Homotopy Type Theory
On the homotopy groups of spheres in homotopy type theory

On the homotopy groups of spheres in homotopy type theory, Guillaume Brunerie,

arxiv

Abstract

The goal of this thesis is to prove that π 4(S 3)/2\pi_4(S^3)\simeq \mathbb{Z}/2\mathbb{Z} 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 π k(S n)\pi_k(S^n) with k<nk\lt 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 nn such that π 4(S 3)/n\pi_4(S^3)\simeq\mathbb{Z}/n\mathbb{Z}. 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 nn to either 11 or 22. The Hopf invariant also allows us to prove that all the groups? of the form π 4n1(S 2n){\pi_{4n-1}}(S^{2n}) are infinite. Finally we construct the Gysin exact sequence?, allowing us to compute the cohomology of P 2\mathbb{C}P^2 and to prove that π 4(S 3)/2\pi_4(S^3)\simeq \mathbb{Z}/2\mathbb{Z} and that more generally π n+1(S n)/2\pi_{n+1}(S^n)\simeq \mathbb{Z}/2\mathbb{Z} for every n3n\ge 3.

category: reference