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 $\mathbb{S}^{2}$ whose fiber over the basepoint is $\mathbb{S}^{1}$ and whose total space is $\mathbb{S}^{3}$ .

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 \autorefsec:long-exact-sequence-homotopy-groups):

$\xymatrix@R=1.2pc{\pi_{k}(\mathbb{S}^{1})\ar[r]&\pi_{k}(\mathbb{S}^{3})\ar[r]&% \pi_{k}(\mathbb{S}^{2})\ar[lld]\\ \vdots&\vdots&\vdots\ar[lld]\\ \pi_{2}(\mathbb{S}^{1})\ar[r]&\pi_{2}(\mathbb{S}^{3})\ar[r]&\pi_{2}(\mathbb{S}% ^{2})\ar[lld]\\ \pi_{1}(\mathbb{S}^{1})\ar[r]&\pi_{1}(\mathbb{S}^{3})\ar[r]&\pi_{1}(\mathbb{S}% ^{2})}$

We’ve already computed all $\pi_{n}(\mathbb{S}^{1})$ , and $\pi_{k}(\mathbb{S}^{n})$ for $k<n$ , so this becomes the following:

$\xymatrix@R=1.2pc{0\ar[r]&\pi_{k}(\mathbb{S}^{3})\ar[r]&\pi_{k}(\mathbb{S}^{2}% )\ar[lld]\\ \vdots&\vdots&\vdots\ar[lld]\\ 0\ar[r]&\pi_{3}(\mathbb{S}^{3})\ar[r]&\pi_{3}(\mathbb{S}^{2})\ar[lld]\\ 0\ar[r]&0\ar[r]&\pi_{2}(\mathbb{S}^{2})\ar[lld]\\ \mathbb{Z}\ar[r]&0\ar[r]&0}$

In particular we get the following result:

Corollary 8.5.2.

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

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

Title	8.5 The Hopf fibration
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