Trisections of 3–manifold bundles over S1

Abstract

Let $X$ be a bundle over $S^1$ with fiber a $3$–manifold $M$ and with monodromy $\phi$. Gay and Kirby showed that if $\phi$ fixes a genus $g$ Heegaard splitting of $M$ then $X$ has a genus $6g+1$ trisection. Genus $3g+1$ trisections have been found in certain special cases, such as the case where $\phi$ is trivial, and it is known that trisections of genus lower than $3g+1$ cannot exist in general. We generalize these results to prove that there exists a trisection of genus $3g+1$ whenever $\phi$ fixes a genus $g$ Heegaard surface of $M$. This means that $\phi$ can be nontrivial, and can preserve or switch the two handlebodies of the Heegaard splitting. We additionally describe an algorithm to draw a diagram for such a trisection given a Heegaard diagram for $M$ and a description of $\phi$.