Isotopy Uniqueness of Self-diffeomorphism of Handlebodies

TL;DR

This paper provides a rigorous proof that the mapping class group of a handlebody embeds into that of its boundary surface, establishing that self-homeomorphisms isotopic on the boundary are ambient isotopic to the identity.

Contribution

It offers a complete proof of the isotopy uniqueness of self-diffeomorphisms of handlebodies, clarifying a fundamental aspect of their mapping class groups.

Findings

Embedding of $MCG(V_g)$ into $MCG( ext{boundary})$ is established.

Self-homeomorphisms isotopic on the boundary are ambient isotopic to the identity.

Provides a rigorous proof filling a gap in the literature.

Abstract

The mapping class group $MCG(\Sigma_g)$ of a surface of genus $g$ has a long-history in topology and group theory. More recently, the mapping class group $MCG(V_g)$ of a handlebody $V_g$ of genus $g$ has become an interesting topic in the study of $3$ manifolds, largely thanks to Heegaard splitting. While $MCG(V_g)$ can be regarded naturally as a sub group of $MCG(\Sigma_g)$, we could not find any complete proof of this fundamental theorem. It is the purpose of this paper that we give a rigorous proof of embedding of $MCG(V_g)$ into $MCG(V_g)$. The key step is: Any self-homeomorphism $f$ of handlebody $V_g$ of genus $g$ is ambient isotopic to identity if the restriction $f|_{\partial V_g}$ is isotopic to identity.