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On Homotopy Invariants of Combings of Three-manifolds

  Published:2014-11-13
 Printed: Feb 2015
  • Christine Lescop,
    Institut Fourier, UJF Grenoble, CNRS , 100 rue des maths, BP 74 , 38402 Saint-Martin d'Hères cedex , France
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Abstract

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 Spin$^c$-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 Spin$^c$-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 $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for $3$-manifolds.
Keywords: Spin$^c$-structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, Casson-Walker invariant, perturbative expansion of Chern-Simons theory, configuration space integrals Spin$^c$-structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, Casson-Walker invariant, perturbative expansion of Chern-Simons theory, configuration space integrals
MSC Classifications: 57M27, 57R20, 57N10 show english descriptions Invariants of knots and 3-manifolds
Characteristic classes and numbers
Topology of general $3$-manifolds [See also 57Mxx]
57M27 - Invariants of knots and 3-manifolds
57R20 - Characteristic classes and numbers
57N10 - Topology of general $3$-manifolds [See also 57Mxx]
 

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