Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

Abstract

We define an infinite sequence of new invariants, δ n , of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ n give obstructions to a 3-manifold fibering over S 1 and to a 3-manifold being Seifert fibered. Moreover, we show that the δ n give computable algebraic obstructions to a 4-manifold of the form X × S 1 admitting a symplectic structure even when the obstructions given by the Seiberg–Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S 3 .