Incompressible surfaces in handlebodies and boundary reducible 3-manifolds

TL;DR

This paper investigates the existence and properties of incompressible surface embeddings within genus two handlebodies and certain 3-manifolds, expanding understanding of their topological structures.

Contribution

It proves that any compact surface can be incompressibly embedded into a genus two handlebody, regardless of orientability, and explores embeddings in related 3-manifolds.

Findings

Every compact surface can be embedded incompressibly into a genus two handlebody.

In the orientable case, embeddings can be separating or non-separating.

The study extends to embeddings in 3-manifolds with boundary components of genus ≥ 2.

Abstract

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.