Noetherian loop spaces

Abstract

The class of loop spaces of which the 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 $\mathbb{C}{P}^{\infty }$ 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 $B\mathbb{C}{P}^{\infty }$. 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 compact Lie groups and sheds new light on how the cohomology of such an object looks like.