Rigidity of maps between configuration spaces

Let $n\geq5$ and $m\geq3$.

Let $\Phi\colon\mathrm{B}_n\to\mathrm{B}_m$ be a homomorphism of braid groups.

We prove that if the image of $\Phi$ is irreducible and not cyclic, then $m=n$ and $\Phi$ 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 $n\geq5$ and $m\geq3$ is affine equivalent to either a constant map or the identity map.

This resolves a conjecture of Farb for $n\neq4$.