Mathematics – Geometric Topology
Scientific paper
2002-07-01
Topology 44 (2005), no. 5, 895--945.
Mathematics
Geometric Topology
Scientific paper
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.
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