Path components of $\mathrm{G}_2$-moduli spaces may be non-aspherical

Abstract

Starting from Joyce's generalised Kummer construction, we exhibit non-trivial families of $\mathrm{G}_2$-manifolds over the two dimensional sphere by resolving singularities with a twisted family of Eguchi-Hanson spaces. We establish that the comparison map $\mathcal{G}_2^{\mathrm{tf}}(M) /\!\!/ \mathrm{Diff}(M)_0 \rightarrow \mathcal{G}_2^{\mathrm{tf}}(M) / \mathrm{Diff}(M)_0$ is a fibration over each path components with Eilenberg Mac Lane spaces as fibres, which allows us to show that these families remain non-trivial in $\mathcal{G}_2^{\mathrm{tf}}(M) / \mathrm{Diff}(M)_0$. In addition, we construct a new invariant based on characteristic classes that allows us to show that different resolutions give rise to different elements in the moduli space.