Homomorphisms of commutator subgroups of braid groups with small number of strings

TL;DR

This paper characterizes all endomorphisms and homomorphisms involving the commutator subgroups of braid groups for small numbers of strings, filling gaps in the existing classification for these cases.

Contribution

It provides a complete description of endomorphisms of $B_n$ and $B'_n$, and homomorphisms from $B'_n$ to $B_n$ for small $n$, using novel approaches.

Findings

Complete classification for small n cases

Different methods used for n=4 and n≥5

Results extend known classifications to small n

Abstract

For any $n$, we describe all endomorphisms of the braid group $B_n$ and of its commutator subgroup $B'_n$, as well as all homomorphisms $B'_n\to B_n$. These results are new only for small $n$ because endomorphisms of $B_n$ are already described by Castel for $n\ge 6$, and homomorphisms $B'_n\to B_n$ and endomorphisms of $B'_n$ are already described by Kordek and Margalit for $n\ge 7$. We use very different approaches for $n=4$ and for $n\ge 5$.