Representing smooth 4-manifolds as loops in the pants complex

Abstract

We show that every smooth, orientable, closed, connected 4-manifold can be represented by a loop in the pants complex. We 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 ${\coprod }_m\mathbb{C}{P}^2{\coprod }_n{\overline{\mathbb{C}P}}^2$. We also use this association to give information about the structure of the pants complex. Namely, given a loop in the pants complex, $L$, which bounds a disk, $D$, we show that the signature of the 4-manifold associated to $L$ gives a lower bound on the number of triangles in $D$.

23 Pages, 23 Figures