Estimating trisection genus via gem theory

Abstract

Gems are a particular type of edge-colored graphs, dual to colored triangulations, which represent compact PL-manifolds of arbitrary dimension, both in the closed and boundary case. In the present paper, gem theory is used to approach trisections of PL 4-manifolds, so as to prove that: - the graph-defined invariant regular genus is an upper bound for the trisection genus of each closed 4-manifold; - a trisection diagram can be directly obtained from any gem of a closed 4-manifold. Moreover, suitable extensions of the above results are presented for compact 4-manifolds with connected boundary.

20 pages, 11 figures. With respect to the previous version: new Remark 6; changes in Propositions 9 and 11 in order to clarify the choices for the systems of curves; new examples of trisection diagrams induced by gems