II CONGRESO IBEROAMERICANO DE TOPOLOGÍA Y SUS APLICACIONES March 20-22, 1997 Morelia, México

Tri-genera of Seifert Manifolds by Víctor Núñez (Centro de Investigación en Matemáticas )

It was recently shown by Gómez, González, and Hoste that if M is a non-orientable closed 3-manifold, then M can be represented as the union of three orientable handlebodies with pairwise disjoint interiors, M=H₁\cup H₂\cup H₃ . The manifold M is said to have tri-genus (g₁,g₂,g₃) if it can be represented as the union of three orientable handlebodies H₁,H₂,H₃ as above, with genera g₁,g₂,g₃ respectively, and such that (g₁,g₂,g₃) is the minimal possible triple among all such triples, with respect to the lexicographic ordering.

If M is a non-orientable Seifert manifold such that M cannot be expressed as an S¹ -bundle with fiber a closed orientable surface, then M has tri-genus of the form either (0,2,g₃) , or (1,1,g₃) with g₃\leq h(M) , where h(M) denotes the Heegaard genus of M .

But if M is a non-orientable Seifert manifold which does admit an S¹ -bundle structure with fiber a closed orientable surface, then M has tri-genus of the form (0,g,g) , with g a very big number. For instance, if M_\alpha=(NnI,3|(1,0),(2,1),(\alpha,1)) , with \alpha an odd positive integer, then M_\alpha has tri-genus (0,g_\alpha,g_\alpha) , with g_\alpha=10\alpha-2 ; the sequence {g_\alpha} goes to infinity if we let \alpha vary in the set of odd numbers; but the Heegaard genus, h(M_\alpha)=4 , for any choice of \alpha .

Althought one could expect a relation between the Heegaard genus of a non-orientable manifold and the number g₃ in the tri-genus, one sees that a relation, if any, cannot be simple.