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

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 ${\mathcal{G}}_2^{\mathrm{t}\mathrm{f}}(M)\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M{)}_0\to {\mathcal{G}}_2^{\mathrm{t}\mathrm{f}}(M)\mathrm{/}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M{)}_0$ 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 ${\mathcal{G}}_2^{\mathrm{t}\mathrm{f}}(M)\mathrm{/}\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M{)}_0$. 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.