On the Heegaard genus and tri-genus of non-orientable Seifert 3-manifolds (Q1807585)

The author computes several geometric invariants of Seifert fiber spaces. He does not assume that the Seifert fiber spaces are orientable. He considers the Heegaard genus, the tri-genus and the minimal genus representation of a given homology class. The Heegaard genus is a well-known invariant. It is a fact that any 3-manifold (orientable or not) may be expressed as the union of three orientable handlebodies. The genera of the three handlebodies ordered lexicographically is known as the tri-genus. The author computes what is essentially the minimal genus of a surface dual to the first Stiefel-Whitney class. He also computes the minimal genus of a fiber, for those Seifert manifolds which fiber over a circle. The rank of \(\pi_1(M)/\langle h\rangle\) is a good lower bound for the genus of a Seifert fiber space. This leads to the Heegaard genus of all but a few Seifert fiber spaces. There is a vertical torus dual to the first Stiefel-Whitney class of a Seifert fiber space. This is the basis for the computation of the Stiefel-Whitney genus. The fibers of a fibration over \(S^1\) are analyzed by projecting to the base orbifold of the Seifert manifold. He studies the vertical torus dual to the Stiefel-Whitney class in order to compute the tri-genus.