Generic flows on 3-manifolds

TL;DR

This paper introduces a combinatorial framework for classifying generic flows on 3-manifolds using polyhedral structures and moves, providing a new way to understand their topological behavior.

Contribution

It presents a novel combinatorial presentation of 3-dimensional generic flows, linking flows to finite polyhedra and moves, and extends from special to general flows.

Findings

Established a surjective correspondence between polyhedral structures and flows.

Constructed a combinatorial model for flows with simple orbit structures.

Extended the model to encompass all generic flows on 3-manifolds.

Abstract

We provide a combinatorial presentation of the set F of 3-dimensional generic flows, namely the set of pairs (M,v) with M a compact oriented 3-manifold and v a nowhere-zero vector field on M having generic behaviour along the boundary of M, with M viewed up to diffeomorphism and v up to homotopy on M fixed on the boundary. To do so we introduce a certain class S of finite 2-dimensional polyhedra with extra combinatorial structures, and some moves on S, exhibiting a surjection f:S->F such that f(P0)=f(P1) if and only if P0 and P1 are related by the moves. To obtain this result we first consider the subset F0 of F consisting of flows having all orbits homeomorphic to closed segments or points, constructing a combinatorial counterpart S0 for F0 and then adapting it to F.