Kyoto Journal of Mathematics

Generic flows on 3-manifolds

Carlo Petronio

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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.

Article information

Kyoto J. Math., Volume 55, Number 1 (2015), 143-167.

First available in Project Euclid: 13 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R25: Vector fields, frame fields
Secondary: 57M20: Two-dimensional complexes 57N10: Topology of general 3-manifolds [See also 57Mxx] 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)

three-manifolds generic flows branched spine


Petronio, Carlo. Generic flows on $3$ -manifolds. Kyoto J. Math. 55 (2015), no. 1, 143--167. doi:10.1215/21562261-2848142.

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