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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.00345 |
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| _version_ | 1866929195702026240 |
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| author | Day, Matthew B. Nakamura, Trevor |
| author_facet | Day, Matthew B. Nakamura, Trevor |
| contents | We construct a 3-dimensional cell complex that is the 3-skeleton for an Eilenberg--MacLane classifying space for the symmetric group $\mathfrak{S}_n$. Our complex starts with the presentation for $\mathfrak{S}_n$ with $n-1$ adjacent transpositions with squaring, commuting, and braid relations, and adds seven classes of 3-cells that fill in certain 2-spheres bounded by these relations. We use a rewriting system and a combinatorial method of K. Brown to prove the correctness of our construction. Our main application is a computation of the second cohomology of $\mathfrak{S}_n$ in certain twisted coefficient modules; we use this computation in a companion paper to study splitting of extensions related to braid groups. As another application, we give a concrete description of the third homology of $\mathfrak{S}_n$ with untwisted coefficients in $\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00345 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A 3-skeleton for a classifying space for the symmetric group Day, Matthew B. Nakamura, Trevor Geometric Topology Group Theory 20B30 (Primary) 20F05, 20J06 (Secondary) We construct a 3-dimensional cell complex that is the 3-skeleton for an Eilenberg--MacLane classifying space for the symmetric group $\mathfrak{S}_n$. Our complex starts with the presentation for $\mathfrak{S}_n$ with $n-1$ adjacent transpositions with squaring, commuting, and braid relations, and adds seven classes of 3-cells that fill in certain 2-spheres bounded by these relations. We use a rewriting system and a combinatorial method of K. Brown to prove the correctness of our construction. Our main application is a computation of the second cohomology of $\mathfrak{S}_n$ in certain twisted coefficient modules; we use this computation in a companion paper to study splitting of extensions related to braid groups. As another application, we give a concrete description of the third homology of $\mathfrak{S}_n$ with untwisted coefficients in $\mathbb{Z}$. |
| title | A 3-skeleton for a classifying space for the symmetric group |
| topic | Geometric Topology Group Theory 20B30 (Primary) 20F05, 20J06 (Secondary) |
| url | https://arxiv.org/abs/2401.00345 |