Distinguishing 4-dimensional geometries via profinite completions

TL;DR

This paper investigates how the geometric structure of 4-dimensional manifolds can be identified from the profinite completion of their fundamental groups, revealing most geometries except for a few special cases.

Contribution

It demonstrates that the geometry of most closed orientable 4-manifolds can be distinguished via their profinite fundamental groups, including Seifert fibred manifolds.

Findings

Most 4D geometries are detectable from profinite completions.

Certain geometries like , , and  are exceptions.

Profinite completion can determine whether a Seifert fibred 4-manifold is geometric.

Abstract

It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a closed orientable 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except when the geometry is $ \mathbb{H}^{4}$, $\mathbb{H}^{2}_{\mathbb{C}}$ or $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some 4-dimensional manifolds with Seifert fibred structures are indeed geometric. For a closed orientable Seifert fibred 4-manifold $M$, we show that whether $M$ is geometric could be detected by the profinite completion of its fundamental group.