Higher order intersection numbers of 2-spheres in 4-manifolds

This is the beginning of an obstruction theory for deciding whether a map f : S → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall’s self-intersection number μ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if μ(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of π1X modulo S3 -symmetry (rather then just one copy modulo S2 -symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant defined in [2] and [12] which corresponds to the Arf-invariant of knots in 3-space. We also give necessary and sufficient conditions for moving three maps f1, f2, f3 : S 2 → X to a position in which they have disjoint images. Again the obstruction λ(f1, f2, f3) generalizes Wall’s intersection number λ(f1, f2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant μ(1, 2, 3), generalizing the Matsumoto triple [10] to the non simply-connected setting. AMS Classification 57N13; 57N35