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

By Shelly L. Harvey. 2002

84 citations recorded by the source corpus.

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 .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6b98d30d2bdd40cd2d3651f34a295b9b5f17bbc9

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In Mathematical Physics

Citations in the corpus

Knot concordance and von Neumann $\rho$-invariants (2004, methodology): “Indeed, in [ Ha ], such bounds were established explicitly for any closed 3-manifold using the Alexander norm.”
A SURVEY OF TWISTED ALEXANDER POLYNOMIALS (2009, methodology): “entation, then we just obtain McMullen’s Alexander norm. The following is proved in [FK08a, Theorem 3.1], but we also refer to the work of McMullen [McM02], Turaev [Tu02a], [Tu02b, Section 6], Harvey [Ha05] and Vidussi [Vi99, Vi03]. The main idea of the proof is to combine Theorem 6.6 with Corollary”
Primary decomposition and the fractal nature of knot concordance (2009, methodology): “The model example is the rational derived series, {G r }, given by G r ≡ G and G r ≡ {x ∈ G r | ∃k > 0, x ∈ [G r , G r ]}, first used systematically in [26].”
The structure of the rational concordance group of knots (2006, methodology): “It is known that G (n) Q is a normal subgroup of G [11, 19].”
Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group (2006, background): “owship. 1 2 SHELLY L. HARVEY series but whose successive quotients G(n) H /G (n+1) H are torsion-free as Z[G/G (n) H ]- modules. This can be compared to the rational derived series G(n) r (studied in [20]) which is a series whose successive quotients are torsion-free as abelian groups. Let Mbe a (4k”
Knot concordance and higher-order Blanchfield duality (2007, background): “Such modules were named higher-order Alexander modules in [12] [7] [24].”