Pulling apart 2-spheres in 4-manifolds
Peter Teichner
Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, GermanyRob Schneiderman
Department of Mathematics and Computer Science, Lehman College, City University of New York, New, York NY 10468, USA

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 emphnon-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 ; 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.
Cite this article
Peter Teichner, Rob Schneiderman, Pulling apart 2-spheres in 4-manifolds. Doc. Math. 19 (2014), pp. 941–992
DOI 10.4171/DM/469