Postnikov pieces and 𝐵ℤ/𝕡-homotopy theory

Abstract

We present a constructive method to compute the cellularization with respect to B m Z / p B^{m}\mathbb {Z}/p for any integer m ≥ 1 m \geq 1 of a large class of H H -spaces, namely all those which have a finite number of non-trivial B m Z / p B^{m}\mathbb {Z}/p -homotopy groups (the pointed mapping space map ∗ ⁡ ( B m Z / p , X ) \operatorname {map}_*(B^{m}\mathbb {Z}/p, X) is a Postnikov piece). We prove in particular that the B m Z / p B^{m}\mathbb {Z}/p -cellularization of an H H -space having a finite number of B m Z / p B^{m}\mathbb {Z}/p -homotopy groups is a p p -torsion Postnikov piece. Along the way, we characterize the B Z / p r B\mathbb {Z}/p^r -cellular classifying spaces of nilpotent groups.