Lifted Heegaard Surfaces and Virtually Haken Manifolds

TL;DR

This paper constructs infinitely many hyperbolic 3-manifolds with genus three Heegaard surfaces that, in some finite covers, can be compressed into incompressible surfaces, advancing understanding of their topological properties.

Contribution

It introduces new examples of non-Haken hyperbolic 3-manifolds with specific Heegaard surface behaviors, extending previous results in the field.

Findings

Existence of infinitely many such manifolds

Finite covers with reducible but compressible Heegaard surfaces

Extension of prior theoretical results

Abstract

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [CG] and extends [MMZ].