A partitioned manifold index theorem for noncompact hypersurfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915417473155072 |
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| author | Hochs, Peter de Kok, Thijs |
| author_facet | Hochs, Peter de Kok, Thijs |
| contents | Roe's partitioned manifold index theorem applies when a complete Riemannian manifold $M$ is cut into two pieces along a compact hypersurface $N$. It states that a version of the index of a Dirac operator on $M$ localized to $N$ equals the index of the corresponding Dirac operator on $N$. This yields obstructions to positive scalar curvature, and implies cobordism invariance of the index of Dirac operators on compact manifolds. We generalize this result to cases where $N$ may be noncompact, under assumptions on the way it is embedded into $M$. This results in an equality between two classes in the $K$-theory of the Roe algebra of $N$. Bunke and Ludewig, and Engel and Wulff, have recently obtained related results based on different approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16591 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A partitioned manifold index theorem for noncompact hypersurfaces Hochs, Peter de Kok, Thijs Differential Geometry K-Theory and Homology Operator Algebras Roe's partitioned manifold index theorem applies when a complete Riemannian manifold $M$ is cut into two pieces along a compact hypersurface $N$. It states that a version of the index of a Dirac operator on $M$ localized to $N$ equals the index of the corresponding Dirac operator on $N$. This yields obstructions to positive scalar curvature, and implies cobordism invariance of the index of Dirac operators on compact manifolds. We generalize this result to cases where $N$ may be noncompact, under assumptions on the way it is embedded into $M$. This results in an equality between two classes in the $K$-theory of the Roe algebra of $N$. Bunke and Ludewig, and Engel and Wulff, have recently obtained related results based on different approaches. |
| title | A partitioned manifold index theorem for noncompact hypersurfaces |
| topic | Differential Geometry K-Theory and Homology Operator Algebras |
| url | https://arxiv.org/abs/2507.16591 |