On homotopy invariants of combings of 3-manifolds

Abstract

Combings of oriented compact 3-manifolds are homotopy classes of nowhere zero vector fields in these manifolds. A first known invariant of a combing is its Euler class, that is the Euler class of the normal bundle to a combing representative in the tangent bundle of the 3-manifold $M$. It only depends on the Spin$^c$-structure represented by the combing. When this Euler class is a torsion element of $H^2(M;Z)$, we say that the combing is a torsion combing. Gompf introduced a $Q$-valued invariant $\theta_G$ of torsion combings of closed 3-manifolds that distinguishes all combings that represent a given Spin$^c$-structure. This invariant provides a grading of the Heegaard Floer homology $\hat{HF}$ for manifolds equipped with torsion Spin$^c$-structures. We give an alternative definition of the Gompf invariant and we express its variation as a linking number. We also define a similar invariant $p_1$ for combings of manifolds bounded by $S^2$. We show that the $\Theta$-invariant, that is the simplest configuration space integral invariant of rational homology spheres, is naturally an invariant of combings of rational homology balls, that reads $(\frac14p_1 + 6 \lambda)$ where $\lambda$ is the Casson-Walker invariant. The article also includes a mostly self-contained presentation of combings.