Higher-Order Polynomial Invariants of 3-Manifolds Giving Lower Bounds for the Thurston Norm

TL;DR

This paper introduces a sequence of new invariants for 3-manifolds that improve bounds on the Thurston norm and provide obstructions to certain geometric structures, with applications to knot theory.

Contribution

It defines delta_n invariants related to the derived series of the fundamental group, offering new lower bounds for the Thurston norm and obstructions to fibering and Seifert fibered structures.

Findings

Delta_n invariants give better Thurston norm bounds than Alexander norm.

Delta_n provide obstructions to fibering over S^1 and Seifert fibered structures.

Obstructions to symplectic structures on 4-manifolds of form X x S^1.

Abstract

We define an infinite sequence of new invariants, delta_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 delta_n give obstructions to a 3-manifold fibering over S^1 and to a 3-manifold being Seifert fibered. Moreover, we show that the delta_n give computable algebraic obstructions to a 4-manifold of the form X 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.