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

Abstract

This is the beginning of an obstruction theory for deciding whether a map $f:S^2\to X^4$ 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 $\mu \left(f\right)$ which tells the whole story in higher dimensions. Our second order obstruction $\tau \left(f\right)$ is defined if $\mu \left(f\right)$ vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of ${\pi }_{1}X$ modulo ${\mathcal{S}}_3$–symmetry (rather then just one copy modulo ${\mathcal{S}}_2$–symmetry). It generalizes to the non-simply connected setting the Kervaire–Milnor invariant which corresponds to the Arf–invariant of knots in 3–space.

We also give necessary and sufficient conditions for moving three maps $f_1,f_2,f_3:S^2\to X^4$ to a position in which they have disjoint images. Again the obstruction $\lambda \left(f_1,f_2,f_3\right)$ generalizes Wall’s intersection number $\lambda \left(f_1,f_2\right)$ 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 $\mu \left(1,2,3\right)$, generalizing the Matsumoto triple to the non simply-connected setting.