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 (connected?) 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: , , and .
It is still an open problem to show that these are the only spaces to have a H-space structure.
Revision on August 5, 2018 at 13:25:52 by Ali Caglayan. See the history of this page for a list of all contributions to it.