Volume 1, issue 1 (2001)

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

Rob Schneiderman and Peter Teichner

Algebraic & Geometric Topology 1 (2001) 1–29
 arXiv: math.GT/0008048
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 ${\mathsc{S}}_{3}$–symmetry (rather then just one copy modulo ${\mathsc{S}}_{2}$–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 ${f}_{1},{f}_{2},{f}_{3}:\phantom{\rule{0.3em}{0ex}}{S}^{2}\to {X}^{4}$ 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.

Keywords
intersection number, 4–manifold, Whitney disk, immersed 2–sphere, cubic form
Primary: 57N13
Secondary: 57N35