Incompressibility and normal minimal surfaces

TL;DR

This paper establishes a new criterion for incompressibility of surfaces in 3-manifolds based on their minimal PL-area normal representatives across all triangulations, linking geometric and topological properties.

Contribution

It introduces a procedure for refining triangulations that preserves normal surfaces and proves the converse relationship between minimal PL-area normal surfaces and incompressibility.

Findings

Incompressible surfaces are isotopic to minimal PL-area normal surfaces in all triangulations.

A new characterization of incompressibility based on minimal PL-area normal surfaces.

A triangulation refinement method that preserves normal surfaces without creating new ones.

Abstract

In this paper we describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces. It is known that an incompressible surface $F$ in a triangulated 3-manifold $M$ is isotopic to a normal surface that is of minimal PL-area in the isotopy class of $F$. Using the above scaling refinement we prove the converse. If $F$ is a surface in a closed 3-manifold $M$ such that for any triangulation $\tau$ of $M$, $F$ is isotopic to a $\tau$-normal surface $F(\tau)$ that is of minimal PL-area in its isotopy class, then we show that $F$ is incompressible.