Graph manifolds with non-empty boundary are covered by surface bundles

Abstract

A classical fact is that Seifert manifolds with non-empty boundary are covered by surface bundles over the circle S1 and closed Seifert manifolds may or may not be covered by surface bundles over S1. Some closed graph manifolds are not covered by surface bundles over S1 ([LW] and [N]). Thurston asked if complete hyperbolic 3-manifolds of finite volume are covered by surface bundles [T]. J. Luecke and Y. Wu asked if graph manifolds with non-empty boundary are covered by surface bundles over S1 ([LW]). In this paper we prove:THEOREM 0·1. Each graph manifold with non-empty boundary is finitely covered by a surface bundle over the circle S1.