Preserving asphericity of virtually nilpotent spaces under P-localization

Abstract

For a set of prime numbers P, we study when the P-localization of an Eilenberg–Mac Lane space with virtually nilpotent fundamental group is again aspherical. While investigating this problem, we devote special attention to infra-nilmanifolds. We prove that the P-localization of an orientable infra-nilmanifold is aspherical if and only if its holonomy group is P-torsion. The same holds for non-orientable infra-nilmanifolds if 2 is in P. We also develop computational techniques to check preservation of asphericity. These are explicitly applied to show that the P-localization of a non-orientable infra-nilmanifold of dimension ⩽3 is always aspherical. We point out that this is no longer true from dimension 4 onwards.