Fibering 3-manifolds that admit free 𝑍_{𝑘} actions

Abstract

Introduction.In a recent paper [3], Kwun showed that closed orientable 3-manifolds which double-cover themselves fiber over the circle (some technical restrictions are placed on the manifolds).In doing so he applied a criterion for fibering due to Stallings [6].In this paper we extend Kwun's approach to show that certain 3-manifolds admitting a free Zk action fiber over the circle.Recall that a free Zk action on a space M is an effective action on M by a cyclic group of order k with only the identity having fixed points.A proper Zk action is one with the property that a generator of the action is homotopic to the identity homeomorphism.M* will be used to denote the orbit space M\Zk.We let/?: A/-> M* be the projection map throughout this paper.Singular homology and cohomology with integer coefficients will be used exclusively.All manifolds are connected.Our goal is to establish the following Theorem.Let M be a compact, connected, orientable, irreducible 3-manifold with Bd M either empty or connected.If M admits a proper free Zk action, for some prime k¡¿2, such that Hi(M*\ Z) has no element of order k then M can be fibered over the circle.Examples of 3-manifolds admitting proper free Zk actions are plentiful.One ready source of nontrivial examples is the class of closed 3-manifolds admitting effective 50(2) actions.In this class we find that the lens space L(ß, -a(mod ß)) admits proper free Zk actions for (k, «)=*= 1. Moreover this space does not fiber over the circle and the first homology group of the orbit space always has elements of order k.Other examples from this class indicate that we cannot drop from our theorem the requirement that the Zk action be proper.For a classification of 50(2) actions on 3-manifolds the reader is referred to [4].2. Partitioning M. In this section we present some lemmas leading to the main result which we prove in § §3 and 4. The first lemma is a collection of well-known facts.