Joyce’s orbifold construction and the twisted connected sums by Kovalev and Corti-Haskins-Nordström-Pacini provide many examples of compact Riemannian 7-manifolds with holonomy $G_2$. We would like to use this wealth of examples to guess further properties of $G_2$-manifolds and to find obstructions against holonomy $G_2$, taking into account the underlying topological $G_2$-structures.
The Crowley-Nordström $\nu$-invariant distinguishes topological $G_2$-structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the $\nu$-invariant can assume all of its 48 possible values. This shows that $G_2$-bordism presents no obstruction against holonomy $G_2$. We also exhibit examples of 7-manifolds with disconnected $G_2$-moduli space. Our computation of the $\nu$-invariants involves integration of the Bismuth-Cheeger $\eta$-forms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind $\eta$-function.