Lifting braids over ${\mathbb{RP}}^2$ and non-orientable Lefschetz fibrations

Abstract

We show that, given a closed connected non-orientable surface $F$ and a simple branched cover $p:F\to {\mathbb{RP}}^2$ of degree $d\ge 5$, every self-diffeomorphism of $F$ is isotopic to the lift of a self-diffeomorphism of ${\mathbb{RP}}^2$ with respect to $p$. As an application, we show that every Lefschetz fibration over a connected surface $B$ with $\partial B\ne \varnothing$ and closed non-orientable regular fiber can be expressed as the composition of a simple cover of $B\times {\mathbb{RP}}^2$ branched over a surface braided over $B$ and projection onto the $B$-factor.