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On the homotopy groups of $$p$$-completed classifying spaces. (English) Zbl 1086.55007
Working one prime at a time has been a successful route into the structure of the homotopy groups of a space. As a consequence of Haynes Miller’s solution to the Sullivan conjecture, McGibbon and Neisendorfer proved that a simply-connected CW-complex $$X$$ with finite mod 2 cohomology has infinitely many homotopy groups $$\pi_i(X)$$ with nontrivial 2-torsion. Generalizations of the algebraic condition, $$H^*(X; {\mathbb F}_2)$$ finite, followed with the development of ideas like Lannes’s $$T$$ functor, $$p$$-local groups, topped off by a theorem of Dwyer and Wilkerson that a 2-connected CW-complex of finite type with nontrivial mod $$p$$ cohomology and for which the indecomposable module of $$H^*(X;{\mathbb F}_p)$$ is locally finite as a module over the Steenrod algebra has nontrivial $$p$$-torsion in infinitely many homotopy groups. The present paper concerns the gap between connected and 2-connected. The authors prove that a $$p$$-complete space $$X$$ for which $$H^*(X;{\mathbb F}_p)$$ is of finite type and for which the indecomposable module of $$H^*(X;{\mathbb F}_p)$$ is locally finite as a module over the Steenrod algebra must satisfy either $$X$$ is aspherical, $$X$$ is a finite product of Eilenberg-Mac Lane spaces of the type $$K({\mathbb Z}_p^{^2})$$, or $$X$$ has infinitely many homotopy groups with nontrivial $$p$$-torsion. The authors further prove the same results for the particular case of $$X = BG$$ for $$G$$ a finite group, a $$p$$-local finite group, a finitely generated virtually nilpotent group, or a quasi-$$p$$-perfect group of finite virtual mod $$p$$ cohomology.

##### MSC:
 55P60 Localization and completion in homotopy theory 55Q05 Homotopy groups, general; sets of homotopy classes 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology
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