Representing smooth 4-manifolds as loops in the pants complex

TL;DR

This paper introduces a novel way to represent smooth, closed 4-manifolds as loops in the pants complex, providing new insights into their cobordism classes and the structure of the complex.

Contribution

It establishes a correspondence between 4-manifolds and loops in the pants complex, offering an elementary proof of cobordism and linking the manifold's signature to the complex's combinatorial properties.

Findings

Every smooth, orientable, closed 4-manifold can be represented as a loop in the pants complex.

The pants complex is simply connected, enabling elementary cobordism proofs.

The signature of the associated 4-manifold bounds the number of triangles in a bounding disk.

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 $\coprod_m \mathbb{C}P^2 \coprod_n \bar{\mathbb{C}P}^2$. 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 $D$.