Description
There are nowadays different types of examples of compact Riemannian manifolds with holonomy group $\mathrm{G}_2$. One may ask if different constructions can lead to diffeomorphic $\mathrm{G}_2$-manifolds, and if so, whether their $\mathrm{G}_2$-metrics belong to different connected components of the corresponding moduli space. Crowley and Nordström's invariant nu and its analytic refinement are able to detect some of these connected components.
After a short review of $\mathrm{G}_2$-geometry, we will compute the nu invariant for extra-twisted connected sums. This involves the computation of eta invariants using gluing formulas and adiabatic limits, which lead us to logarithms of special values of the Dedekind eta function at arguments in certain imaginary quadratic number fields. Finally, we see that many of the possible values of the nu invariant are attained by our examples.