Kyoto Journal of Mathematics
- Kyoto J. Math.
- Volume 55, Number 1 (2015), 143-167.
Generic flows on -manifolds
A -dimensional generic flow is a pair with a smooth compact oriented -manifold and a smooth nowhere-zero vector field on having generic behavior along ; 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 the quotient set. In this paper we provide a combinatorial presentation of . To do so we introduce a certain class of finite -dimensional polyhedra with extra combinatorial structures, and some moves on , exhibiting a surjection such that if and only if and are related by the moves. To obtain this result we first consider the subset of consisting of flows having all orbits homeomorphic to closed segments or points, constructing a combinatorial counterpart for , and then adapting it to .
Kyoto J. Math., Volume 55, Number 1 (2015), 143-167.
First available in Project Euclid: 13 March 2015
Permanent link to this document
Digital Object Identifier
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.)
Petronio, Carlo. Generic flows on $3$ -manifolds. Kyoto J. Math. 55 (2015), no. 1, 143--167. doi:10.1215/21562261-2848142. https://projecteuclid.org/euclid.kjm/1426252133