Cellular properties of nilpotent spaces

Chachólski, Wojciech¹ ; Farjoun, Emmanuel Dror² ; Flores, Ramón³ ; Scherer, Jérôme⁴

¹ Department of Mathematics, KTH Stockholm, Lindstedtsvägen 25, 10044 Stockholm, Sweden
² Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel
³ Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
⁴ Department of Mathematics, EPFL Lausanne, Station 8, 1015 Lausanne, Switzerland

Geometry & topology, Tome 19 (2015) no. 5, pp. 2741-2766.

Résumé

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 $z_{k}X$ whose terms we prove are all $X$–cellular for any $X$. As straightforward consequences, we show that if $X$ is $\mathcal{K}$–acyclic and nilpotent for a given homology theory $\mathcal{K}$, then so are all its Postnikov sections $P_{n}X$, and that any nilpotent space for which the space of pointed self-maps ${map}_{\ast }\left(X,X\right)$ is “canonically” discrete must be aspherical.

Détail

DOI : 10.2140/gt.2015.19.2741

Classification : 55P60, 20F18, 55N20, 55R35
Keywords: cellular approximation, nilpotent group, generalized homology theory, classifying spaces of groups, Eilenberg–Mac Lane space

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