In classical algebraic topology we have four Hopf fibrations (of spheres):
These can be constructed in HoTT as part of a more general construction:
A H-space structure on a pointed type gives a fibration over via the hopf construction. This fibration can be written classically as: where is the join of and . This is all done in the HoTT book. Note that can be written as a homotopy pushout , and there is a lemma in the HoTT book allowing you to construct a fibration on a pushout (the equivalence needed is simply the multiplication from the H-space ).
Thus the problem of constructing a hopf fibration reduces to finding a H-space structure on the spheres: the , and .
The space is not connected? so we cannot perform the construction from the book on it. However it is very easy to construct a family with fiber by induction on . (Note: loop maps to where is the equivalence of negation and is the univaence axiom?.
For 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: , and where is also defined by circle induction?: and . denotes functional extensionality?.
For Buchholtz-Rijke 16 solved this through a homotopy theoretic version of the Cayley-Dickson construction. This has been formalised in Lean.
For 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 which has been defined in On the homotopy groups of spheres in homotopy type theory.
Ulrik Buchholtz, Egbert Rijke, The Cayley-Dickson Construction in Homotopy Type Theory (arXiv:1610.01134)
Revision on January 2, 2019 at 02:08:54 by Ali Caglayan. See the history of this page for a list of all contributions to it.