#### Volume 19, issue 5 (2015)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060

### Diarmuid Crowley and Johannes Nordström

Geometry & Topology 19 (2015) 2949–2992
##### Abstract

We define a ${ℤ}_{48}$–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.

##### Mathematical Subject Classification 2010
Primary: 53C10, 57R15
Secondary: 53C25, 53C27