Formalising and Computing the Fourth Homotopy Group of the $3$-Sphere in Cubical Agda

TL;DR

This paper formalises Brunerie's synthetic proof in Homotopy Type Theory that the fourth homotopy group of the 3-sphere is /2, using Cubical Agda to achieve a fully computer-verified proof.

Contribution

It provides the first complete formalisation of Brunerie's proof in Cubical Agda, including a new simplified proof that the Brunerie number /2 normalises to   2.

Findings

Successfully formalised Brunerie's proof in Cubical Agda.

Developed a new simplified proof for the Brunerie number /2.

Achieved a fully computer-verified proof of /2 for  3-sphere.

Abstract

Brunerie's 2016 PhD thesis contains the first synthetic proof in Homotopy Type Theory (HoTT) of the classical result that the fourth homotopy group of the 3-sphere is $\mathbb{Z}/2\mathbb{Z}$. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" $\beta$ can be normalised to $\pm 2$. The question of whether Brunerie's proof could be formalised in a proof assistant, either by computing this number or by formalising the pen-and-paper proof, has since remained open. In this paper, we present a complete formalisation in Cubical Agda. We do this by modifying Brunerie's proof so that a key technical result, whose proof Brunerie only sketched in his thesis, can be avoided. We also present a formalisation of a new and much simpler proof that $\beta$ is $\pm 2$. This formalisation provides us with a sequence of simpler Brunerie numbers, one of which normalises very quickly to $-2$ in Cubical Agda, resulting in a fully formalised computer-assisted proof that $\pi_4(\mathbb{S}^3) \cong \mathbb{Z}/2\mathbb{Z}$.