[[!redirects hopf construction]] [[!redirects Hopf construction]] In classical [[nLab:algebraic topology]] we have four [[nLab:Hopf fibrations]] (of spheres): 1. $S^0 \hookrightarrow S^1 \to S^1$ The [[nLab:real Hopf fibration]] 2. $S^1 \hookrightarrow S^3 \to S^2$ The usual [[nLab:complex Hopf fibration]] 3. $S^3 \hookrightarrow S^7 \to S^4$ The [[nLab:quaternionic Hopf fibration]] 4. $S^7 \hookrightarrow S^15 \to S^8$ The [[nLab:octonionic Hopf fibration]] These can be constructed in HoTT as part of a more general construction: A [[H-space]] structure on a pointed (connected?) type $A$ gives a fibration over $\Sigma A$ via the hopf construction. This fibration can be written classically as: $A \to A\ast A \to \Sigma A$ where $A\ast A$ is the join of $A$ and $A$. This is all done in the HoTT book. Note that $\Sigma A$ can be written as a homotopy pushout $\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 $A \to A$ needed is simply the multiplication from the H-space $\mu(a,-)$). Thus the problem of constructing a hopf fibration reduces to finding a H-space structure on the [[nLab:spheres]]: the $S^0$, $S^1$, $S^3$ and $S^7$. * For $S^0=\mathbf 2$ this is a trivial exercise and it is in the book. * For $S^1$ Lumsdaine gave the construction in 2012 and Brunerie proved it was correct in 2013. * For $S^3$ [Buchholtz-Rijke 16](#BuchholtzRijke16) solved this through a [[nLab:homotopy theory|homotopy theoretic]] version of the [[nLab:Cayley-Dickson construction]]. * For $S^7$ this is still an open problem. It is still an open problem to show that these are the only spaces to have a H-space structure. ## References * {#BuchholtzRijke16} [[nLab:Ulrik Buchholtz]], [[nLab:Egbert Rijke]], _The Cayley-Dickson Construction in Homotopy Type Theory_ ([arXiv:1610.01134](https://arxiv.org/abs/1610.01134)) category: homotopy theory