Idempotent Symmetries in Algebra and Topology

Abstract

The proofs of Ravenel’s conjectures [22] and their reinterpretations in the form of the classification of the homotopy idempotent functors of spectra commuting with telescopes by Devintaz–Hopkins–Smith [7, 12], were a culmination of a few decades of progress achieved in stable homotopy theory. The simplicity of this classification is remarkable. For each prime p, the category of restrictions of these functors to p-local finite spectra is isomorphic to the poset of natural numbers. The obtained invariant is called the Morava–Hopkins type. The stable classification was generalized to the classification of so-called Bousfield localizations of finite p-local spaces; see Bousfield [3]. This unstable classification is also remarkably simple. Bousfield showed that, in addition to the Morava–Hopkins type invariant, connectivity determines such functors.