Rational SU(3)-equivariant cohomology theories

TL;DR

This paper constructs a detailed algebraic framework for rational SU(3)-equivariant cohomology theories by analyzing conjugacy classes of subgroups and establishing a Quillen equivalence with differential graded objects.

Contribution

It introduces the category A(SU(3)) that captures the structure of rational SU(3)-spectra and proves its Quillen equivalence to the category of rational SU(3)-spectra.

Findings

Classification of conjugacy classes of subgroups of SU(3).

Identification of sheaf of rings and component structure for each block.

Establishment of Quillen equivalence between rational SU(3)-spectra and differential graded objects.

Abstract

We describe the spectral space of conjugacy classes of subgroups of SU(3), together with the additional structure of a sheaf of rings and a component structure. It is a disjoint union of 18 blocks each dominated by a subgroup. For each of these blocks we identify a sheaf of rings and component structure. Taken together, this gives an abelian category A(SU(3)) designed to reflect the structure of rational SU(3)-equivariant cohomology theories, and we assemble the results from elsewhere to show that the category of rational SU(3)-spectra is Quillen equivalent to the category of differential graded objects of A(SU(3)).