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Whitney towers and the Kontsevich integral

Rob Schneiderman and Peter Teichner

Geometry & Topology Monographs 7 (2004) 101–134

DOI: 10.2140/gtm.2004.7.101


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.


2–sphere, 4–manifold, link concordance, Kontsevich integral, Milnor invariants, Whitney tower

Mathematical Subject Classification

Primary: 57M99

Secondary: 57M25


Received: 4 December 2003
Revised: 24 July 2004
Accepted: 17 June 2004
Published: 18 September 2004

Rob Schneiderman
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York NY 10012-1185
Peter Teichner
Department of Mathematics
University of California
Berkeley CA 94720-3840