Rational Equivariant Rigidity

TL;DR

This paper proves that for circle and profinite groups, the rational G-equivariant stable homotopy category's structure is entirely determined by its triangulated structure, establishing a form of rigidity.

Contribution

It demonstrates the rigidity of the rational G-equivariant stable homotopy category for S1 and profinite groups, linking intrinsic formality to this rigidity.

Findings

The rational G-equivariant stable homotopy category is rigid for G = S1 or profinite groups.

Rigidity is derived from an intrinsic formality property of associated differential graded algebras.

The work connects formality concepts with homotopical rigidity in equivariant stable homotopy theory.

Abstract

We prove that if G is the circle group or a profinite group, then the all of the homotopical information of the category of rational G-spectra is captured by triangulated structure of the rational G-equivariant stable homotopy category. That is, for G profinite or S1, the rational G-equivariant stable homotopy category is rigid. For the case of profinite groups this rigidity comes from an intrinsic formality statement, so we carefully relate the notion of intrinsic formality of a differential graded algebra to rigidity.