Homotopy Type Theory Hopf construction > history (Rev #6, 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 $S^0$, $S^1$, $S^3$ 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 solved this through a homotopy theoretic version of the 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. This would be done by showing these are the only space with hopf invariant $1$ which has been defined in On the homotopy groups of spheres in homotopy type theory.

References

• Ulrik Buchholtz ,

  HoTT book

  Egbert Rijke , The Cayley-Dickson Construction in Homotopy Type Theory (arXiv:1610.01134)

• Ulrik Buchholtz, Egbert Rijke, The Cayley-Dickson Construction in Homotopy Type Theory (arXiv:1610.01134)