Filtering smooth concordance classes of topologically slice knots

Abstract

We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball 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, {B_n}, 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_{n+1} has infinite rank. But our primary interest is in the induced filtration, {T_n}, 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 tau, s, and epsilon-invariants; while the non-triviliality 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_{n+1} has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.

41 pages, slightly revised introduction, minor corrections and up-dated references, this is the final version to appear in Geometry and Topology

References in corpus (3)

• The knot Floer complex and the smooth concordance group
• Gauge theory and Rasmussen's invariant
• Homology and Derived p-Series of Groups

Cited by in corpus (26)

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• Structure in the bipolar filtration of topologically slice knots
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• Satellite operators with distinct iterates in smooth concordance
• The (n)-solvable filtration of the link concordance group and Milnor's invariants
• Positive Links
• Grope metrics on the knot concordance set
• Topologically slice knots that are not smoothly slice in any definite 4-manifold
• On rational sliceness of Miyazaki's fibered, $-$amphicheiral knots
• Relative genus bounds in indefinite four-manifolds
• Deep and shallow slice knots in 4-manifolds
• On independence of iterated Whitehead doubles in the knot concordance group
• Polynomial splittings of correction terms and doubly slice knots
• From zero surgeries to candidates for exotic definite four-manifolds
• Rational homology 3-spheres and simply connected definite bounding
• Linear independence of cables in the knot concordance group
• Links with nontrivial Alexander polynomial which are topologically concordant to the Hopf link
• On the slice genus of quasipositive knots in indefinite 4-manifolds
• A note on 0-bipolar knots of concordance order two
• Smoothly slice boundary links whose derivative links have nonvanishing Milnor invariants
• One-bipolar topologically slice knots and primary decomposition
• Concordance invariants of doubled knots and blowing up
• Action of the Mazur pattern up to topological concordance
• Knots that are not slice either in positons or in negatons