Journal of the European Mathematical Society

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Volume 13, Issue 5, 2011, pp. 1225–1244
DOI: 10.4171/JEMS/279

Published online: 2011-07-13

Noetherian loop spaces

Natàlia Castellana[1], Juan A. Crespo[2] and Jérôme Scherer[3]

(1) Universidad Autonoma de Barcelona, Bellaterra, Spain
(2) Universidad Autónoma de Madrid, Spain
(3) EPFL, Lausanne, Switzerland

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.

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Castellana Natàlia, Crespo Juan, Scherer Jérôme: Noetherian loop spaces. J. Eur. Math. Soc. 13 (2011), 1225-1244. doi: 10.4171/JEMS/279