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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on open 3-manifolds supporting foliations by planes


Authors: Carlos Biasi and Carlos Maquera
Journal: Proc. Amer. Math. Soc. 140 (2012), 961-969
MSC (2010): Primary 37C85; Secondary 57R30
Published electronically: July 18, 2011
MathSciNet review: 2869080
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Abstract: We show that if $ N$, an open connected $ n$-manifold with finitely generated fundamental group, is $ C^{2}$ foliated by closed planes, then $ \pi_{1}(N)$ is a free group. This implies that if $ \pi_{1}(N)$ has an abelian subgroup of rank greater than one, then $ \mathcal{F}$ has at least a nonclosed leaf. Next, we show that if $ N$ is three dimensional with fundamental group abelian of rank greater than one, then $ N$ is homeomorphic to $ \mathbb{T}^2\times \mathbb{R}.$ Furthermore, in this case we give a complete description of the foliation.


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Additional Information

Carlos Biasi
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-São Carlos, Av. do Trabalhador São-Carlense 400, 13560-970 São Carlos, SP, Brazil
Email: biasi@icmc.usp.br

Carlos Maquera
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-São Carlos, Av. do Trabalhador São-Carlense 400, 13560-970 São Carlos, SP, Brazil
Email: cmaquera@icmc.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10960-2
PII: S 0002-9939(2011)10960-2
Keywords: Foliation by planes, open manifolds, incompressible torus, fundamental group, free group
Received by editor(s): June 15, 2009
Received by editor(s) in revised form: May 28, 2010, August 28, 2010, and December 18, 2010
Published electronically: July 18, 2011
Additional Notes: The first author was supported by FAPESP Grant 2008/57607-6.
The second author was supported by CNPq and FAPESP Grants 2008/57607-6 and 2009/17493-4.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2011 American Mathematical Society