Global classification of generic multi-vector fields of top degree

Abstract

For any compact oriented manifold $M$, we show that that the top degree multi-vector fields transverse to the zero section of $\wedge^{\text{top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of the arrangement of its zero locus and a finite number of numerical invariants. The group governing the infinitesimal deformations of such multi-vector fields is computed, and an explicit set of generators exhibited. For the spheres $S^n$, a correspondence between certain isotopy classes of multi-vector fields and classes of weighted signed trees is established.