8.5 The Hopf fibration

In this section we will define the Hopf fibration.

Theorem 8.5.1 (Hopf Fibration).

There is a fibration $H$ over ${\mathrm{S}}^{\mathrm{2}}$ whose fiber over the basepoint is ${\mathrm{S}}^{\mathrm{1}}$ and whose total space is ${\mathrm{S}}^{\mathrm{3}}$.

The Hopf fibration will allow us to compute several homotopy groups of spheres. Indeed, it yields the following long exact sequence of homotopy groups (see sec:long-exact-sequence-homotopy-groups):

We’ve already computed all ${\pi }_n({𝕊}^1)$, and ${\pi }_k({𝕊}^n)$ for , so this becomes the following:

In particular we get the following result:

Corollary 8.5.2.

We have ${\pi }_{\mathrm{2}}\mathit{}\mathrm{(}{\mathrm{S}}^{\mathrm{2}}\mathrm{)}\mathrm{\simeq }\mathrm{Z}$ and ${\pi }_k\mathit{}\mathrm{(}{\mathrm{S}}^{\mathrm{3}}\mathrm{)}\mathrm{\simeq }{\pi }_k\mathit{}\mathrm{(}{\mathrm{S}}^{\mathrm{2}}\mathrm{)}$ for every $k\mathrm{\ge }\mathrm{3}$ (where the map is induced by the Hopf fibration, seen as a map from the total space ${\mathrm{S}}^{\mathrm{3}}$ to the base space ${\mathrm{S}}^{\mathrm{2}}$).

In fact, we can say more: the fiber sequence of the Hopf fibration will show that ${\mathrm{\Omega }}^3({𝕊}^3)$ is the fiber of a map from ${\mathrm{\Omega }}^3({𝕊}^2)$ to ${\mathrm{\Omega }}^2({𝕊}^1)$. Since ${\mathrm{\Omega }}^2({𝕊}^1)$ is contractible, we have ${\mathrm{\Omega }}^3({𝕊}^3)\simeq {\mathrm{\Omega }}^3({𝕊}^2)$. In classical homotopy theory, this fact would be a consequence of cor:pis2-hopf and Whitehead’s theorem, but Whitehead’s theorem is not necessarily valid in homotopy type theory (see sec:whitehead). We will not use the more precise version here though.

Title	8.5 The Hopf fibration