Noetherian loop spaces

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 ℂP∞ 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∞. 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. © European Mathematical Society 2011.