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Whitney tower concordance of classical links

James Conant, Rob Schneiderman and Peter Teichner

Geometry & Topology 16 (2012) 1419–1479

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.

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
Received: 4 February 2011
Revised: 28 May 2012
Accepted: 28 May 2012
Published: 24 July 2012
Proposed: Rob Kirby
Seconded: Ronald J Stern, David Gabai
James Conant
Department of Mathematics
University of Tennessee
Knoxville TN 37996
Rob Schneiderman
Department of Mathematics and Computer Science
Lehman College, CUNY
Bronx NY 10468-1589
Peter Teichner
Department of Mathematics
University of California, Berkeley
Berkeley CA 94720-3840
Max-Planck Institut für Mathematik, Bonn, Germany