3-manifolds with subgroups Z ⊕ Z ⊕ Z in their fundamental groups

Erhard Luft and Denis Karmen Sjerve

Vol. 114 (1984), No. 1, 191–205

DOI: 10.2140/pjm.1984.114.191

Abstract

In this paper we characterize those 3-manifolds M³ satisfying Z ⊕Z ⊕Z ⊆ π₁(M). All such manifolds M arise in one of the following ways: (I) M = M₀#R, (II) M = M₀#R^∗, (III) M = M₀ ∪_∂R^∗. Here M₀ is any 3-manifold in (I), (II) and any 3-manifold having P² components in its boundary in (III). R is a flat space form and R^∗ is obtained from R and some involution i : R → R with fixed points, but only finitely many, as follows: if C₁,…,C_n are disjoint 3-cells around the fixed points then R^∗ is the 3-manifold obtained from (R− int(C₁ ∪⋯∪C_n))∕i by identifying some pairs of projective planes in the boundary.

Mathematical Subject Classification 2000

Primary: 57N10

Milestones

Received: 3 June 1982

Published: 1 September 1984