KW-Euler Classes via Twisted Symplectic Bundles
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916476598878208 |
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| author | D'Angelo, Alessandro |
| author_facet | D'Angelo, Alessandro |
| contents | In this paper we are going to compute the $ \mathrm{KW} $-Euler classes for rank 2 vector bundles on the classifying stack $ \mathcal{B}N $, where $N$ is the normaliser of the standard torus in $SL_2$ and $\mathrm{KW}$ represents Balmer's derived Witt groups. Using these computations we will recover, through a new and different strategy, the formulas previously obtained by Levine in Witt-sheaf cohomology. In order to obtain our results, we will prove Künneth formulas for products of $GL_n$'s and $SL_n$'s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in $SL$-oriented theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06504 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | KW-Euler Classes via Twisted Symplectic Bundles D'Angelo, Alessandro Algebraic Geometry Algebraic Topology K-Theory and Homology In this paper we are going to compute the $ \mathrm{KW} $-Euler classes for rank 2 vector bundles on the classifying stack $ \mathcal{B}N $, where $N$ is the normaliser of the standard torus in $SL_2$ and $\mathrm{KW}$ represents Balmer's derived Witt groups. Using these computations we will recover, through a new and different strategy, the formulas previously obtained by Levine in Witt-sheaf cohomology. In order to obtain our results, we will prove Künneth formulas for products of $GL_n$'s and $SL_n$'s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in $SL$-oriented theories. |
| title | KW-Euler Classes via Twisted Symplectic Bundles |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2411.06504 |