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Cellular properties of nilpotent spaces

Wojciech Chachólski, Emmanuel Dror Farjoun, Ramón Flores and Jérôme Scherer

Geometry & Topology 19 (2015) 2741–2766

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield–Kan homology completion tower zkX whose terms we prove are all X–cellular for any X. As straightforward consequences, we show that if X is K–acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections PnX, and that any nilpotent space for which the space of pointed self-maps map(X,X) is “canonically” discrete must be aspherical.

cellular approximation, nilpotent group, generalized homology theory, classifying spaces of groups, Eilenberg–Mac Lane space
Mathematical Subject Classification 2010
Primary: 55P60, 20F18
Secondary: 55N20, 55R35
Received: 16 January 2014
Revised: 8 October 2014
Accepted: 12 November 2014
Published: 20 October 2015
Proposed: Mark Behrens
Seconded: Haynes Miller, Jesper Grodal
Wojciech Chachólski
Department of Mathematics
KTH Stockholm
Lindstedtsvägen 25
10044 Stockholm
Emmanuel Dror Farjoun
Department of Mathematics
Hebrew University of Jerusalem
Givat Ram
Jerusalem 91904
Ramón Flores
Departamento de Matemáticas
Facultad de Ciencias
Universidad Autónoma de Madrid
28049 Madrid
Jérôme Scherer
Department of Mathematics
EPFL Lausanne
Station 8
1015 Lausanne