Disconnecting the G_2 Moduli Space

Johannes Nordstrom
Tue, Jul 7, 2015
PIMS, University of British Columbia
PIMS Symposium on the Geometry and Topology of Manifolds

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.