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On Homotopy Invariants of Combings of Three-manifolds

Published online by Cambridge University Press:  20 November 2018

Christine Lescop
Christine Lescop*
Affiliation:
Institut Fourier, UJF Grenoble, CNRS, 100 rue des maths, BP 74, 38402 Saint-Martin d'Héres cedex, France email: [email protected]
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Abstract

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Combings of compact, oriented, 3-dimensional manifolds $M$ are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. A combing is called torsion if this Euler class is a torsion element of ${{H}^{2}}(M;\,\mathbb{Z})$. Gompf introduced a $\mathbb{Q}$-valued invariant ${{\theta }_{G}}$ of torsion combings on closed 3-manifolds, and he showed that ${{\theta }_{G}}$ distinguishes all torsion combings with the same $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. We give an alternative definition for ${{\theta }_{G}}$ and we express its variation as a linking number. We define a similar invariant ${{p}_{1}}$ of combings for manifolds bounded by ${{S}^{2}}$. We relate ${{p}_{1}}$ to the $\Theta$-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula $\Theta \,=\,\frac{1}{4}{{p}_{1}}\,+\,6\text{ }\!\!\lambda\!\!\text{ }\left( {\hat{M}} \right)$, where $\text{ }\!\!\lambda\!\!\text{ }$ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

Keywords

57M2757R2057N10Spinc-structurenowhere zero vector fieldsfirst Pontrjagin classEuler classHeegaard Floer homology gradingGompf invariantTheta invariantCasson-Walker invariantperturbative expansion of Chern-Simons theoryconfiguration space integrals
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 1 , 01 February 2015 , pp. 152 - 183
Copyright
Copyright © Canadian Mathematical Society 2015

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