Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies

Every closed nanorientable 3-manifold M can be obtained as the union of three orientable handlebodies V₁, V₂, V₃ whose interiors are pairwise disjoint. If g_i denotes the genus of V_i, g₁g₂g₃, we say that M has tri-genus (g₁, g₂, g₃), if in terms of lexicographical ordering, the triple (g₁, g₂, g₃) is minimal among all such decompositions of M into orientable handlebodies. We relate the tri-genus of M to the genus of a surface that represents the dual of the first Stiefel-Whitney class of M. This is used to determine g₁ and g₂.