2010/11/28 by Jim Conant, Rob Schneiderman, Peter Teichner
Mathematics · #Advanced Combinatorial Mathematics #Ball (mathematics) #Combinatorics #Geometric and Algebraic Topology #Geometry #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Iterated function #Mathematical analysis #Mathematics #Pure mathematics #Topology (electrical circuits) #math.GT #msc:57M25 #msc:57M27 #msc:57Q45
paper · pdf · doi:10.1073/pnas.1018581108
arxiv created 2010/11/28 · openalex publication_date 2011/04/25 · arxiv updated 2016/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato-Levine, and Arf invariants. We also define higher-order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.