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, 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.

Figures (16)

• Figure 1: The landscape of examples in Table \ref{['nuxxtcs:table:matchings']}
• Figure 2: Fundamental domains of $T$ and $\widetilde{T}_\pm$.
• Figure 3: angle pi/4
• Figure 4: angle pi/6
• Figure 5: angle pi/3

Theorems & Definitions (110)

• Theorem 2
• Theorem 3
• Remark 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• Proposition 1.8