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<title>Spherical braid group</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Spherical braid group</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>spherical braid group</b> or <b>Hurwitz braid group</b> is a <a href="Braid_group" title="Braid group">braid group</a> on <span class="texhtml mvar" style="font-style:italic;">n</span> strands. In comparison with the usual braid group, it has an additional group relation that comes from the strands being on the <a href="Sphere" title="Sphere">sphere</a>. The group also has relations to the <a href="Inverse_Galois_problem" title="Inverse Galois problem">inverse Galois problem</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>The spherical braid group on <span class="texhtml mvar" style="font-style:italic;">n</span> strands, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SB_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle SB_{n}}</annotation>

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</math></span><img src="./97870136ef93fdef8fb23dff032555d58afa2594.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.482ex; height:2.509ex;" alt="{\displaystyle SB_{n}}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}(S^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle B_{n}(S^{2})}</annotation>

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</math></span><img src="./700075f6d78ef9b0e7d20cb24b8195f3d4fa74de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.368ex; height:3.176ex;" alt="{\displaystyle B_{n}(S^{2})}" loading="lazy"></span>, is defined as the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of the <a href="Configuration_space_(mathematics)" title="Configuration space (mathematics)">configuration space</a> of the sphere:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle B_{n}(S^{2})=\pi _{1}(\mathrm {Conf} _{n}(S^{2})).}</annotation>

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The spherical braid group has a <a href="Group_presentation" class="mw-redirect" title="Group presentation">presentation</a> in terms of generators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}}</annotation>

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</math></span><img src="./ee74a8ea3e942e3b99768b19c082593527ddbae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.622ex; height:2.009ex;" alt="{\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n-1}}" loading="lazy"></span> with the following relations:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}</annotation>

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</math></span><img src="./708c02826f49f8a68897517be43947d257972333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.827ex; height:2.343ex;" alt="{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |i-j|\geq 2}">

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<annotation encoding="application/x-tex">{\displaystyle |i-j|\geq 2}</annotation>

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</math></span><img src="./805c831797c6565cd81a477b65d45c21bafff89c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.156ex; height:2.843ex;" alt="{\displaystyle |i-j|\geq 2}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}}</annotation>

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</math></span><img src="./f4a2394fef2bf43989a132f985bb87e089874cd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.162ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n-2}">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq n-2}</annotation>

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</math></span><img src="./a67c59789ffbf44e14952a663853c6c28e1f71de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.559ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq n-2}" loading="lazy"></span> (the <a href="Yang%E2%80%93Baxter_equation" title="Yang–Baxter equation">Yang–Baxter equation</a>)</li>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{1}\sigma _{2}\cdots \sigma _{n-1}\sigma _{n-1}\sigma _{n-2}\cdots \sigma _{1}=1}</annotation>

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</math></span><img src="./c71fa188c5d3cf6e7d80c40f46168e00604142a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.34ex; height:2.509ex;" alt="{\displaystyle \sigma _{1}\sigma _{2}\cdots \sigma _{n-1}\sigma _{n-1}\sigma _{n-2}\cdots \sigma _{1}=1}" loading="lazy"></span></li></ul>

<p>The last relation distinguishes the group from the usual braid group.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFIhara2007" class="citation cs2">Ihara, Yasutaka (2007), Cartier, Pierre; Katz, Nicholas M.; Manin, Yuri I.; Illusie, Luc (eds.), <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-0-8176-4575-5_8">"Automorphisms of Pure Sphere Braid Groups and Galois Representations"</a></span>, <i>The Grothendieck Festschrift: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck</i>, Modern Birkhäuser Classics, Boston, MA: Birkhäuser, pp. <span class="nowrap">353–</span>373, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-8176-4575-5_8">10.1007/978-0-8176-4575-5_8</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4575-5</bdi><span class="reference-accessdate">, retrieved <span class="nowrap">2023-11-24</span></span></cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFChenSalter2020" class="citation journal cs1">Chen, Lei; Salter, Nick (2020). <a rel="nofollow" class="external text" href="https://www.mendeley.com/catalogue/75cd35b0-fb2a-3986-a5b6-e486dcc723ee/">"Section problems for configurations of points on the Riemann sphere"</a>. <i>Algebraic and Geometric Topology</i>. <b>20</b> (6): <span class="nowrap">3047–</span>3082. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1807.10171">1807.10171</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fagt.2020.20.3047">10.2140/agt.2020.20.3047</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119669926">119669926</a>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFFadellBuskirk1962" class="citation journal cs1">Fadell, Edward; Buskirk, James Van (1962). <a rel="nofollow" class="external text" href="https://www.mendeley.com/catalogue/5fe4ec1d-e4e9-3dc4-98af-cfbe28d6ce24/">"The braid groups of E2 and S2"</a>. <i>Duke Mathematical Journal</i>. <b>29</b> (2): <span class="nowrap">243–</span>257. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-62-02925-3">10.1215/S0012-7094-62-02925-3</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFKlassenKopeliovich2004" class="citation journal cs1">Klassen, Eric P.; Kopeliovich, Yaacov (2004). <a rel="nofollow" class="external text" href="https://www.mendeley.com/catalogue/040a97b9-ac59-3411-abf4-e02c0191e13f/">"Hurwitz spaces and braid group representations"</a>. <i>Rocky Mountain Journal of Mathematics</i>. <b>34</b> (3): <span class="nowrap">1005–</span>1030. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1216%2Frmjm%2F1181069840">10.1216/rmjm/1181069840</a></span>.</cite></span>

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