Open Access
April 2015 Generic flows on 3-manifolds
Carlo Petronio
Kyoto J. Math. 55(1): 143-167 (April 2015). DOI: 10.1215/21562261-2848142


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 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 F the quotient set. In this paper we provide a combinatorial presentation of F. 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 φ:SF such that φ(P0)=φ(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.


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Carlo Petronio. "Generic flows on 3-manifolds." Kyoto J. Math. 55 (1) 143 - 167, April 2015.


Published: April 2015
First available in Project Euclid: 13 March 2015

zbMATH: 1323.57015
MathSciNet: MR3323530
Digital Object Identifier: 10.1215/21562261-2848142

Primary: 57R25
Secondary: 57M20 , 57N10 , 57R15

Keywords: branched spine , generic flows , Three-manifolds

Rights: Copyright © 2015 Kyoto University

Vol.55 • No. 1 • April 2015
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