Unstable Periodic Homotopy Theory

Yuqing Shi

Research output: ThesisDoctoral thesis 1 (Research UU / Graduation UU)


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.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Utrecht University
  • Moerdijk, Ieke, Supervisor
  • Heuts, Gijs, Co-supervisor
Award date11 Oct 2023
Place of PublicationUtrecht
Publication statusPublished - 11 Oct 2023


  • Homotopy theory
  • Algebraic Topology
  • Unstable vh-periodic Homotopy Theory
  • Monochromatic Layer
  • Localisation of Topological Spaces
  • Homological and Homotopical Equivalences
  • H-Space
  • Spectral Lie Algebra
  • En-algebras
  • Costabilisation


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