Pulling apart 2-spheres in 4-manifolds

  • Rob Schneiderman

    Department of Mathematics Max-Planck Institut and Computer Science, für Mathematik Lehman College, Vivatsgasse 7, 53111 Bonn City University of New York, Germany New York NY 10468 USA
  • Peter Teichner

    Department of Mathematics Max-Planck Institut and Computer Science, für Mathematik Lehman College, Vivatsgasse 7, 53111 Bonn City University of New York, Germany New York NY 10468 USA
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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

Rob Schneiderman, Peter Teichner, Pulling apart 2-spheres in 4-manifolds. Doc. Math. 19 (2014), pp. 941–992

DOI 10.4171/DM/469