8.5 The Hopf fibration
In this section we will define the Hopf fibration.
Theorem 8.5.1 (Hopf Fibration).
There is a fibration over whose fiber over the basepoint is andwhose total space is .
The Hopf fibration will allow us to compute several homotopy groups ofspheres.Indeed, it yields the following long exact sequenceof homotopy groups(see\\autorefsec:long-exact-sequence-homotopy-groups):
We’ve already computed all , and for , so thisbecomes the following:
In particular we get the following result:
Corollary 8.5.2.
We have and forevery (where the map is induced by the Hopf fibration, seen as a mapfrom the total space to the base space ).
In fact, we can say more: the fiber sequence of the Hopf fibration will show that is the fiber of a map from to .Since is contractible, we have .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.