# Noetherian loop spaces

### Jérôme Scherer

EPFL, Lausanne, Switzerland### Natàlia Castellana Vila

Universidad Autonoma de Barcelona, Bellaterra, Spain### Juan A. Crespo

Universidad Autónoma de Madrid, Spain

## 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.

## Cite this article

Jérôme Scherer, Natàlia Castellana Vila, Juan A. Crespo, Noetherian loop spaces. J. Eur. Math. Soc. 13 (2011), no. 5, pp. 1225–1244

DOI 10.4171/JEMS/279