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A formula for the $\Theta$–invariant from Heegaard diagrams

Christine Lescop

Geometry & Topology 19 (2015) 1205–1248

The Θ–invariant is the simplest 3–manifold invariant defined by configuration space integrals. It is actually an invariant of rational homology spheres equipped with a combing over the complement of a point. It can be computed as the algebraic intersection of three propagators associated to a given combing X in the 2–point configuration space of a –sphere M. These propagators represent the linking form of M so that Θ(M,X) can be thought of as the cube of the linking form of M with respect to the combing X. The invariant Θ is the sum of 6λ(M) and p1(X)4, where λ denotes the Casson–Walker invariant, and p1 is an invariant of combings, which is an extension of a first relative Pontrjagin class. In this article, we present explicit propagators associated with Heegaard diagrams of a manifold, and we use these “Morse propagators”, constructed with Greg Kuperberg, to prove a combinatorial formula for the Θ–invariant in terms of Heegaard diagrams.

configuration space integrals, finite type invariants of $3$–manifolds, homology spheres, Heegaard splittings, Heegaard diagrams, combings, Casson–Walker invariant, perturbative expansion of Chern-Simons theory, $\Theta$–invariant
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 55R80, 57R20
Received: 17 September 2012
Revised: 24 July 2014
Accepted: 6 August 2014
Published: 21 May 2015
Proposed: Peter Teichner
Seconded: Robion Kirby, Simon Donaldson
Christine Lescop
Institut Fourier
Université de Grenoble, CNRS
100 rue des maths
BP 74
38402 Saint-Martin d’Hères cedex