TITLE: open book decompositions of $\Sigma\times S^1$
QUESTION [6 upvotes]: Let $\Sigma$ be a closed orientable surface. Is there a standard open book decomposition on the $3$-manifold $M=\Sigma\times S^1$?
If the binding is allowed to be empty in the definition of an open book decomposition, then this is obvious, since $M$ is the mapping torus of the identity of $\Sigma$. The literature is not clear on this, however.
If the binding must be non-empty, then since it is contained in every page, it seems that the obvious fibering $M\to S^1$ is never the fibering of an open book.

REPLY [5 votes]: I'm not sure about "preferred", but here is the simplest open book decomposition of $\Sigma_g \times S^1$ that I know:
Take the open book $(\Sigma_{g,1},\rm{id})$ with page the genus-$g$ surface with one boundary component and trivial monodromy. The result of page-framed surgery (ie $0$-surgery) on the binding component will give us $\Sigma_g \times S^1$. There are several ways to implement that surgery on the open book, but the simplest (AFAIK) is to do a positive stabilisation (along a boundary-parallel arc) to get $(\Sigma_{g,2},T_{\partial_1})$ and then add a negative Dehn twist to $\partial_2$ (which is the same knot as we started with, but now with page framing $-1$).
So $(\Sigma_{g,2},T_{\partial_1}T^{-1}_{\partial_2})$ will give you an open book for $\Sigma_g \times S^1$. Notice that when $g=0$, this recovers the standard open book $(S^1 \times [0,1],\rm{id})$ for $S^2 \times S^1$.