Periodic phenomena in equivariant stable homotopy theory

TL;DR

This paper develops an equivariant version of chromatic homotopy theory for finite abelian p-groups, extending classical non-equivariant results and exploring computational examples, especially for A = C_2.

Contribution

It introduces equivariant analogs of v_n-self maps, the chromatic tower, and key theorems, advancing the understanding of A-equivariant stable homotopy theory.

Findings

Structure of the Balmer spectrum for A-spectra analyzed.

Equivariant v_n-self maps and chromatic tower constructed.

Connections made between equivariant chromatic theory and Mahowald invariants.

Abstract

Building off of many recent advances in the subject by many different researchers, we describe a picture of A-equivariant chromatic homotopy theory which mirrors the now classical non-equivariant picture of Morava, Miller-Ravenel-Wilson, and Devinatz-Hopkins-Smith, where A is a finite abelian p-group. Specifically, we review the structure of the Balmer spectrum of the category of A-spectra, and the work of Hausmann-Meier connecting this to MU_A and equivariant formal group laws. Generalizing work of Bhattacharya-Guillou-Li, we introduce equivariant analogs of v_n-self maps, and generalizing work of Carrick and Balderrama, we introduce equivariant analogs of the chromatic tower, and give equivariant analogs of the smash product and chromatic convergence theorems. The equivariant monochromatic theory is also discussed. We explore computational examples of this theory in the case of A = C_2, where we connect equivariant chromatic theory with redshift phenomena in Mahowald invariants.