For $n \in \mathbb{N}$ with $n \gt 1$, consider continuous functions between spheres of the form
The homotopy cofiber of $\phi$
Hence for $\alpha, \beta$ generators of the cohomology groups in degree $n$ and $2n$ (unique up to choice of sign), respectively, there exists an integer $h(\phi)$ which expresses the cup product square of $\alpha$ as a multiple of $\beta$:
This integer $h(\phi)$ is called the Hopf invariant of $\phi$.
It depends on the choices made only up to sign. In particular it has a well-defined image $[h(\phi)] \in \mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$, and as such it is the Steenrod square
For $n$ odd, then the Hopf invariant necessarily vanishes. For $n$ even however, then there is a homomorphism
whose image contains at least the even integers.
Hence a famous open question in the 1950s was for which maps $\phi$ one has Hopf invariant one, $h(\phi) = 1$.
The Hopf invariant one theorem (Adams60) states that the only maps of Hopf invariant one, $h(\phi) = 1$, are the Hopf constructions on the four real normed division algebras:
the real Hopf fibration;
Wikipedia, Hopf invariant
John Michael Boardman, B. Steer, On Hopf Invariants (pdf)
Michael Crabb, Andrew Ranicki, The geometric Hopf invariant (pdf)
Last revised on February 17, 2019 at 06:10:27. See the history of this page for a list of all contributions to it.