A gluing formula for the $Z_2$-valued index of odd symmetric operators
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912370328076288 |
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| author | Braverman, Maxim Sadegh, Ahmad Reza Haj Saeedi Yan, Junrong |
| author_facet | Braverman, Maxim Sadegh, Ahmad Reza Haj Saeedi Yan, Junrong |
| contents | We investigate Dirac-type operator $D$ on involutive manifolds with boundary with symmetry, which forces the index of $D$ to vanish. We study the secondary $Z_2$-valued index of elliptic boundary value problems for such operators. We prove a $Z_2$-valued analog of the splitting theorem: the $Z_2$-valued index of an operator on a closed manifold $M$ equals the $Z_2$-valued index of a boundary value problem on a manifold obtained by cutting $M$ along a hypersurface $N$. When $N$ divides $M$ into two disjoint submanifolds $M_1$ and $M_2$, the $Z_2$-valued index on $M$ is equal to the mod 2 reduction of the usual $Z$-valued index of the Atiyah-Patodi-Singer boundary value problem on $M_1$. This leads to a cohomological formula for the $Z_2$-valued index. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07094 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A gluing formula for the $Z_2$-valued index of odd symmetric operators Braverman, Maxim Sadegh, Ahmad Reza Haj Saeedi Yan, Junrong Differential Geometry 58J20, 19K56, 58J22, 58J32 We investigate Dirac-type operator $D$ on involutive manifolds with boundary with symmetry, which forces the index of $D$ to vanish. We study the secondary $Z_2$-valued index of elliptic boundary value problems for such operators. We prove a $Z_2$-valued analog of the splitting theorem: the $Z_2$-valued index of an operator on a closed manifold $M$ equals the $Z_2$-valued index of a boundary value problem on a manifold obtained by cutting $M$ along a hypersurface $N$. When $N$ divides $M$ into two disjoint submanifolds $M_1$ and $M_2$, the $Z_2$-valued index on $M$ is equal to the mod 2 reduction of the usual $Z$-valued index of the Atiyah-Patodi-Singer boundary value problem on $M_1$. This leads to a cohomological formula for the $Z_2$-valued index. |
| title | A gluing formula for the $Z_2$-valued index of odd symmetric operators |
| topic | Differential Geometry 58J20, 19K56, 58J22, 58J32 |
| url | https://arxiv.org/abs/2505.07094 |