8.5 The Hopf fibration
Theorem 8.5.1 (Hopf Fibration).
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 , and for , so this becomes the following:
In particular we get the following result:
We have and for every (where the map is induced by the Hopf fibration, seen as a map from 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.
|Title||8.5 The Hopf fibration|