Hopf invariant

Summary

The Hopf invariant is an integer $H(f)$ attached to a map $f \colon S^{2n-1} \to S^n$. It is read off from the cup product in the cohomology of the space obtained by attaching a $2n$-cell to $S^n$ along $f$. It depends only on the homotopy class of $f$ and gives a homomorphism $\pi_{2n-1}(S^n) \to \Z$. It showed that $\pi_3(S^2)$ is infinite, and the question of which $n$ admit maps of Hopf invariant one is tied to division algebras.