Braids, mapping class groups, and categorical delooping
Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map φ: β2g → τ g,1 from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2...
| मुख्य लेखकों: | , |
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| स्वरूप: | Journal article |
| भाषा: | English |
| प्रकाशित: |
2007
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| _version_ | 1826285198293073920 |
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| author | Song, Y Tillmann, U |
| author_facet | Song, Y Tillmann, U |
| author_sort | Song, Y |
| collection | OXFORD |
| description | Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map φ: β2g → τ g,1 from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257-175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, φ is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology. © 2007 Springer-Verlag. |
| first_indexed | 2024-03-07T01:25:15Z |
| format | Journal article |
| id | oxford-uuid:91be6932-3912-4f53-a456-76c0981dad5e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:25:15Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:91be6932-3912-4f53-a456-76c0981dad5e2022-03-26T23:20:45ZBraids, mapping class groups, and categorical deloopingJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:91be6932-3912-4f53-a456-76c0981dad5eEnglishSymplectic Elements at Oxford2007Song, YTillmann, UDehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map φ: β2g → τ g,1 from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257-175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, φ is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology. © 2007 Springer-Verlag. |
| spellingShingle | Song, Y Tillmann, U Braids, mapping class groups, and categorical delooping |
| title | Braids, mapping class groups, and categorical delooping |
| title_full | Braids, mapping class groups, and categorical delooping |
| title_fullStr | Braids, mapping class groups, and categorical delooping |
| title_full_unstemmed | Braids, mapping class groups, and categorical delooping |
| title_short | Braids, mapping class groups, and categorical delooping |
| title_sort | braids mapping class groups and categorical delooping |
| work_keys_str_mv | AT songy braidsmappingclassgroupsandcategoricaldelooping AT tillmannu braidsmappingclassgroupsandcategoricaldelooping |