Normal tori in ♯n(S2×S1)♯n(S2×S1)

The fundamental group of M=♯n(S2×S1)M=♯n(S2×S1) is FnFn, the free group with n   generators. There is a 1–1 correspondence between the equivalence classes of ZZ-splittings of FnFn and homotopy classes of imbedded essential tori in M. We define and prove a local notion of minimal intersection of a torus with respect to a maximal sphere system in M, which is inspired by Hatcherʼs work (Hatcher, 1995 [7]) on 2-spheres in the same manifold.