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.