Rigidity of maps between configuration spaces

Abstract

Let $n\ge 5$ and $m\ge 3$. Let $\hat{I}¦:{\mathrm{B}}_n\to {\mathrm{B}}_m$ be a homomorphism of braid groups. We prove that if the image of $\hat{I}¦$ is irreducible and not cyclic, then $m=n$ and $\hat{I}¦$ agrees with an automorphism modulo the center $Z({\mathrm{B}}_m)$. This resolves in the affirmative a conjecture of Chen, Kordek, and Margalit. It also provides a partial resolution to a problem on the K3 problem list. As a consequence, we prove that every holomorphic map ${\mathrm{UConf}}_n(\mathbb{C})\to {\mathrm{UConf}}_m(\mathbb{C})$ for $n\ge 5$ and $m\ge 3$ is affine equivalent to either a constant map or the identity map. This resolves a conjecture of Farb for $n\ne 4$.

63 pages, 18 figures