Nilpotence and Periodicity in Stable Homotopy Theory

An overview of abelian varieties in homotopy theory

Through an investigation of properties of Chern classes, Quillen discovered a connection between stable homotopy theory and 1-dimensional formal group laws [31]. After almost 40 years, the impacts of this connection are still being felt. The stratification of formal group laws in finite characteristic gives rise to the chromatic filtration in stable homotopy theory [32], and has definite calcul...

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.

The Periodicity Theorem , Talk 1 Vitaly

Note: These are my notes on part of chapter 6 of Ravenel’s Orange Book (Nilpotence and Periodicity in Stable Homotopy Theory), prepared for a talk in the graduate student topology seminar at Johns Hopkins. Jon Beardsley gave a talk on the first part of chapter 6, which reduces the proof of the Periodicity Theorem to constructing an example of a single type n spectrum with a vn-self map. For not...

Nilpotence and Stable Homotopy Theory II

Topology Seminar Non-nilpotent elements in motivic homotopy theory

Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite p-local spectra induce nonzero homomorphisms on BP -homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and whilst η : S1,1 → S0,0 induces zero on BP -homology, it is nonnilpotent. Recent work with Haynes Miller has led to a calculation of ηπ∗...