Quantitative K-theory, positive scalar curvature, and band width
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| Format: | Preprint |
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2020
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| _version_ | 1866909300994080768 |
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| author | Guo, Hao Xie, Zhizhang Yu, Guoliang |
| author_facet | Guo, Hao Xie, Zhizhang Yu, Guoliang |
| contents | We develop two connections between the quantitative framework of operator $K$-theory for geometric $C^*$-algebras and the problem of positive scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a refinement of the well-known obstruction of Rosenberg to positive scalar curvature on closed spin manifolds coming from the higher index of the Dirac operator. We show that on a manifold with uniformly positive scalar curvature, the propagation at which the index of the Dirac operator vanishes is related inversely to the curvature lower bound. Second, we give an approach, using related techniques, to Gromov's band width conjecture, which has been the subject of recent work by Zeidler and Cecchini from a different point of view. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_01749 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Quantitative K-theory, positive scalar curvature, and band width Guo, Hao Xie, Zhizhang Yu, Guoliang K-Theory and Homology Differential Geometry 19K56, 53C21 We develop two connections between the quantitative framework of operator $K$-theory for geometric $C^*$-algebras and the problem of positive scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a refinement of the well-known obstruction of Rosenberg to positive scalar curvature on closed spin manifolds coming from the higher index of the Dirac operator. We show that on a manifold with uniformly positive scalar curvature, the propagation at which the index of the Dirac operator vanishes is related inversely to the curvature lower bound. Second, we give an approach, using related techniques, to Gromov's band width conjecture, which has been the subject of recent work by Zeidler and Cecchini from a different point of view. |
| title | Quantitative K-theory, positive scalar curvature, and band width |
| topic | K-Theory and Homology Differential Geometry 19K56, 53C21 |
| url | https://arxiv.org/abs/2010.01749 |