The Exotic $K(2)$-Local Picard Group at the Prime $2$
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910026171416576 |
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| author | Beaudry, Agnes Bobkova, Irina Goerss, Paul G. Henn, Hans-Werner Pham, Viet-Cuong Stojanoska, Vesna |
| author_facet | Beaudry, Agnes Bobkova, Irina Goerss, Paul G. Henn, Hans-Werner Pham, Viet-Cuong Stojanoska, Vesna |
| contents | We calculate the group $κ_2$ of exotic elements in the $K(2)$-local Picard group at the prime $2$ and find it is a group of order $2^9$ isomorphic to $(\mathbb{Z}/8)^2 \times (\mathbb{Z}/2)^3$. In order to do this we must define and exploit a variety of different ways of constructing elements in the Picard group, and this requires a significant exploration of the theory. The most innovative technique, which so far has worked best at the prime $2$, is the use of a $J$-homomorphism from the group of real representations of finite quotients of the Morava stabilizer group to the $K(n)$-local Picard group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07858 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Exotic $K(2)$-Local Picard Group at the Prime $2$ Beaudry, Agnes Bobkova, Irina Goerss, Paul G. Henn, Hans-Werner Pham, Viet-Cuong Stojanoska, Vesna Algebraic Topology 55P42, 55Q51, 55Q10, 55P60, 14C22, 55Q50 We calculate the group $κ_2$ of exotic elements in the $K(2)$-local Picard group at the prime $2$ and find it is a group of order $2^9$ isomorphic to $(\mathbb{Z}/8)^2 \times (\mathbb{Z}/2)^3$. In order to do this we must define and exploit a variety of different ways of constructing elements in the Picard group, and this requires a significant exploration of the theory. The most innovative technique, which so far has worked best at the prime $2$, is the use of a $J$-homomorphism from the group of real representations of finite quotients of the Morava stabilizer group to the $K(n)$-local Picard group. |
| title | The Exotic $K(2)$-Local Picard Group at the Prime $2$ |
| topic | Algebraic Topology 55P42, 55Q51, 55Q10, 55P60, 14C22, 55Q50 |
| url | https://arxiv.org/abs/2212.07858 |