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2015 A formula for the $\Theta$–invariant from Heegaard diagrams
Christine Lescop
Geom. Topol. 19(3): 1205-1248 (2015). DOI: 10.2140/gt.2015.19.1205


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.


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Christine Lescop. "A formula for the $\Theta$–invariant from Heegaard diagrams." Geom. Topol. 19 (3) 1205 - 1248, 2015.


Received: 17 September 2012; Revised: 24 July 2014; Accepted: 6 August 2014; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1344.57007
MathSciNet: MR3352234
Digital Object Identifier: 10.2140/gt.2015.19.1205

Primary: 57M27
Secondary: 55R80, 57R20

Rights: Copyright © 2015 Mathematical Sciences Publishers


Vol.19 • No. 3 • 2015
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