Book Review: Nilpotence and periodicity in stable homotopy theory

Nilpotence and Periodicity in Stable Homotopy Theory

Contents Preface xi Introduction xiii 1 The main theorems 1 1.

Nilpotence in Stable Homotopy Theory

This talk covers most of section 4 in the Mathew-Naumann-Noel paper [MNN15]. We first discuss nilpotence in an arbitrary symmetric monoidal stable ∞-category. We then discuss the historical origins of nilpotence in the stable homotopy category, namely the Ravenel conjectures and the Nilpotence theorem proved by Devinatz-Hopkins-Smith.

Nilpotence and Stable Homotopy Theory II

Localization and Periodicity in Unstable Homotopy Theory

In this paper, we develop a hierarchy of natural localizations of spaces, called the vn-periodizations for n 2: 0, which may be used to expose and study periodic phenomena in unstable homotopy theory. These vn-periodizations act less radically than the corresponding homological localizations [2] and respect fibrations to a very considerable extent. A major part of this paper is devoted to devel...

Stable A1-homotopy Theory

Definition 1. A T -prespectrum E is a sequence En of pointed spaces together with structure morphisms σn : En ∧ T → En+1 for all n. A morphism λ : E → F of T -prespectra is a sequence of maps λn : En → Fn commuting with the structure morphisms. We will denote the category of T -prespectra by PSpT (k). Example 1. For any based space X , we have the suspension prespectrum Σ∞(X ) of X given by Σ∞(...