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nLab > Latest Changes: spherical braid group

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeFeb 16th 2025
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Author: Urs
Format: MarkdownItexstarting something, for the moment just to record the basic characterization by [Fadell & VanBuskirk 1961](https://ncatlab.org/nlab/show/spherical+braid+group#FadellVanBuskirk61) <a href="https://ncatlab.org/nlab/revision/spherical+braid+group/1">v1</a>, <a href="https://ncatlab.org/nlab/show/spherical+braid+group">current</a>

starting something, for the moment just to record the basic characterization by Fadell & VanBuskirk 1961

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeApr 4th 2025
- (edited Apr 4th 2025)
- PermaLink
Author: Urs
Format: MarkdownItexadded the example ([here](https://ncatlab.org/nlab/show/spherical+braid+group#SphericalBraidsOnTwoStrands)) that $Br_2(S^2) \simeq C_2$. <a href="https://ncatlab.org/nlab/revision/diff/spherical+braid+group/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/spherical+braid+group/2">v2</a>, <a href="https://ncatlab.org/nlab/show/spherical+braid+group">current</a>

added the example (here) that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Br</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Br_2(S^2) \simeq C_2</annotation></semantics></math>$ .

diff, v2, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeApr 4th 2025
- (edited Apr 4th 2025)
- PermaLink
Author: Urs
Format: MarkdownItexadded statement that the order $\vert Br_n(S^2) \vert$ is 1. 2 for $n = 2$ 1. 12 for $n =3$ 1. $\infty$ for $n \geq 4$. quoted from [Fadell & VanBuskirk 1961 p 255](https://ncatlab.org/nlab/show/spherical+braid+group#FadellVanBuskirk61) <a href="https://ncatlab.org/nlab/revision/diff/spherical+braid+group/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/spherical+braid+group/2">v2</a>, <a href="https://ncatlab.org/nlab/show/spherical+braid+group">current</a>
added statement that the order $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">|</mo><msub><mi>Br</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\vert Br_n(S^2) \vert</annotation></semantics></math>$ is
1. 2 for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math>$
2. 12 for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n =3</annotation></semantics></math>$
3. $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>$ for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n \geq 4</annotation></semantics></math>$ .
quoted from Fadell & VanBuskirk 1961 p 255

diff, v2, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeApr 5th 2025
- PermaLink
Author: Urs
Format: MarkdownItexadded a warning remark ([here](https://ncatlab.org/nlab/show/spherical+braid+group#NotAnMCG)) that the spherical $n$-braid groups are not in general isomorphic to the mapping class groups of the $n$-punctured spheres. <a href="https://ncatlab.org/nlab/revision/diff/spherical+braid+group/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/spherical+braid+group/4">v4</a>, <a href="https://ncatlab.org/nlab/show/spherical+braid+group">current</a>

added a warning remark (here) that the spherical $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -braid groups are not in general isomorphic to the mapping class groups of the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -punctured spheres.

diff, v4, current

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nLab > Latest Changes: spherical braid group