Gem-induced trisections of compact PL $4$-manifolds

The idea of studying trisections of closed smooth $4$-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to colored triangulations associated to the so called simple crystallizations of standard simply-connected $4$-manifolds. The present paper performs a generalization of these ideas along two different directions: first, we take in consideration also compact PL $4$-manifolds with connected boundary, introducing a possible extension of trisections to the boundary case; then, we analyze the trisections induced not only by simple crystallizations, but by any 5-colored graph encoding a simply-connected $4$-manifold. This extended notion is referred to as gem-induced trisection, and gives rise to the G-trisection genus, generalizing the well-known trisection genus. Both in the closed and boundary case, we give conditions on a 5-colored graph which ensure one of its gem-induced trisections - if any - to realize the G-trisection genus, and prove how to determine it directly from the graph itself. Moreover, the existence of gem-induced trisections and an estimation of the G-trisection genus via surgery description is obtained, for each compact simply-connected PL 4-manifold admitting a handle decomposition lacking in 1-handles and 3-handles. As a consequence, we prove that the G-trisection genus equals $1$ for all $\mathbb D^2$-bundles of $\mathbb S^2$, and hence it is not finite-to-one.