#### Volume 16, issue 3 (2012)

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Whitney tower concordance of classical links

### James Conant, Rob Schneiderman and Peter Teichner

Geometry & Topology 16 (2012) 1419–1479
##### Abstract

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato–Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the $4$–ball bounded by a link in the $3$–sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor’s link invariants.

##### Keywords
Whitney tower, grope, link concordance, tree, higher-order Arf invariant, higher-order Sato–Levine invariant, twisted Whitney disk
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57Q60
Secondary: 57N10