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 $S^3$ 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, $\left\{{\mathcal{ℬ}}_n\right\}$, 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 ${\mathcal{ℬ}}_n∕{\mathcal{ℬ}}_{n+1}$ has infinite rank. But our primary interest is in the induced filtration, $\left\{{\mathcal{T}}_n\right\}$, on the subgroup, $\mathcal{T}$, of knots that are topologically slice. We prove that $\mathcal{T}∕{\mathcal{T}}_0$ is large, detected by gauge-theoretic invariants and the $\tau$, $s$, $ϵ$–invariants, while the nontriviality of ${\mathcal{T}}_0∕{\mathcal{T}}_1$ can be detected by certain $d$–invariants. All of these concordance obstructions vanish for knots in ${\mathcal{T}}_1$. Nonetheless, going beyond this, our main result is that ${\mathcal{T}}_1∕{\mathcal{T}}_2$ has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each ${\mathcal{T}}_n∕{\mathcal{T}}_{n+1}$ has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.