Generic flows on 3-manifolds

A 3-dimensional emph{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 behaviour 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 $calF$ the quotient set. In this paper we provide a combinatorial presentation of $calF$. To do so we introduce a certain class $calS$ of finite 2-dimensional polyhedra with extra combinatorial structures, and some moves on $calS$, exhibiting a surjection $arphi:calS ocalF$ such that $arphi(P_0)=arphi(P_1)$ if and only if $P_0$ and $P_1$ are related by the moves. To obtain this result we first consider the subset $calF_0$ of $calF$ consisting of flows having all orbits homeomorphic to closed segments or points, constructing a combinatorial counterpart $calS_0$ for $calF_0$ and then adapting it to $calF$. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM.