Pulling apart 2-spheres in 4-manifolds
Rob Schneiderman (CUNY)
Abstract:
An obstruction theory for representing homotopy classes of surfaces
in 4--manifolds by immersions with pairwise disjoint images is
developed, using a 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 (order 0) 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 about systems of Diophantine quadratic equations coupled by the intersection form. This is joint work with Peter Teichner (UC Berkeley/MPIM).