Essential curves in handlebodies and topological contractions

TL;DR

This paper constructs smooth topological contractions in 3D handlebodies of genus at least 2, demonstrating the existence of essential images, and provides a criterion for simple curves to be essential in handlebodies.

Contribution

It introduces a criterion for essentiality of simple curves in handlebodies and constructs specific topological contractions with essential images in higher genus handlebodies.

Findings

Existence of smooth topological contractions with essential images in genus ≥ 2 handlebodies

A new criterion for a simple curve to be essential in a handlebody

Construction of contractions demonstrating essentiality in 3D handlebodies

Abstract

If $X$ is a compact set, a {\it topological contraction} is a self-embedding $f$ such that the intersection of the successive images $f^k(X)$, $k>0$, consists of one point. In dimension 3, we prove that there are smooth topological contractions of the handlebodies of genus $\geq 2$ whose image is essential. Our proof is based on an easy criterion for a simple curve to be essential in a handlebody.