On homotopy invariants of combings of 3-manifolds
arXiv:1209.2785 · doi:10.4153/CJM-2014-031-4
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 . It only depends on the Spin-structure represented by the combing. When this Euler class is a torsion element of , we say that the combing is a torsion combing. Gompf introduced a -valued invariant of torsion combings of closed 3-manifolds that distinguishes all combings that represent a given Spin-structure. This invariant provides a grading of the Heegaard Floer homology for manifolds equipped with torsion Spin-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 for combings of manifolds bounded by . We show that the -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 where is the Casson-Walker invariant. The article also includes a mostly self-contained presentation of combings.
31 pages + 1 page at the end that summarizes the changes with respect to the first version
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