On the Heegaard genus and tri-genus of non-orientable Seifert 3-manifolds

We calculate the Heegaard genus, h(M), of the closed non-orientable Seifert manifolds. -manifold M admits a decomposition M=H1∪H2∪H3 into three orientable handlebodies of genera g1,g2,g3, respectively, and g1≤g2≤g3, we call the triple (g1,g2,g3) the tri-genus of M if (g1,g2,g3) is minimal among all such triples ordered lexicographically. pute the tri-genus (g1,g2,g3) of all non-orientable Seifert manifolds M which admit an S1-bundle structure with fiber an orientable surface. In this case the number g3 is much bigger than h(M) for a fixed M. ain also that h(M) is an upper bound for the number g3 in case M is a non-orientable Seifert manifold which does not admit an S1-bundle structure. that, although one could expect a relation between the number g3 and the Heegaard genus, this relation, if any, can not be simple.