The Splitting of Generalisations of the Fadell-Neuwirth short exact sequence

Abstract

We study some generalisations to mixed braid groups of the Fadell-Neuwirth short exact sequence and the possible splitting of this sequence. In certain cases, we determine conditions under which the projection from the mixed braid group $B_{n_1,\dots ,n_k}(M)$ to $B_{n_1,\dots ,n_{k-q}}(M)$ admits a section, where $M$ is either the torus or the Klein bottle, $n_1,\dots ,n_k,q\in \mathbb{N}$, and $1\le q\le k-1$. For $k\ge 2$ and $q=k-1$, we show that this projection admits a section if and only if $n_1$ divides $n_i$ for all $i=2,\dots ,k$. We present some partial conclusions in the case $k\ge 3$ and $q=1$. To obtain our results, we compute and make use of suitable mixed braid groups of $M$, as well as certain key quotients that play a central rôle in our analysis.