FINITELY DOMINATED COVERING SPACES OF 3- AND 4-MANIFOLDS

Abstract

AbstractIf P is a closed 3-manifold the covering space associated to a finitely presentable subgroup ν of infinite index in π1(P) is finitely dominated if and only if P is aspherical or $\raisebox {11pt}{}\widetilde {P}\simeq {S^2}$. There is a corresponding result in dimension 4, under further hypotheses on π and ν. In particular, if M is a closed 4-manifold, ν is an ascendant, FP3, finitely-ended subgroup of infinite index in π1(M), π is virtually torsion free and the associated covering space is finitely dominated then either M is aspherical or $\widetilde {M}\simeq {S^2}$ or S3. In the aspherical case such an ascendant subgroup is usually Z, a surface group or a PD3-group.