Abstract view
On Homotopy Invariants of Combings of Threemanifolds


Published:20141113
Printed: Feb 2015
Christine Lescop,
Institut Fourier, UJF Grenoble, CNRS , 100 rue des maths, BP 74 , 38402 SaintMartin d'HÃ¨res cedex , France
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 CassonWalker invariant.
The article also includes a selfcontained 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, CassonWalker invariant, perturbative expansion of ChernSimons theory, configuration space integrals
Spin$^c$structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, CassonWalker invariant, perturbative expansion of ChernSimons theory, configuration space integrals
