Primary decomposition in the smooth concordance group of topologically slice knots

TL;DR

This paper investigates the primary decomposition conjectures in the smooth concordance group of topologically slice knots, providing evidence that supports these conjectures through new subgroup constructions and analogues for graded groups.

Contribution

It demonstrates the existence of large subgroups where the primary decomposition conjectures hold and establishes analogues for associated graded groups, advancing understanding of knot concordance.

Findings

Large subgroup where conjectures are true

Infinitely many primary parts with infinite rank

Supports conjectures for topologically slice knots

Abstract

We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts each of which has infinite rank. This supports the conjectures for topologically slice knots. We also prove analogues for the associated graded groups of the bipolar filtration of topologically slice knots. Among ingredients of the proof, we use amenable $L^2$-signatures, Ozsv\'ath-Szab\'o $d$-invariants and N\'emethi's result on Heegaard Floer homology of Seifert 3-manifolds. In an appendix, we present a general formulation of the notion of primary decomposition.