On ${\mathbb{Z}}_2$-Thurston norms and pseudo-horizontal surfaces in orientable Seifert $3$-manifolds

Abstract

We describe a general method to compute the ${\mathbb{Z}}_2$-Thurston norm for every ${\mathbb{Z}}_2$-homology class in an orientable Seifert manifold with orientable orbit surface. Our main tools are pseudo-horizontal surfaces. We give a necessary and sufficient criterion for the existence of pseudo-horizontal surfaces, calculate the non-orientable genera for such surfaces, and detect their ${\mathbb{Z}}_2$-homology classes. We then describe an algorithm to calculate the ${\mathbb{Z}}_2$-Thurston norm of each ${\mathbb{Z}}_2$-homology classes. We also present several interesting examples.

24 pages, 3 figures