Mathematics Colloquia and Seminars

A trisection diagram $(\Sigma, \alpha, \beta, \gamma)$ is a 4-dimensional analog of a Heegaard diagram of a 3-manifold, where $\Sigma$ is a surface and $\alpha, \beta, \gamma \subset \Sigma$ are collections of disjoint, simple, closed curves such that each pair of collections can be made to look "standard." To each trisection diagram we can associate a unique smooth, compact, connected, oriented 4-manifold $X$; if $\Sigma$ is closed, then so is $X$. In this talk we will define trisections of $4$-manifolds and their diagrams, quickly restricting our attention to the case when $\partial X \neq \empty.$ We will show how to uniquely construct a $4$--manifold with boundary from a relative trisection diagram and, if time permits, we will show how such a diagram uniquely determines an open book decomposition of the bounding $3$--manifold. This is joint with David Gay and Juanita Pinzón.