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 orie ntable handlebodies V-1, V-2, V-3 whose interiors are pairwise disjoint. If g(i) denotes the genus of partial derivative V-i and g(1) less than or equ al to g(2) less than or equal to g(3), then the tri-genus of M is the minim um triple (g(1), g(2), g(3)), ordered lexicographically. If the Bockstein o f the first Stiefel-Whitney class beta w(1)(M) = 0, then M has tri-genus (0 , 2g, g3), where g is the minimal genus of a 2-sided Stiefel Whitney surfac e of M. In this paper it is shown that, if beta w1(M) not equal 0, then M h as 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.