# Homotopy Type Theory hopf fibration (changes)

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

1. $S^0 \hookrightarrow S^1 \to S^1$ The real Hopf fibration
2. $S^1 \hookrightarrow S^3 \to S^2$ The usual complex Hopf fibration
3. $S^3 \hookrightarrow S^7 \to S^4$ The quaternionic Hopf fibration
4. $S^7 \hookrightarrow S^15 \to S^8$ 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 $A$ 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 spheres: the  S^0 S^1,  S^1 S^3 , and S^3 S^7 and $S^7$.

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

• For $S^1$ Lumsdaine gave the construction in 2012 and Brunerie proved it was correct in 2013.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: $\mu(base)\equiv id_{S^1}$, and $ap_\mu(loop)\equiv funext(h)$ where $h : (x : S^1) \to (x = x)$ is also defined by circle induction?: $h(base) = loop$ and $ap_h(loop) = refl$. $funext$ denotes functional extensionality?.

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

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

It is still an open problem to show that these are the only spaces spheres to have a H-space structure. This would be done by showing these are the only space spheres with hopf invariant$1$ which has been defined in On the homotopy groups of spheres in homotopy type theory.

category: homotopy theory