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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Presentations of $ 3$-manifolds arising from vector fields


Author: Peter Percell
Journal: Trans. Amer. Math. Soc. 221 (1976), 361-377
MSC: Primary 57D25; Secondary 58C25, 57A10
MathSciNet review: 0407857
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Abstract: A method is given for constructing a smooth, closed, orientable 3-manifold from the information contained in a combinatorial object called an abstract intersection sequence. An abstract intersection sequence of length n is just a cyclic ordering of the set $ \{ \pm 1, \ldots , \pm n\} $ plus a map $ \nu :\{ 1, \ldots ,n\} \to \{ \pm 1\} $. It is shown that up to diffeomorphism every closed, connected, orientable 3-manifold can be constructed by the method. This is proved by showing that compact, connected, orientable 3-manifolds with boundary the 2-sphere admit vector fields of a certain type. The intersection sequences arise as descriptions of the vector fields.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0407857-X
PII: S 0002-9947(1976)0407857-X
Keywords: Presentation, 3-manifold, intersection sequence, generic transient vector field, branched manifold
Article copyright: © Copyright 1976 American Mathematical Society