No abstract

In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot concordance group C,

For each sequence of polynomials, P=(p_1(t),p_2(t),...), we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S^3, such a sequence of polynomials arises naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These group series yield new filtrations of the knot concordance group that refine the (n)-solvable filtration of Cochran-Orr-Teichner. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no Cochran-Orr-Teichner knot is concordant to any Cochran-Harvey-Leidy knot.Comment: 60 pages, added 4 pages to introduction, minor corrections otherwise; Math. Annalen 201

We propose and analyze a structure with which to organize the difference between a knot in S 3 bounding a topologically embedded 2-disk in B 4 and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n-solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, fB n g, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each B n =B nC1 has infinite rank. But our primary interest is in the induced filtration, fT n g, on the subgroup, T , of knots that are topologically slice. We prove that T =T 0 is large, detected by gauge-theoretic invariants and the , s , -invariants, while the nontriviality of T 0 =T 1 can be detected by certain d -invariants. All of these concordance obstructions vanish for knots in T 1 . Nonetheless, going beyond this, our main result is that T 1 =T 2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each T n =T nC1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity. 57M25

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 CheegerGromov -invariant, we obtain new real-valued homology cobordism invariants n for closed .4k 1/-dimensional manifolds. For 3-dimensional manifolds, we show that f n jn 2 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 (m D 1), 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.