Sectional surface of dimension 1 foliation on 3 manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T14:59:45Z http://mathoverflow.net/feeds/question/80201 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80201/sectional-surface-of-dimension-1-foliation-on-3-manifolds Sectional surface of dimension 1 foliation on 3 manifolds Bin Yu 2011-11-06T09:03:06Z 2011-11-06T15:23:51Z <p>In <a href="http://arxiv.org/PS_cache/arxiv/pdf/1007/1007.3333v1.pdf" rel="nofollow">link text</a>, I have considered a kind of sectional surfaces (i.e., regular level sets) about a spectial type of nonsingular flows (i.e., Smale flow) on three manifolds. I obtained that a 3-manifold $M$ admits a nonsingular Smale ﬂow which has a regular level set homeomorphic to $(n + 1)T^2$ if and only if $M$ admit at least $n$ $S^1 \times S^2$ factors. Here $(n + 1)T^2$ is homeomorphic to the connected sum of $n+1$ tori.</p> <p>I wander whether it is still true for more general case. More precisely, whether the following is true:</p> <p><strong>A 3-manifold $M$ admits a dimension 1 foliation which has a sectional surface homeomorphic to $(n + 1)T^2$ if and only if $M$ admit at least $n$ $S^1 \times S^2$ factors.</strong> </p> <p>Moreover, are there some simlar results for higher dimensions?</p> <p>Remark: we only consider closed orientable 3 manifolds.</p>