# Mapping classes as Lefschetz fibrations over surfaces with positive genus

Let $$\Sigma_{g,r}$$ be the surface of genus $$g$$ and $$r$$ boundary components. It is known that, from a positive factorization of a mapping class $$\phi$$ in the mapping class group $$MCG(\Sigma_{g,r}, \partial \Sigma_{g,r})$$, one can construct a Lefschetz fibration (with corners) over the disc $$D^2$$ so that there are as many critical points/values as Dehn twists take place in the factorization and such that the restriction of the Lefschetz fibration to $$\partial D^2$$ is a surface bundle with monodromy $$\phi$$. Conversely, from any such fibration one can recover a mapping class factorized by right-handed Dehn twists.

My question is:

Is it known for which mapping classes $$\phi \in MCG(\Sigma_{g,r}, \partial \Sigma_{g,r})$$ there exists $$h \in \mathbb{N}$$ and a Lefschetz fibration over the surface $$\Sigma_{h,1}$$ such that the monodromy of the restriction of the Lefschetz fibration to $$\partial \Sigma_{h,1}$$ is $$\phi$$?