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The fundamental group of $M = \sharp_n (S^2\times S^1)$ is $F_n$, the freegroup with $n$ generators. There is a 1-1 correspondence between theequivalence classes of $\mathbb{Z}$-- splittings of $F_n$ and homotopy classesof embedded essential tori in $M$. We define and prove a local notion ofminimal intersection of a torus with respect to a maximal sphere system in $M$,which generalizes Hatcher's work \cite{H1} on 2-spheres in the same manifold.

Normal Tori in $\sharp_n (S^2\times S^1)$