2016/06/19 by Guillaume Brunerie, Brunerie, Guillaume
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Logic in Computer Science (cs.LO)
paper · pdf · doi:10.48550/arxiv.1606.05916
openalex publication_date 2016/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.