Homotopy Type Theory
hopf fibration (Rev #7)

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.

  • The space S 0=2(Bool)S^0=\mathbf 2(\equiv Bool) is not connected? so we cannot perform the construction from the book on it. However it is very easy to construct a family S 1𝒰S^1 \to \mathcal{U} with fiber BoolBool by induction on S 1S^1. (Note: loop maps to ua(neg)ua(neg) where negneg is the equivalence of negation and uaua is the univaence axiom?.

  • For S 1S^1 Peter Lumsdaine gave the construction in 2012 and Guillaume Brunerie proved it was correct in 2013. By induction? on the circle we can define the multiplication: μ(base)id S 1\mu(base)\equiv id_{S^1}, and ap μ(loop)funext(h)ap_\mu(loop)\equiv funext(h) where h:(x:S 1)(x=x)h : (x : S^1) \to (x = x) is also defined by circle induction?: h(base)=looph(base) = loop and ap h(loop)=reflap_h(loop) = refl. funextfunext denotes functional extensionality?.

  • For S 3S^3 Buchholtz-Rijke 16 solved this through a homotopy theoretic version of the Cayley-Dickson construction. This has been formalised in Lean.

  • For S 7S^7 this is still an open problem.

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.


category: homotopy theory

Revision on January 1, 2019 at 21:08:54 by Ali Caglayan. See the history of this page for a list of all contributions to it.