On the homotopy groups of spheres in homotopy type theory

TL;DR

This thesis provides a constructive, purely homotopy-theoretic proof in homotopy type theory that the fourth homotopy group of the 3-sphere is isomorphic to Z/2Z, extending to a general result for spheres.

Contribution

It introduces a new constructive proof of homotopy groups of spheres within homotopy type theory, including the computation of cb4(S^3) and cb4(S^n) for n b7 3.

Findings

Proves cb4(S^3) b1 Z/2Z.

Establishes cb4(S^{n+1}) b1 Z/2Z for all n b7 3.

Develops tools like James construction and Gysin sequence within homotopy type theory.

Abstract

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