Coverings of dehn fillings of surface bundles

Abstract

This paper concern the study of 3-manifolds which are obtained by Dehn filling on a surface bundle over S 1 (every closed 3-manifold is so representable) and directed toward the question: which 3-manifolds, M, have fundamental group, π 1 ( M), virtually Z -representable (have a finite sheeted covering space M̃ → M with β 1( M ̃ ) = rank H 1( M ̃ )>0) ? For Dehn fillings of a bundle with periodic or reducible monodromy (or of any nonsimple 3-manifold) this is shown to be generally the case with possible exceptions of a specified form. So, for example, nontrivial surgery on a composite knot always produces a 3-manifold with virtually Z -representable fundamental group. We note that any sheeted covering, M̃, of a Dehn filling, M, of a surface bundle, N, is a Dehn filling of a surface bundle, Ñ, covering N. We determine a lower bound for {β 1(M̃): M̃ a filling of Ñ}, and combine these results with some calculations to show that a/ b surgery on the figure eight knot produces manifolds with virtually Z -representable fundamental groups for any a and either b≡±2 a mod 7 or b≡± a mod 13