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On the homotopy groups of spheres in homotopy type theory
< , On the homotopy groups of spheres in homotopy type theoryGuillaume Brunerie,
The goal of this thesis is to prove that 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 with , 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 such that . 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 to either or . The Hopf invariant also allows us to prove that all the groups? of the form are infinite. Finally we construct the Gysin exact sequence?, allowing us to compute the cohomology of and to prove that and that more generally for every .
Last revised on June 9, 2022 at 17:56:22. See the history of this page for a list of all contributions to it.