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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2411.06570 |
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| _version_ | 1866910693030100992 |
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| author | D'Angelo, Alessandro |
| author_facet | D'Angelo, Alessandro |
| contents | In this paper, we extend the Virtual Localization Formula of Levine to a wide class of motivic ring spectra, obtaining in particular a localization formula for virtual fundamental classes in Witt theory $ \mathrm{KW} $. Applying standard tools of $\mathbb A^1$-intersection theory to any $ SL_η $-oriented spectra $ \mathrm A $, we obtain an additive presentation of $ \mathrm A(BN) $, for $ N $ the normaliser of the torus in $ SL_2 $. Then we establish an equivariant Atiyah-Bott localization theorem for $ \mathrm A_N^{\mathrm{BM}}(X) $ and we conclude with the $N$-equivariant virtual localisation formula. Of independent interest, we also describe the ring structure of $ \mathrm{KW}(BN) $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06570 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Virtual Localisation Formula for $ SL_η $-Oriented Theories D'Angelo, Alessandro Algebraic Geometry Algebraic Topology In this paper, we extend the Virtual Localization Formula of Levine to a wide class of motivic ring spectra, obtaining in particular a localization formula for virtual fundamental classes in Witt theory $ \mathrm{KW} $. Applying standard tools of $\mathbb A^1$-intersection theory to any $ SL_η $-oriented spectra $ \mathrm A $, we obtain an additive presentation of $ \mathrm A(BN) $, for $ N $ the normaliser of the torus in $ SL_2 $. Then we establish an equivariant Atiyah-Bott localization theorem for $ \mathrm A_N^{\mathrm{BM}}(X) $ and we conclude with the $N$-equivariant virtual localisation formula. Of independent interest, we also describe the ring structure of $ \mathrm{KW}(BN) $. |
| title | Virtual Localisation Formula for $ SL_η $-Oriented Theories |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2411.06570 |