Filtering smooth concordance classes of topologically slice knots

Recall that the n-solvable filtration of the smooth knot concordance group, suggested by work of Cochran-Orr-Teichner, has the property that any topologically slice knot lies in every term of the filtration. With the aim of studing topologically slice knots, 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. As is the case for the n-solvable filtration, 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-triviality 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.