The Splitting of Generalisations of the Fadell-Neuwirth short exact sequence
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arXiv
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| Format: | Preprint |
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2025
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| author | Gonçalves, Daciberg Lima Guaschi, John Pereiro, Carolina de Miranda e |
| author_facet | Gonçalves, Daciberg Lima Guaschi, John Pereiro, Carolina de Miranda e |
| contents | 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 $B_{n_{1},\ldots,n_{k}}(M)$ to $B_{n_{1},\ldots, n_{k-q}}(M)$ admits a section, where $M$ is either the torus or the Klein bottle, $n_{1}, \ldots, n_{k},q \in \mathbb{N}$, and $1\leq q \leq k-1$. For $k\geq 2$ and $q=k-1$, we show that this projection admits a section if and only if $n_{1}$ divides $n_{i}$ for all $i=2,\ldots, k$. We present some partial conclusions in the case $k\geq 3$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_02707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Splitting of Generalisations of the Fadell-Neuwirth short exact sequence Gonçalves, Daciberg Lima Guaschi, John Pereiro, Carolina de Miranda e Geometric Topology Algebraic Topology 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 $B_{n_{1},\ldots,n_{k}}(M)$ to $B_{n_{1},\ldots, n_{k-q}}(M)$ admits a section, where $M$ is either the torus or the Klein bottle, $n_{1}, \ldots, n_{k},q \in \mathbb{N}$, and $1\leq q \leq k-1$. For $k\geq 2$ and $q=k-1$, we show that this projection admits a section if and only if $n_{1}$ divides $n_{i}$ for all $i=2,\ldots, k$. We present some partial conclusions in the case $k\geq 3$ 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. |
| title | The Splitting of Generalisations of the Fadell-Neuwirth short exact sequence |
| topic | Geometric Topology Algebraic Topology |
| url | https://arxiv.org/abs/2509.02707 |