to the vector fields on spheres problem.