Thorsten Hertl: Resolution Bundles with Applications to G_2-Moduli Spaces

Abstract: Like Kähler-metrics, Riemannian metrics with holonomy groups \(\mathrm{G}_2\) constitute a special geometry, and it is a central theme in differential geometry to understand these \(\mathrm{G}_2\)-manifolds. However, in contrast to their complex cousins, \(\mathrm{G}_2\)-geometry is specific to dimension 7 and finding \(\mathrm{G}_2\)-manifolds is considerably harder. Indeed, the first example of a closed \(\mathrm{G}_2\)-manifold was constructed by Dominic Joyce in 1996. In the same landmark article, he also proved that the moduli space of \(\mathrm{G}_2\)-holonomy metrics on a fixed closed manifold \(M\) is itself a (possibly empty) finite-dimensional smooth manifold.

However, this structural result of Joyce is only local in nature, and knowledge about global topological properties of these moduli spaces has remained rather limited. Indeed, to date we only know that they may be disconnected by the work of Crowley, Goette and Nordström; whether all path components are contractible remains open.

In this talk, I will outline a construction of elements in the second homotopy group of the moduli space of \(\mathrm{G}_2\)-metrics on Joyce’s first example, and present along the way a homotopy-theoretical result guaranteeing that their order is infinite. This talk is based on joint work with Diarmuid Crowley and Sebastian Goette.