Document Zbl 1348.55009 - zbMATH Open

This work is concerned with the notion of cellular space introduced by E. Dror Farjoun [Cellular spaces, null spaces and homotopy localization. Berlin: Springer-Verlag (1995; Zbl 0842.55001)]. A class of pointed spaces closed both under weak equivalences and homotopy colimits is called cellular and the smallest cellular class containing a given pointed space \(A\) is denoted \(\mathcal{C}(A)\). The relation \(X\in{\mathcal C}(A)\) is written \(X \gg A\).
With this notion, the authors change the perspective between a nilpotent space \(X\) and its Postnikov tower \(P_n X\). It is usual to consider \(X\) as homotopy limit of \(P_n X\) but here the authors prove that the Postnikov sections can be constructed out of \(X\) by means of wedges, homotopy pushouts and telescopes. In the previous setting that means \(P_nX\gg X\) if \(P_n X\) is nilpotent.
This theorem is established by using a modified Bousfield-Kan completion tower \(\ldots\to z_kX\to\ldots\to z_0X\), already defined by E. Dror Farjoun [Contemp. Math. 265, 27–39 (2000; Zbl 0971.55017)], and the property \(z_k X\gg X\).
The authors present also some consequences of this main result. First, mention the following extension of the “key lemma” of A. K. Bousfield [Am. J. Math. 119, No. 6, 1321–1354 (1997; Zbl 0886.55011)]: let \(X\) be a connected space whose fundamental group \(\pi_1 X\) is nilpotent. If the map \(\pi_1: {\mathrm{map}}_*(X,X)\to {\mathrm{Hom}}(\pi_1X,\pi_1X)\) is a weak equivalence, then \(X\) is weakly equivalent to \(K(\pi_1X,1)\). They provide also an extension of the classical Serre class statements. In particular, for any reduced homology theory \(\mathcal K\), if \(X\) is nilpotent, they prove that \(\prod_{k\geq 1} K(\pi_kX,k)\) is \(\mathcal K\)-acyclic if and only if \(\prod_{k\geq 1} K(H_k(X;\mathbb Z),k)\) is \(\mathcal K\)-acyclic.

MSC:

55P60	Localization and completion in homotopy theory
20F18	Nilpotent groups
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology
55R35	Classifying spaces of groups and \(H\)-spaces in algebraic topology

References:

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