Unstable Periodic Homotopy Theory

Abstract

In this thesis we study unstable vₕ-periodic homotopy theory, where h is a natural number; here ``unstable'' refers to the homotopy theory of topological spaces. The work consists of two parts. In Part I we give a detailed exposition of the foundations of unstable vₕ-periodic homotopy theory, sharpen an existing result about vₕ-periodic equivalences of H-spaces, and pose concrete questions and conjectures for future studies. The expository part follows paper by Bousfield, Dror Farjoun and Heuts and aims to assemble in one place the central notions and theorems of unstable localisations with a focus on unstable periodic homotopy theory. The goal of Part II is to understand unstable vₕ-periodic phenomena from the point of view of Lie algebras in the stable vₕ-periodic homotopy category. We analyse the costabilisation of vₕ-periodic homotopy types and obtain a universal property of the Bousfield--Kuhn functor.