Open Books $( \Sigma, \Phi) $ living in Lefschetz Fibrations over the disk $D^2$

Question

I have a question about open books and Lefschetz fibrations over the 2-disk $D^2$. Please let me set it up first, before going on.

Setup: Say we have a Lefschetz fibration $f: W^4 \rightarrow D^2 $ , i.e., over the 2-disk $D^2 $ . Then f restricted to the boundary of $W^4 $ is an open book $( \Sigma, \Phi)$ for $\partial W^4$, where $\Phi$ is the monodromy and $\Sigma$ is the fiber surface. .

Now, I want to go in the opposite direction and "embed" a given open book $(\Sigma, \Phi)$ in a Lefschetz fibration with singularities, say $x_1,..,x_n$ , i.e., given this open book, I want to find a Lefschetz fibration whose boundary is $( \Sigma, \Phi)$ . A necessary ( and I think sufficient) condition is that the total monodromy of the fibration, given as the composition of Dehn twists $D_1 \circ \ D_2 \circ ...\circ D_n$ about vanishing cycles $\gamma_i$ in the critical surfaces $f^{-1}(x_i)$ must agree with the monodromy $\Phi $ of the open book.

Question ( Phew): Is it always possible to do this, i.e., given $\Phi$ in MCG( $\Sigma $) , is it always possible to express $\Phi $ as the composition of Dehn twists about vanishing cycles?

Thanks in Advance.

Answer 1

Indeed, it is necessary and sufficient that the monodromy be a product of (positive) symplectic Dehn twists.

However, a general mapping class $\Phi\in\operatorname{MCG}(\Sigma)$ need not be expressible as a product of positive Dehn twists. There are probably more elementary ways of seeing this, but here is one: every contact three-manifold has an open book (by Giroux) and if its monodromy is expressible as a product of positive Dehn twists, then it is Stein fillable (by the existence of a Lefschetz fibration), however there certainly exist contact three-manifolds which are not Stein fillable (e.g. anything overtwisted).

You may find this note of Auroux interesting. He mainly discusses Lefschetz fibrations over $S^2$, but of course much of what he says is applicable over $D^2$ as well.