Representing smooth \(4\)-manifolds as loops in the pants complex (Q2168340)

Simplicial complexes associated to curves on a surface play a central role in \(2\)- and \(3\)-manifold topology, particularly in the study of mapping class groups and Heegaard splittings. Recently, \textit{R. Kirby} and \textit{A. Thompson} [Proc. Natl. Acad. Sci. USA 115, No. 43, 10857--10860 (2018; Zbl 1421.57031)] pushed these techniques into dimension four, assigning a loop in the cut complex to a trisected \(4\)-manifold. The aim of this paper is to, in some sense, reverse this. The authors show that every smooth, orientable, closed, connected \(4\)- manifold can be represented by a loop in the pants complex. In particular, given a loop in the pants complex, they show how to uniquely build a closed smooth \(4\)-manifold. They use this representation, together with the fact that the pants complex is simply connected, to provide an elementary proof that such \(4\)-manifolds are smoothly cobordant to a connected sum of complex projective planes, with either orientation. The authors also use their correspondence to gain insight into the structure of the pants complex. In particular, given a loop \(L\) in the pants complex, they define an invariant \(\sigma(L)\), which is the signature of the \(4\)-manifold associated to \(L\). This may be calculated using information only of the \(1\)-skeleton of the pants complex, but contains information about possible disks that this loop can bound. Namely, given a loop in the pants complex, \(L\), which bounds a disk, \(D\), they show that the signature of the \(4\)-manifold associated to \(L\) gives a lower bound on the number of triangles in \(D\).