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In link text, 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.

I wander whether it is still true for more general case. More precisely, whether the following is true:

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.

Moreover, are there some simlar results for higher dimensions?

Remark: we only consider closed orientable 3 manifolds.

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    $\begingroup$ If by "more general case" you simply mean an arbitrary 1-dimensional foliation of M, without the assumptions of a Smale flow nor any other type of dynamical assumptions, then the product flow on $(n+1)T^2 \times S^1$ is a counterexample. $\endgroup$
    – Lee Mosher
    Mar 6, 2012 at 4:16

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