Higher Order Intersections in Low-Dimensional Topology

Abstract

We show how to measure the failure of the Whitney trick 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 like 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 non- triviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described. This article is an announcement and summary of results to be published in several forthcoming papers.

References in corpus (4)

• Stable concordance of knots in 3-manifolds
• Geometric Filtrations of Classical Link Concordance
• Universal Quadratic Forms and Untwisting Whitney Towers
• Pulling Apart 2-spheres in 4-manifolds

Cited by in corpus (9)

• Universal quadratic forms and Whitney tower intersection invariants
• Tree homology and a conjecture of Levine
• Concordance of links with identical Alexander invariants
• Geometric Filtrations of Classical Link Concordance
• Universal Quadratic Forms and Untwisting Whitney Towers
• Classification of string links up to $2n$-moves and link-homotopy
• Classification of Links Up to 0-Solvability
• Whitney towers, Gropes and Casson-Gordon style invariants of links
• Pulling Apart 2-spheres in 4-manifolds