New invariants of $G_2$–structures

Abstract

We define a ${\mathbb{Z}}_{48}$–valued homotopy invariant $\nu \left(\phi \right)$ of a $G_2$–structure $\phi$ on the tangent bundle of a closed $7$–manifold in terms of the signature and Euler characteristic of a coboundary with a $Spin\left(7\right)$–structure. For manifolds of holonomy $G_2$ obtained by the twisted connected sum construction, the associated torsion-free $G_2$–structure always has $\nu \left(\phi \right)=24$. Some holonomy $G_2$ examples constructed by Joyce by desingularising orbifolds have odd $\nu$.

We define a further homotopy invariant $\xi \left(\phi \right)$ such that if $M$ is $2$–connected then the pair $\left(\nu ,\xi \right)$ determines a $G_2$–structure up to homotopy and diffeomorphism. The class of a $G_2$–structure is determined by $\nu$ on its own when the greatest divisor of $p_1\left(M\right)$ modulo torsion divides 224; this sufficient condition holds for many twisted connected sum $G_2$–manifolds.

We also prove that the parametric $h$–principle holds for coclosed $G_2$–structures.