Tri-genera of Seifert Manifolds
by
Ví ctor 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 <= 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.

Date received: February 18, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caak-48.