Local index theory and $\mathbb{Z}/k\mathbb{Z}$ $K$-theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910082608922624 |
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| author | Ho, Man-Ho |
| author_facet | Ho, Man-Ho |
| contents | For any given submersion $π:X\to B$ with closed, oriented and spin$^c$ fibers of even dimension, equipped with a Riemannian and differential spin$^c$ structure, we apply the Atiyah-Singer-Gorokhovsky-Lott approach to the local family index theorem without the kernel bundle assumption to construct an analytic index $\textrm{ind}^a_k$ in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory at the cocycle level. This is achieved by associating to every cocycle $(\mathbf{E}, \mathbf{F}, α)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $X$ a cocycle $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, α)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $B$. We also prove a Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory, which expresses the Cheeger-Chern-Simons form of $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, α)$ in terms of that of $(\mathbf{E}, \mathbf{F}, α)$. Furthermore, we show that the analytic index $\textrm{ind}^a_k$ and the Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory refine the underlying geometric bundle of the analytic index and the Riemann-Roch-Grothendieck theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local index theory and $\mathbb{Z}/k\mathbb{Z}$ $K$-theory Ho, Man-Ho K-Theory and Homology Differential Geometry 19K56, 58J20, 19L50, 19L10 For any given submersion $π:X\to B$ with closed, oriented and spin$^c$ fibers of even dimension, equipped with a Riemannian and differential spin$^c$ structure, we apply the Atiyah-Singer-Gorokhovsky-Lott approach to the local family index theorem without the kernel bundle assumption to construct an analytic index $\textrm{ind}^a_k$ in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory at the cocycle level. This is achieved by associating to every cocycle $(\mathbf{E}, \mathbf{F}, α)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $X$ a cocycle $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, α)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $B$. We also prove a Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory, which expresses the Cheeger-Chern-Simons form of $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, α)$ in terms of that of $(\mathbf{E}, \mathbf{F}, α)$. Furthermore, we show that the analytic index $\textrm{ind}^a_k$ and the Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory refine the underlying geometric bundle of the analytic index and the Riemann-Roch-Grothendieck theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory, respectively. |
| title | Local index theory and $\mathbb{Z}/k\mathbb{Z}$ $K$-theory |
| topic | K-Theory and Homology Differential Geometry 19K56, 58J20, 19L50, 19L10 |
| url | https://arxiv.org/abs/2410.16399 |