Disconnecting the G_2 Moduli Space

Little is currently known about the global properties of the $G_2$ moduli space of a closed 7-manifold, ie the space of Riemannian metrics with holonomy $G_2$ modulo diffeomorphisms. A holonomy $G_2$ metric has an associated $G_2$-structure, and I will define a Z/48 valued homotopy invariant of a $G_2$-structure in terms of the signature and Euler characteristic of a Spin(7)-coboundary. I will describe examples of manifolds with holonomy $G_2$ metrics where the invariant is amenable to computation in terms of eta invariants, and which are candidates for having a disconnected moduli space. This is joint work in progress with Diarmuid Crowley and Sebastian Goette.