Higher Order Intersections in Low-Dimensional Topology

TL;DR

This paper introduces higher-order intersection invariants for Whitney towers in 4-manifolds, linking them to classical link invariants and using them to classify Whitney tower existence, with implications for string links and homology cylinders.

Contribution

It develops new higher-order invariants for Whitney towers, connecting them to known link invariants and providing a classification framework for Whitney tower obstructions.

Findings

Identification of new invariants with Milnor, Sato-Levine, and Arf invariants

Definition of higher-order Sato-Levine and Arf invariants

Classification of Whitney towers using these invariants

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.