Pulling Apart 2–Spheres in 4–Manifolds

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 2–sphere 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 × S; 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. 2010 Mathematics Subject Classification: Primary 57M99; Secondary 57M25.