Rational Sp(2)-equivariant cohomology theories I: dominant subgroups

TL;DR

This paper describes the structure of rational Sp(2)-equivariant cohomology theories using a sheaf-based categorical framework, linking spectra to differential graded objects.

Contribution

It introduces a new abelian category A(Sp(2)) that captures the structure of rational Sp(2)-spectra and establishes a Quillen equivalence with differential graded objects.

Findings

Spectral space of conjugacy classes decomposes into blocks with specific dimensions.

Construction of the abelian category A(Sp(2)) reflecting cohomology theory structure.

Quillen equivalence between rational Sp(2)-spectra and differential graded objects of A(Sp(2)).

Abstract

We give a general description of the spectral space of conjugacy classes of subgroups of Sp(2): it is a disjoint union of finitely many blocks, each dominated by a subgroup: of these blocks, 26 are of dimension 1, 6 are of dimension 2 and the remainder are isolated points. On each of these blocks there is a sheaf of polynomial rings and a component structure. These are the ingredients for constructing an abelian category A(Sp(2)) designed to reflect the structure of rational Sp(2)-equivariant cohomology theories. We assemble the results from earlier papers in the series to show that the category of rational Sp(2)-spectra is Quillen equivalent to the category of differential graded objects of A(Sp(2)). In the sequel we will make the fine structure of A(Sp(2)) explicit, and make calculations based upon it.