Pulling Apart 2-spheres in 4-manifolds

TL;DR

This paper develops an obstruction theory using Whitney towers to determine when 2-spheres in 4-manifolds can be separated, generalizing Milnor invariants and providing complete obstructions in certain cases.

Contribution

It introduces higher-order intersection invariants based on Whitney towers that generalize Milnor's link invariants and fully characterize when 2-spheres can be pulled apart in 4-manifolds.

Findings

Complete obstruction to pulling apart 2-spheres in certain 4-manifolds.

Any number of parallel immersed surfaces with vanishing self-intersection can be pulled apart in simply connected 4-manifolds.

Order 2 invariants lead to interesting number theoretic questions.

Abstract

An obstruction theory for representing homotopy classes of surfaces in 4-manifolds by immersions with pairwise disjoint images is developed, using the theory of non-repeating Whitney towers. The accompanying higher-order intersection invariants provide a geometric generalization of Milnor's link-homotopy invariants, and can give the complete obstruction to pulling apart 2-spheres in certain families of 4-manifolds. It is also shown that in an arbitrary simply connected 4-manifold any number of parallel copies of an immersed surface with vanishing self-intersection number can be pulled apart, and that this is not always possible in the non-simply connected setting. The order 1 intersection invariant is shown to be the complete obstruction to pulling apart 2-spheres in any 4-manifold after taking connected sums with finitely many copies of S^2\times S^2; and the order 2 intersection indeterminacies for quadruples of immersed 2-spheres in a simply connected 4-manifold are shown to lead to interesting number theoretic questions.