Homotopy Type Theory
hopf fibration

In classical algebraic topology we have four Hopf fibrations (of spheres):

  1. S 0S 1S 1S^0 \hookrightarrow S^1 \to S^1 The real Hopf fibration
  2. S 1S 3S 2S^1 \hookrightarrow S^3 \to S^2 The usual complex Hopf fibration
  3. S 3S 7S 4S^3 \hookrightarrow S^7 \to S^4 The quaternionic Hopf fibration
  4. S 7S 15S 8S^7 \hookrightarrow S^15 \to S^8 The octonionic Hopf fibration

These can be constructed in HoTT as part of a more general construction:

A H-space structure on a pointed type AA gives a fibration over ΣA\Sigma A via the hopf construction. This fibration can be written classically as: AA*AΣAA \to A\ast A \to \Sigma A where A*AA\ast A is the join of AA and AA. This is all done in the HoTT book. Note that ΣA\Sigma A can be written as a homotopy pushout ΣA:=1 A1\Sigma A := \mathbf 1 \sqcup^A \mathbf 1 , and there is a lemma in the HoTT book allowing you to construct a fibration on a pushout (the equivalence AAA \to A needed is simply the multiplication from the H-space μ(a,)\mu(a,-)).

Thus the problem of constructing a hopf fibration reduces to finding a H-space structure on the spheres: the S 1S^1, S 3S^3 and S 7S^7.

It is still an open problem to show that these are the only spheres to have a H-space structure. This would be done by showing these are the only spheres with hopf invariant 11 which has been defined in On the homotopy groups of spheres in homotopy type theory.

References

category: homotopy theory