Abstract
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, Nordstr\"om, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordstr\"om 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.
| Original language | English |
|---|---|
| Journal | Acta Mathematica |
| DOIs | |
| Publication status | Submitted - 4 Apr 2025 |
Bibliographical note
73 pages; v2: Improved exposition and minor corrections, adjusted authorshipFunding
We would like to thank the Simons foundation for its support of their research under the Simons Collaboration on “Special Holonomy in Geometry, Analysis and Physics” (grants #488617, Sebastian Goette, and #488631, Johannes Nordström). We gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University at which some of the research for this paper was carried out.
| Funders | Funder number |
|---|---|
| Simons Foundation |
Keywords
- math.GT
- math.AG
- math.DG
- 57R20 (Primary) 53C29, 58J28, 11F20 (Secondary)