A characterization of virtually embedded subsurfaces in 3-manifolds

TL;DR

This paper introduces the spirality character to analyze when essentially immersed subsurfaces in 3-manifolds are virtually embedded, linking this property to taut foliations and providing new examples in non-geometric 3-manifolds.

Contribution

It generalizes Rubinstein and Wang's invariant by defining the spirality character, characterizes virtual embedding via aspirality, and presents examples in non-geometric 3-manifolds.

Findings

Virtually embedded subsurfaces correspond to aspiral almost fiber parts.

Non-geometric 3-manifolds can contain essentially immersed but not virtually embedded subsurfaces.

The spirality character generalizes previous invariants for subsurface analysis.

Abstract

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces.