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 8-dimensional coboundaries, and a Z-valued refinement in terms of eta invariants. I will describe examples of manifolds with holonomy G_2 metrics where these invariants are amenable to computation and can be used to prove that the moduli space is disconnected. This is joint work with Diarmuid Crowley and Sebastian Goette

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