Amenable groups and smooth topology of 4-manifolds

TL;DR

This paper investigates how amenable groups influence the stabilization process in 4-manifold topology, revealing that the minimal stabilization needed becomes negligible relative to covers, through the lens of sweepout width and coarse geometry.

Contribution

It introduces the concept of sweepout width as a link between 4-dimensional topology and coarse geometry, and shows that for amenable fundamental groups, the stabilization number grows subextensively in covers.

Findings

Minimal stabilization n is subextensive in covers for amenable groups.

Sweepout width connects 4D topology with coarse geometric properties.

Stabilization complexity diminishes relative to cover size in amenable cases.

Abstract

A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-dimensional topology and coarse geometry.