Title
On the bipolar filtration of topologically slice knots
Abstract
Celebrated theorems of Freedman and Donaldson have an immediate corollary that there is a topologically slice knot which is not smoothly slice. Let $\mathcal{T}$ be the subgroup of the smooth knot concordance group of topologically slice knots. Understanding the structure of $\mathcal{T}$ is of fundamental importance since $\mathcal{T}$ measures the subtle difference between topological and smooth category in dimension 4. Cochran, Harvey and Horn proposed a beautiful framework to study $\mathcal{T}$ systematically by introducing a geometrically defined filtration on $\mathcal{T}$ which is called the bipolar filtration. Cochran, Harvey and Horn could interpret many knot concordance invariants in terms of the bipolar filtration. Up to now, the non-triviality of this filtration was settled only at the zeroth level and the first level. In this talk, we prove that the bipolar filtration on $\mathcal{T}$ is highly non-trivial at every level. The proof involves both Cheeger-Gromov $L^2$ $\rho$-invariants and Heegaard Floer $d$-invariants. This is joint work with Professor Jae Choon Cha.