On Homotopy Invariants of Combings of Three-manifolds

Source abstract

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 ; Z). Gompf introduced a Q-valued invariant θ G of torsion combings on closed 3-manifolds, and he showed that θ G distinguishes all torsion combings with the same Spin c -structure. We give an alternative definition for θ 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 p1 to the Θ-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula Θ = ¼ P 1 + 6λ , where λ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.