Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group

By Shelly L. Harvey. 2006

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For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger‐ Gromov ‐invariant, we obtain new real-valued homology cobordism invariants n for closed .4k 1/‐dimensional manifolds. For 3‐dimensional manifolds, we show thatf njn2 Ng is a linearly independent set and for each n 0, the image of n is an infinitely generated and dense subset of R. In their seminal work on knot concordance, T Cochran, K Orr and P Teichner define a filtration F m .n/ of the m‐component (string) link concordance group, called the .n/‐solvable filtration. They also define a grope filtration G m n . We show that n vanishes for .nC1/‐solvable links. Using this, and the nontriviality of n , we show that for each m 2, the successive quotients of the .n/‐solvable filtration of the link concordance group contain an infinitely generated subgroup. We also establish a similar result for the grope filtration. We remark that for knots (mD1), the successive quotients of the .n/‐solvable filtration are known to be infinite. However, for knots, it is unknown if these quotients have infinite rank when n 3. 57M27; 20F14

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/54dbf17677d45f94d376a315f2d24899eb4c3e15

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Citations in the corpus

Knot concordance and higher-order Blanchfield duality (2007, background): “Thus in this special case (which was the case treated in [11; 12; 24]), the size of rational .”
Primary decomposition and the fractal nature of knot concordance (2009, background): “In this section we generalize a philosophy introduced by the second author [29] to define various commutator series by killing (via the πn) selected torsion elements of G ∗ /[G ∗ ,G ∗ ], where the latter is viewed as a module (see below).”
A SURVEY OF TWISTED ALEXANDER POLYNOMIALS (2009, unknown): “he notion of higher order Alexander polynomials was extended to 3–manifolds by Harvey [Ha05] and Turaev [Tu02b]. This theory is diﬀerent in spirit to the fore mentioned L2–invariants, but we refer to [Ha08] and [FLM09, Proposition 2.4] for some connections. 10.1. Non–commutative Alexander polynomial”