8.5 The Hopf fibration

In this sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath we will define the Hopf fibration.

Theorem 8.5.1 (Hopf Fibration).

There is a fibrationMathworldPlanetmath H over S2 whose fiber over the basepoint is S1 and whose total space is S3.

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):


We’ve already computed all πn(𝕊1), and πk(𝕊n) for k<n, so this becomes the following:


In particular we get the following result:

Corollary 8.5.2.

We have π2(S2)Z and πk(S3)πk(S2) for every k3 (where the map is induced by the Hopf fibration, seen as a map from the total space S3 to the base space S2).

In fact, we can say more: the fiber sequence of the Hopf fibration will show that Ω3(𝕊3) is the fiber of a map from Ω3(𝕊2) to Ω2(𝕊1). Since Ω2(𝕊1) is contractibleMathworldPlanetmath, we have Ω3(𝕊3)Ω3(𝕊2). 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