Tautological characteristic classes II: the Witt class
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914963100008448 |
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| author | Dymara, Jan Januszkiewicz, Tadeusz |
| author_facet | Dymara, Jan Januszkiewicz, Tadeusz |
| contents | Let $K$ be an arbitrary infinite field. The cohomology group $H^2(SL(2,K), H_2\,SL(2,K))$ contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in $SL(2,K)$ it is useful to have classes stable under deformations (Fenchel--Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovar. The Milnor--Wood inequality asserts that an $SL(2,{\bf R})$-bundle over a surface of genus $g$ admits a flat structure if and only if its Euler number is $\leq (g-1)$. We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tautological characteristic classes II: the Witt class Dymara, Jan Januszkiewicz, Tadeusz K-Theory and Homology Group Theory 20G10 Let $K$ be an arbitrary infinite field. The cohomology group $H^2(SL(2,K), H_2\,SL(2,K))$ contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in $SL(2,K)$ it is useful to have classes stable under deformations (Fenchel--Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovar. The Milnor--Wood inequality asserts that an $SL(2,{\bf R})$-bundle over a surface of genus $g$ admits a flat structure if and only if its Euler number is $\leq (g-1)$. We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general. |
| title | Tautological characteristic classes II: the Witt class |
| topic | K-Theory and Homology Group Theory 20G10 |
| url | https://arxiv.org/abs/2403.05255 |