On the non-realizability of braid groups by diffeomorphisms

TL;DR

This paper proves that for large enough n, the surface braid group cannot be realized by diffeomorphisms on the surface, extending known non-lifting results and applying to various embedding and motion groups.

Contribution

It establishes non-realizability of surface braid groups by diffeomorphisms for large n and extends the results to codimension-2 embedding spaces and motion groups.

Findings

No lift of $B_n(S)$ to $ ext{Diff}(S,n)$ exists for large n.

Provides a new proof of Morita's non-lifting theorem.

Extends techniques to spherical and string motion groups.

Abstract

For every compact surface $S$ of finite type (possibly with boundary components but without punctures), we show that when $n$ is sufficiently large there is no lift $\sigma$ of the surface braid group $B_n(S)$ to $\operatorname{Diff}(S,n)$, the group of $C^1$ diffeomorphisms preserving $n$ marked points and restricting to the identity on the boundary. Our methods are applied to give a new proof of Morita's non-lifting theorem in the best possible range. These techniques extend to the more general setting of spaces of codimension-$2$ embeddings, and we obtain corresponding results for spherical motion groups, including the string motion group.