Fibrewise Orbifold Resolutions with Applications to ${\mathrm{G}}_2$-Moduli Spaces

By resolving the singularities of tailor-made orbifolds via twisted families of blow-ups, we construct manifold bundles $M\to E\to S^2$. Using tools from real homotopy theory, we show that these bundles determine a free subgroup in ${\pi }_2(B\mathrm{h}\mathrm{A}\mathrm{u}\mathrm{t}(M{)}_0)$. The proof relies on a generalisation of Sullivan’s result, which describes the real homotopy groups of the monoid of homotopy automorphisms $\mathrm{h}\mathrm{A}\mathrm{u}\mathrm{t}(X)$ in terms of derivations of the minimal model of $X$, to the monoid ${\mathrm{h}\mathrm{A}\mathrm{u}\mathrm{t}}_A(X)$ of relative homotopy automorphisms. As an application, we prove that the moduli space of torsion-free ${\mathrm{G}}_2$-structures arising from many generalised Kummer constructions contains a free subgroup of positive rank in its second homotopy group.