Shelly Harvey

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

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

Knot concordance and higher-order Blanchfield duality

The filtration is important because of its strong connection to the classification of topological 4‐manifolds. Here we introduce new techniques for studying C and use them to prove that, for each n2 N0 , the group Fn=Fn:5 has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson‐Gordon and Gilmer, contain slice knots. 57M25; 57M10

Primary decomposition and the fractal nature of knot concordance

For each sequence

Filtering smooth concordance classes of topologically slice knots

{\mathcal{P}=(p_1(t),p_2(t),\dots)}

Monotonicity of degrees of generalized Alexander polynomials of groups and 3-manifolds

of polynomials we define a characteristic series of groups, called the derived series localized at

2-torsion in the n-solvable filtration of the knot concordance group

{\mathcal{P}}