Borel cohomology and the relative Gorenstein condition for classifying spaces of compact Lie groups

TL;DR

This paper investigates the properties of Borel cohomology spectra for compact Lie groups, establishing conditions under which these spectra generate module categories and analyzing the Gorenstein property of certain maps, with a focus on connectedness.

Contribution

It demonstrates that for connected compact Lie groups, the Borel cohomology spectrum generates its module category, and it characterizes the Gorenstein property of the map between cochains of classifying spaces for subgroups.

Findings

Borel cohomology spectrum generates its module category for connected groups

The map C^*(BG)→C^*(BH) is relatively Gorenstein under certain conditions

Connectedness of groups is crucial for the results

Abstract

For a compact Lie group G we show that if the representing spectrum for Borel cohomology generates its category of modules if G is connected. For a closed subgroup H of G we consider the map C^*(BG)--->C^*(BH) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups.