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

Abstract

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.