Hopf invariant one problem

Remarks

The maps for $n = 2, 4, 8$ are the Hopf fibrations built from the complex numbers, the quaternions and the octonions. For odd $n$, every Hopf invariant is $0$, and for even $n$, the Whitehead product $[\iota_n, \iota_n]$ has Hopf invariant $\pm 2$, so only the value $\pm 1$ is in question. An argument with Steenrod squares shows that $n$ must be a power of $2$. J. F. Adams excluded all remaining $n \ge 16$ with secondary cohomology operations, which is Adams' theorem on Hopf invariant one. Adams and Atiyah later gave a short proof with Adams operations in topological K-theory. Consequences: $\R^n$ carries the structure of a real division algebra, with a bilinear product and no zero divisors, only for $n = 1, 2, 4, 8$, and $S^{n-1}$ is an H-space only for $n = 1, 2, 4, 8$.

Going deeper

For a student

The theorem explains why the list $\R, \C, \mathbb{H}, \mathbb{O}$ of division algebras stops. A bilinear product without zero divisors on $\R^n$ makes $S^{n-1}$ an H-space, and the Hopf construction turns this multiplication into a map $S^{2n-1} \to S^n$ of Hopf invariant $\pm 1$. So the topological theorem implies the algebraic one. The problem also led to the Adams spectral sequence and to the use of K-theory operations in homotopy theory.