Nu-invariants of extra-twisted connected sums
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arXiv
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| Format: | Preprint |
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2020
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| author | Goette, Sebastian Nordström, Johannes Zagier, Don |
| author_facet | Goette, Sebastian Nordström, Johannes Zagier, Don |
| contents | We analyse the possible ways of gluing twisted products of circles with asymptotically cylindrical Calabi-Yau manifolds to produce manifolds with holonomy G_2, thus generalising the twisted connected sum construction of Kovalev and Corti, Haskins, Nordström, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordström arXiv:1505.02734 in terms of fixpoint and gluing contributions, which include different types of (generalised) Dedekind sums. Surprisingly, the calculations involve some non-trivial number-theoretical arguments connected with special values of the Dedekind eta-function and the theory of complex multiplication. One consequence of our computations is that there exist compact G_2-manifolds that are not G_2-nullbordant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_16367 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Nu-invariants of extra-twisted connected sums Goette, Sebastian Nordström, Johannes Zagier, Don Geometric Topology Algebraic Geometry Differential Geometry 57R20 (Primary) 53C29, 58J28, 11F20 (Secondary) We analyse the possible ways of gluing twisted products of circles with asymptotically cylindrical Calabi-Yau manifolds to produce manifolds with holonomy G_2, thus generalising the twisted connected sum construction of Kovalev and Corti, Haskins, Nordström, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordström arXiv:1505.02734 in terms of fixpoint and gluing contributions, which include different types of (generalised) Dedekind sums. Surprisingly, the calculations involve some non-trivial number-theoretical arguments connected with special values of the Dedekind eta-function and the theory of complex multiplication. One consequence of our computations is that there exist compact G_2-manifolds that are not G_2-nullbordant. |
| title | Nu-invariants of extra-twisted connected sums |
| topic | Geometric Topology Algebraic Geometry Differential Geometry 57R20 (Primary) 53C29, 58J28, 11F20 (Secondary) |
| url | https://arxiv.org/abs/2010.16367 |