Realization of the mapping class group of handlebody by diffeomorphisms

TL;DR

This paper proves that for handlebodies of genus at least 6, the natural surjection from the diffeomorphism group to the mapping class group does not admit a section, highlighting a fundamental limitation in their relationship.

Contribution

It demonstrates the non-existence of a section for the natural surjection in handlebodies of genus at least 6, revealing new insights into their topological structure.

Findings

No section exists for the surjection when genus ≥ 6

The result clarifies the relationship between diffeomorphisms and mapping class groups

Advances understanding of handlebody automorphisms

Abstract

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms of it to the mapping class group of it has no section when g is at least 6.