Geometric filtrations of string links and homology cylinders

TL;DR

This paper explores the structure of string links and homology cylinders through filtrations, establishing a new epimorphism and classifying associated graded groups up to 2-torsion, advancing understanding of their algebraic and geometric properties.

Contribution

It introduces a well-defined epimorphism from the Artin representation on string links and classifies the associated graded groups of homology cylinders up to 2-torsion.

Findings

Epimorphism induced by Artin representation on string links

Kernel generated by band sums of Bing-doubles of knots with nonzero Arf invariant

Complete classification of graded groups up to 2-torsion for Goussarov-Habiro filtration

Abstract

We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also continue J. Levine's work [20, 21, 22] comparing two filtrations of the group of homology cobordism classes of 3-dimensional homology cylinders, one defined in terms of an Artin-type representation (the Johnson filtration) and one defined using clasper surgery (the Goussarov-Habiro filtration). In particular, the associated graded groups are completely classified up to an unknown 2-torsion summand for the Goussarov-Habiro filtration, for which we obtain an upper bound, in a precisely analogous fashion to the classification of the Whitney tower filtration of link concordance.