# Dehn's Lemma for Immersed Loops

## TL;DR

This paper proves a version of Dehn's Lemma for immersed loops on the boundary of 3-manifolds, showing they can be isotoped to bound disks in the manifold.

## Contribution

It extends Dehn's Lemma to immersed loops on boundary surfaces, providing a new isotopy technique for null-homotopic immersed curves.

## Key findings

- Immersed boundary loops can be displaced to bound disks in the manifold.
- The result applies to generic immersed curves, not just embedded ones.
- Provides a method to simplify immersed loops in 3-manifold boundaries.

## Abstract

Suppose $\delta$ is a generic immersed closed curve in the boundary of a 3-manifold M and $\delta$ is null-homotopic in M. Then $\delta$ can be displaced by a height function in a collar of the boundary so that the resulting simple closed curve in the collar bounds a disk in M.