Detecting embedded surfaces using finite quotients

TL;DR

This paper establishes conditions under which certain 3-manifolds with equivalent profinite fundamental groups share Haken properties, and provides bounds on Betti numbers for covers of aspherical homology spheres.

Contribution

It introduces new criteria linking profinite fundamental groups to Haken properties and Betti number bounds in 3-manifold topology.

Findings

Conditions for Haken property transfer via profinite groups

Lower bound of four on Betti numbers for covers of aspherical homology spheres

Sharpness of the Betti number lower bound

Abstract

We give conditions on a Haken hyperbolic rational homology three sphere that imply that any other 3-manifold with profinitely equivalent fundamental group must also be Haken. In the appendix, we show that a regular finite-sheeted cover of an aspherical integral homology three-sphere with positive first Betti number must have first Betti number at least four. We also show that this lower bound is sharp.