Lefschetz fibrations with handlebody monodromy

Abstract

We study relatively minimal Lefschetz fibrations over closed oriented surfaces whose monodromy extends over a fixed 3--dimensional handlebody. If $X$ is the total space and $\ell$ is the number of critical points, we prove that $σ(X)=-\ell$. Such fibrations with $\ell>0$ exist precisely for fiber genus $g\geq3$ and base genus $h\geq2$, and one can take $\ell=1$ for every such pair. We also show that hyperelliptic handlebody monodromy forces the fibration to be a surface bundle and that every meridian twist, separating or nonseparating, has commutator length two in the handlebody group for $g\geq3$.