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

We define an infinite sequence of new invariants, δn, 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 δn give obstructions to a 3-manifold fibering over S1 and to a 3-manifold being Seifert fibered. Moreover, we show that the δn give computable algebraic obstructions to a 4-manifold of the form X×S1 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 S3.