Noetherian loop spaces

TL;DR

This paper investigates a broad class of loop spaces with Noetherian mod p cohomology, analyzing their classifying spaces and revealing key differences from p-compact groups, especially in their homotopy groups.

Contribution

It extends the understanding of loop spaces with Noetherian cohomology beyond p-compact groups, characterizing their classifying spaces and homotopy properties.

Findings

Classifying space cohomology is comparable to BCP^∞.

BX differs from p-compact group classifying spaces in one homotopy group.

Includes analysis of 4-connected covers of Lie group classifying spaces.

Abstract

The class of loop spaces whose mod p cohomology is Noetherian is much larger than the class of p-compact groups (for which the mod p cohomology is required to be finite). It contains Eilenberg-Mac Lane spaces such as the infinite complex projective space and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space BX of such an object and prove it is as small as expected, that is, comparable to that of BCP^\infty. We also show that BX differs basically from the classifying space of a p-compact group in a single homotopy group. This applies in particular to 4-connected covers of classifying spaces of Lie groups and sheds new light on how the cohomology of such an object looks like.