Stiefel–Whitney surfaces and the tri-genus of non-orientable 3-manifolds

Every non-orientable 3-manifold M can be expressed as a union of three orientable handlebodies V ₁,V ₂,V ₃ whose interiors are pairwise disjoint. If g _i denotes the genus of ∂V _i and g ₃≤g ₂≤g ₃, then the tri-genus of M is the minimum triple (g ₁,g ₂,g ₃), ordered lexicographically. If the Bockstein of the first Stiefel–Whitney class βw ₁(M)=0, then M has tri-genus (0,2g,g ₃), where g is the minimal genus of a 2-sided Stiefel Whitney surface of M. In this paper it is shown that, if βw ₁(M)&\ne;0, then M has tri-genus (1,2g−1,g ₃), where g is the minimal genus of a (1-sided) Stiefel–Whitney surface. As an application the tri-genus of certain graph manifolds is computed. Received: 28 April 1999