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.