Comparing 4–manifolds in the pants complex via trisections

Abstract

Given two smooth, oriented, closed 4-manifolds M 1 and M 2 , we construct two invariants, D P (M 1 , M 2 ) and D(M 1 , M 2 ), coming from distances in the pants complex and the dual curve complex respectively. To do this, we adapt work of Johnson on Heegaard splittings of 3-manifolds to the trisections of 4-manifolds introduced by Gay and Kirby. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of "nearby" manifolds. This naturally leads to various graphs of 4-manifolds coming from unbalanced trisections, and we briefly explore their properties. arXiv:1707.07108v1 [math.GT]

Trisections and spun four-manifolds

“…The theory of trisections was introduced by Gay and Kirby as a novel way of studying the smooth topology of 4-manifolds [10]. Since then, the theory has developed in a number of directions: Extensions of the theory to the settings of manifolds with boundary [6,7,8], knotted surfaces [22], algebraic objects [1], and higher dimensional manifolds [29] have been established; programs offering connections with singularity theory [9,10,11,12], and Dehn surgery [20,23], have been initiated; some classification results have been obtained [20,24]; interpretations of constructions and cutand-paste operation have been explored [13]; and new invariants have been proposed [15,18]. The purpose of this note is two-fold: motivate an extension of the classification program and generate a rich set of examples of manifolds with trisection diagrams that are simple enough to be amenable to study.…”