Papers › On the homotopy groups of spheres in homotopy type theory
On the homotopy groups of spheres in homotopy type theory
Guillaume Brunerie
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
The goal of this thesis is to prove that π₄(S³) ≃ℤ/2ℤ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πₖ(Sⁿ) with k < n, and the construction of the Hopf fibration. We then move to more advanced tools. In particular, we define the James construction which allows us to prove the Freudenthal suspension theorem and the fact that there exists a natural number n such that π₄(S³) ≃ℤ/nℤ. Then we study the smash product of spheres, we construct the cohomology ring of a space, and we introduce the Hopf invariant, allowing us to narrow down the n to either $1$ or $2$. The Hopf invariant also allows us to prove that all the groups of the form π₄ₙ₋₁(S²ⁿ) are infinite. Finally we construct the Gysin exact sequence, allowing us to compute the cohomology of ℂP² and to prove that π₄(S³) ≃ℤ/2ℤ and that more generally πₙ₊₁(Sⁿ) ≃ℤ/2ℤ for every n ≥3.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections