Geometry & Topology Monographs, Vol. 7 (2004),
Proceedings of the Casson Fest,
Paper no. 4, pages 101--134.

Whitney towers and the Kontsevich integral

Rob Schneiderman, Peter Teichner

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.

Keywords. 2-sphere, 4-manifold, link concordance, Kontsevich integral, Milnor invariants, Whitney tower

AMS subject classification. Primary: 57M99. Secondary: 57M25.

E-print: arXiv:math.GT/0401441

Submitted to GT on 4 December 2003. (Revised 24 July 2004.) Paper accepted 17 June 2004. Paper published 18 September 2004.

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

Email: schneiderman@courant.nyu.edu, teichner@math.berkeley.edu

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