3-manifolds with subgroups<i>Z</i>⊕<i>Z</i>⊕<i>Z</i>in their fundamental groups

Abstract

In this paper we characterize those 3-manifolds M 3 satisfying ZZZC ^i(f). All such manifolds M arise in one of the following ways: (I) M = M o # R, (II) M= M o # R*, (III) M = M o U R*. Here f 0 is any 3-manifold in (I), (II) and any 3-manifold having P 2 components in its boundary in (III). R is a flat space form and R* is obtained from R and some involution t: R -> R with fixed points, but only finitely many, as follows: if C,,..., C n are disjoint 3-cells around the fixed points then R* is the 3-manifold obtained from (R -int(C, U UQ))/ by identifying some pairs of projective planes in the boundary.