Generic flows on 3-manifolds

Abstract

A $3$-dimensional generic flow is a pair $(M,v)$ with $M$ a smooth compact oriented $3$-manifold and $v$ a smooth nowhere-zero vector field on $M$ having generic behavior along $\partial M$; on the set of such pairs we consider the equivalence relation generated by topological equivalence (homeomorphism mapping oriented orbits to oriented orbits) and by homotopy with fixed configuration on the boundary, and we denote by $\mathcal{F}$ the quotient set. In this paper we provide a combinatorial presentation of $\mathcal{F}$. To do so we introduce a certain class $\mathcal{S}$ of finite $2$-dimensional polyhedra with extra combinatorial structures, and some moves on $\mathcal{S}$, exhibiting a surjection $\phi :\mathcal{S}\to \mathcal{F}$ such that $\phi \left(P_0\right)=\phi \left(P_1\right)$ if and only if $P_0$ and $P_1$ are related by the moves. To obtain this result we first consider the subset ${\mathcal{F}}_0$ of $\mathcal{F}$ consisting of flows having all orbits homeomorphic to closed segments or points, constructing a combinatorial counterpart ${\mathcal{S}}_0$ for ${\mathcal{F}}_0$, and then adapting it to $\mathcal{F}$.