On Homotopy Invariants of Combings of Three-manifolds

Abstract

AbstractCombings of compact, oriented, 3-dimensional manifoldsMare 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 Spinc-structure. A combing is called torsion if this Euler class is a torsion element of H2(M; Z). Gompf introduced a Q-valued invariant θGof torsion combings on closed 3-manifolds, and he showed that θGdistinguishes all torsion combings with the same Spinc-structure. We give an alternative definition for θGand we express its variation as a linking number. We define a similar invariantp1of combings for manifolds bounded by S2. We relate p1 to the Θ-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula Θ = ¼P1+ 6λ, where λ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.