Orientation double covers of non-orientable Lefschetz fibrations

TL;DR

This work links non-orientable Lefschetz fibrations to orientable ones via the orientation double cover. It proves that composing a non-orientable fibration with the standard double cover yields an achiral Lefschetz fibration on the cover, with genus reduced by one and preserved counts of positive and negative critical points, and it shows how isomorphism classes correspond to Hurwitz-type equivalence classes of monodromy factorizations through a lift map $\eta$. The authors establish explicit monodromy transformations that preserve isomorphism classes and prove a classification theorem for genus-$g$ non-orientable fibrations ( $g\ge 3$ ) in terms of these Hurwitz-type moves and an orientation-preserving base diffeomorphism. Collectively, the results provide a combinatorial framework for constructing and classifying non-orientable Lefschetz fibrations via their orientable lifts and monodromy data.

Abstract

In this paper, we prove that the composition of the standard orientation double covering map and a non-orientable Lefschetz fibration is an achiral Lefschetz fibration and specify a monodromy factorization of this composition. As an application of these results, we also give three transformations with respect to monodromy factorizations of non-orientable Lefschetz fibrations which do not change their isomorphism classes via the similar result to orientable Lefschetz fibrations.

Figures (3)

• Figure 1: Cross caps and two-sided simple closed curves on $N_{2g+2}$.
• Figure 2: Vanishimg cycles of a genus-$g$ non-orientable Lefschetz fibration $f$ and the genus-($g-1$) achiral Lefschetz fibration $f \circ \pi$, where $g$ is odd.
• Figure 3: A right handed Dehn twist $T_c$ about $c$.

Theorems & Definitions (22)

• Proposition 1.1
• Theorem 1.2
• Definition 2.1
• Example 2.2
• Example 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.6
• Remark 2.7
• proof : Proof of Proposition \ref{['prop1.1']}.