  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.
