Formalizing π4(S3) ≅Z/2Z and Computing a Brunerie Number in Cubical Agda

• Published in: 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) pp. 1 - 13
• Main Authors: Ljungstrom, Axel, Mortberg, Anders
• Format: Conference Proceeding
• Language: English
• Published: IEEE 26.06.2023
• Subjects: Algebra; Codes; Computer science; Libraries; Machinery; Stress; Topology
• DOI: 10.1109/LICS56636.2023.10175833

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 ℤ/2ℤ. 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" β can be normalized to ±2. The question of whether Brunerie's proof could be formalized in a proof assistant, either by computing this number or by formalizing the pen-and-paper proof, has since remained open. In this paper, we present a complete formalization 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 formalization of a new and much simpler proof that β is ±2. This formalization provides us with a sequence of simpler Brunerie numbers, one of which normalizes very quickly to −2 in Cubical Agda, resulting in a fully formalized computer-assisted proof that {\pi _4}({\mathbb{S}^3}) \cong \mathbb{Z}/2\mathbb{Z}.