Essential disks and semi-essential surfaces in 3-manifolds

TL;DR

This paper develops a new normal surface theory to analyze essential disks and semi-essential surfaces in 3-manifolds with compressible boundary, especially handlebodies, aiming to understand their limits and invariants.

Contribution

It introduces a novel normal surface theory applicable to essential disks and semi-essential surfaces in 3-manifolds with compressible boundary, facilitating the study of their limits and automorphism invariants.

Findings

New normal surface theory for handlebodies and compression bodies

Potential to describe limits of essential disks in 3-manifolds

Application to automorphisms and invariant laminations

Abstract

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory. We hope the normal surface theory will be used in other papers to describe objects representing limits of essential disks in a handlebody or a 3-manifold with compressible boundary. For certain automorphisms of handlebodies, these disk limits should serve as invariant objects akin to laminations and analogous to the invariant laminations for pseudo-Anosov automorphisms of surfaces.