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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2505.12561 |
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| _version_ | 1866908655175073792 |
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| author | Minami, Haruo |
| author_facet | Minami, Haruo |
| contents | Let $G$ be a simply connected compact Lie group and $\mathscr{L}$ be the left invarinat framing of $G$. Let $\mathcal{L}^λ$ be the framing obtained by twisting $\mathscr{L}$ by a faithful representation $λ$. Given a torus subgroup $T''$ of $G$ we have a framing $(\mathcal{L}^λ)_{T''}$ of the quotient $G/T''$ induced from $\mathcal{L}^λ$. In this note we show that under a certain dimensional condition the $e_\mathbb{C}$-invariant of $G/T''$ with this framing provides a generator of the $J$-homomorphism or twice that. Thereby we also give a unified proof of the results for $SU(2n)$, $Spin(4n+1)$ and $Spin(8n-2)$ $(n\ge 1)$ previously proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12561 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $e$-invariants of quotients of Lie groups Minami, Haruo Algebraic Topology 22E46, 55Q45 Let $G$ be a simply connected compact Lie group and $\mathscr{L}$ be the left invarinat framing of $G$. Let $\mathcal{L}^λ$ be the framing obtained by twisting $\mathscr{L}$ by a faithful representation $λ$. Given a torus subgroup $T''$ of $G$ we have a framing $(\mathcal{L}^λ)_{T''}$ of the quotient $G/T''$ induced from $\mathcal{L}^λ$. In this note we show that under a certain dimensional condition the $e_\mathbb{C}$-invariant of $G/T''$ with this framing provides a generator of the $J$-homomorphism or twice that. Thereby we also give a unified proof of the results for $SU(2n)$, $Spin(4n+1)$ and $Spin(8n-2)$ $(n\ge 1)$ previously proved. |
| title | $e$-invariants of quotients of Lie groups |
| topic | Algebraic Topology 22E46, 55Q45 |
| url | https://arxiv.org/abs/2505.12561 |