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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.13234 |
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| _version_ | 1866912710823772160 |
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| author | Mathai, Varghese Melrose, Richard B. |
| author_facet | Mathai, Varghese Melrose, Richard B. |
| contents | By a small bundle gerbe we mean a bundle gerbe in the sense of Murray defined on a smooth, finite-dimensional, fibre bundle over a manifold. We construct such gerbes over compact oriented aspherical 3-manifolds, as well as in higher dimensions, generalizing the construction of decomposable bundle gerbes in earlier work with Singer. For these small bundle gerbes there is a direct index map given in terms of either fibrewise pseudodifferential operators, or more conveniently fibrewise semiclassical smoothing operators, twisted by the simplicial line bundle. We prove the Atiyah-Singer type theorem that this realizes the push-forward into twisted K-theory. We also give an application via the index of projective families of Spin_c Dirac operators, to show the existence of obstructions to metrics with large positive scalar curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13234 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Index and small bundle gerbes Mathai, Varghese Melrose, Richard B. Differential Geometry Analysis of PDEs By a small bundle gerbe we mean a bundle gerbe in the sense of Murray defined on a smooth, finite-dimensional, fibre bundle over a manifold. We construct such gerbes over compact oriented aspherical 3-manifolds, as well as in higher dimensions, generalizing the construction of decomposable bundle gerbes in earlier work with Singer. For these small bundle gerbes there is a direct index map given in terms of either fibrewise pseudodifferential operators, or more conveniently fibrewise semiclassical smoothing operators, twisted by the simplicial line bundle. We prove the Atiyah-Singer type theorem that this realizes the push-forward into twisted K-theory. We also give an application via the index of projective families of Spin_c Dirac operators, to show the existence of obstructions to metrics with large positive scalar curvature. |
| title | Index and small bundle gerbes |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2306.13234 |