Diagrams for relative trisections (Q1703650)

The secondly named author and \textit{R. Kirby} [Geom. Topol. 20, No. 6, 3097--3132 (2016; Zbl 1372.57033)] defined and proved the existence of trisections of mainly closed smooth oriented 4-manifolds, and also referred to the case of compact 4-manifolds with connected boundary. Here, \textit{a trisection} of a closed smooth oriented 4-manifold \(X\), or, more precisely, a \((g,k)\)-trisection, is a decomposition of \(X\) as a union of three mutually diffeomorphic 1-handlebodies \(X_i \cong \natural^k (S^1\times B^3)\), the boundary connected sum of \(k\) copies of the 4-dimensional solid tori, such that their intersection \(X_1 \cap X_2\cap X_3\) is a closed connected oriented surface \(\Sigma\) of genus \(g\) and gives a Heegaard splitting of the boundary \(\partial X_i\) of \(X_i\) for each \(i=1,2,3\). To describe trisections, a \textit{trisection diagram} was defined as \((\Sigma; \alpha,\beta,\gamma)\), where \(\alpha\) (and \(\beta, \gamma\)) is a collection of disjoint \(g\) simple closed curves in the surface \(\Sigma\). In the paper under review, the relative case, i.e., trisections of 4-manifolds with connected boundary, is considered. In this case, the surface \(\Sigma\) can have boundary components. The structure gives an open book structure on the boundary 3-manifold \(\partial X\). Let \(p, b\) denote the genus of the page (the fiber surface) and the number of components of the binding (the fibered link) of the open book structure, respectively. The boundary \(X_i\cap X_{i +1}\) is a compression body starting from a genus \(g\) surface \(\Sigma\) down to a genus \(p\) one, rather than a genus \(g\) handlebody in the closed case above. These are called \((g,k; p,b)\)-trisections. The parameter \(k\) has to satisfy that \[ 2p-b-1\leq k \leq g + p + b -1. \] The authors ``establish a correspondence between trisections of smooth, compact, oriented 4-manifolds with connected boundary and diagrams describing these trisected manifolds.'' These ``should be thought of as the 4-dimensional analog of a sutured Heegaard diagram for a sutured 3-manifold''. Some examples are demonstrated: (1) the total space of a disk bundle over the 2-sphere, (2) Lefschetz fibrations, (3) plumbed manifolds for weighted trees, (4) product spaces of a knot exterior and the circle.