Whitney towers and the Kontsevich integral

Abstract

We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and with relations well known from the 3-dimensional theory of finite type invariants. Surprisingly, the same exact relations arise in 4 dimensions, for example the Jacobi (or IHX) relation comes in our context from the freedom of choosing Whitney arcs. We use the finite type theory to show that our invariants agree with the (leading term of the tree part of the) Kontsevich integral in the case where the 4-manifold is obtained from the 4-ball by attaching handles along a link in the 3-sphere.