Realization problems for diffeomorphism groups

TL;DR

This paper surveys realization problems for diffeomorphism groups, exploring when group surjections admit sections, highlighting open questions, recent results, and techniques across various mathematical contexts.

Contribution

It provides a comprehensive overview of recent developments, open problems, and new illustrative proofs related to realization problems in diffeomorphism groups.

Findings

Identification of key open problems in realization theory

Presentation of new proofs illustrating core techniques

Drawing parallels between different areas of mathematical realization problems

Abstract

We discuss recent results and open questions on the broad theme of (Nielsen) realization problems. Beyond realizing subgroups of mapping class groups, there are many other natural instances where one can ask if a surjection from a group of diffeomorphisms of a manifold to another group admits a section over particular subgroups. This survey includes many open problems, and some short proofs of new results that are illustrative of key techniques; drawing attention to parallels between problems arising in different areas.