Geometric filtrations of string links and homology cylinders

Abstract

We show that the group of string links modulo order $n$ twisted Whitney tower concordance is an extension of the image of the nilpotent Artin representation by a finite 2-group. Moreover, this 2-group is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also analyze the Goussarov–Habiro clasper filtration of the group of 3-dimensional homology cylinders modulo homology cobordism, importing techniques from our work on Whitney towers to improve on results of J. Levine. In particular, we classify the graded group associated to the Goussarov–Habiro filtration in all orders except $4n+1$. In this last case, it is classified up to unknown 2-torsion with a precise upper bound. These calculations confirm conjectures of Levine in the even cases, and improve on his conjectures in the odd cases. In the last section of this paper we connect the settings of string links and homology cylinders by analyzing a geometric map, originally formulated by N. Habegger.