Nu-invariants of extra-twisted connected sums
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.
Cite
@article{arxiv.2010.16367,
title = {Nu-invariants of extra-twisted connected sums},
author = {Sebastian Goette and Johannes Nordström and Don Zagier},
journal= {arXiv preprint arXiv:2010.16367},
year = {2025}
}
Comments
75 pages; v3: Added discussion of Kovalev-Lefschetz fibrations