Topological representations of motion groups and mapping class groups -- a unified functorial construction
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866916551114883072 |
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| author | Palmer, Martin Soulié, Arthur |
| author_facet | Palmer, Martin Soulié, Arthur |
| contents | For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved themselves especially important for the question of linearity, a key example being the family of topologically-defined representations introduced by Lawrence and Bigelow, and used by Bigelow and Krammer to prove that braid groups are linear. In this paper, we give a unified foundation for the construction of homological representations using a functorial approach. Namely, we introduce homological representation functors encoding a large class of homological representations, defined on categories containing all mapping class groups and motion groups in a fixed dimension. These source categories are defined using a topological enrichment of the Quillen bracket construction applied to categories of decorated manifolds. This approach unifies many previously-known constructions, including those of Lawrence-Bigelow, and yields many new representations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_13423 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Topological representations of motion groups and mapping class groups -- a unified functorial construction Palmer, Martin Soulié, Arthur Algebraic Topology Geometric Topology 20C12, 20F36, 57K20, 18B40, 20C07, 20J05, 55R80, 57M07, 57M10 For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved themselves especially important for the question of linearity, a key example being the family of topologically-defined representations introduced by Lawrence and Bigelow, and used by Bigelow and Krammer to prove that braid groups are linear. In this paper, we give a unified foundation for the construction of homological representations using a functorial approach. Namely, we introduce homological representation functors encoding a large class of homological representations, defined on categories containing all mapping class groups and motion groups in a fixed dimension. These source categories are defined using a topological enrichment of the Quillen bracket construction applied to categories of decorated manifolds. This approach unifies many previously-known constructions, including those of Lawrence-Bigelow, and yields many new representations. |
| title | Topological representations of motion groups and mapping class groups -- a unified functorial construction |
| topic | Algebraic Topology Geometric Topology 20C12, 20F36, 57K20, 18B40, 20C07, 20J05, 55R80, 57M07, 57M10 |
| url | https://arxiv.org/abs/1910.13423 |