The Equivariant Generating Hypothesis

TL;DR

This paper investigates the generating hypothesis within the homotopy category of G-spectra, establishing its implications for finite groups and providing a counterexample for rational S^1-equivariant spectra.

Contribution

It proves that for finite groups, the generating hypothesis implies the strong version, and presents a counterexample in the rational S^1-equivariant spectra case.

Findings

Generating hypothesis implies strong generating hypothesis for finite G.

Counterexample to the generating hypothesis exists in rational S^1-equivariant spectra.

Clarifies the limits of the generating hypothesis in equivariant stable homotopy theory.

Abstract

We state the generating hypothesis in the homotopy category of G-spectra for a compact Lie group G, and prove that if G is finite, then the generating hypothesis implies the strong generating hypothesis, just as in the non-equivariant case. We also give an explicit counterexample to the generating hypothesis in the category of rational S^1-equivariant spectra.