The bipolar filtration of topologically slice knots

TL;DR

This paper investigates the structure of the smooth concordance group of topologically slice knots using the bipolar filtration, revealing infinite rank in its graded quotients and employing advanced invariants for detection.

Contribution

It demonstrates that the graded quotients of the bipolar filtration have infinite rank at each stage beyond one, using higher order invariants for detection.

Findings

Graded quotients of the bipolar filtration have infinite rank.

Higher order amenable Cheeger-Gromov $L^2$ $ ho$-invariants are effective in detection.

Heegaard Floer correction term $d$-invariants complement the analysis.

Abstract

The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of deeper structures in the smooth concordance group of topologically slice knots. We show that the graded quotient of the bipolar filtration of topologically slice knots has infinite rank at each stage greater than one. To detect nontrivial elements in the quotient, the proof simultaneously uses higher order amenable Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term $d$-invariants.