Pré-Publication, Document De Travail Année : 2012

A formula for the Theta invariant from Heegaard diagrams

Christine Lescop

Résumé

The Theta invariant is the simplest 3-manifold invariant defined with 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 Q-sphere M. These propagators represent the linking form of M so that $\Theta(M,X)$ can be thought of as the cube of the linking form of M with respect to the combing X. The Theta invariant is the sum of $6 \lambda(M)$ and $p_1(X)/4$, where $\lambda$ denotes the Casson-Walker invariant, and $p_1$ is an invariant of combings that 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 Theta invariant in terms of Heegaard diagrams.

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Dates et versions

hal-00732323 , version 1 (14-09-2012)
hal-00732323 , version 2 (05-02-2014)
hal-00732323 , version 3 (09-06-2015)

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Christine Lescop. A formula for the Theta invariant from Heegaard diagrams. 2012. ⟨hal-00732323v1⟩
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