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 \emph{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 $S^2\times S^2$; 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.

2010 Mathematics Subject Classification: Primary 57M99; Secondary 57M25.

Keywords and Phrases: 2--sphere, 4--manifold, disjoint immersion, homotopy invariant, non-repeating Whitney tower.

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