Hopf invariant one

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

and proved that ${\displaystyle \eta }$ is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles

It was later shown that the homotopy group ${\displaystyle \pi _{3}(S^{2})}$ is the infinite cyclic group generated by ${\displaystyle \eta }$. In 1951, Jean-Pierre Serre proved that the rational homotopy groups

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