### Abstract

We define a Z48–valued homotopy invariant ν(φ) of a G2–structure φ on the tangent bundle of a closed 7–manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)–structure. For manifolds of holonomy G2 obtained by the twisted connected sum construction, the associated torsion-free G2–structure always has ν(φ)=24. Some holonomy G2 examples constructed by Joyce by desingularising orbifolds have odd ν.

We define a further homotopy invariant ξ(φ) such that if M is 2–connected then the pair (ν,ξ) determines a G2–structure up to homotopy and diffeomorphism. The class of a G2–structure is determined by ν on its own when the greatest divisor of p1(M) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G2–manifolds.

We also prove that the parametric h–principle holds for coclosed G2–structures.

We define a further homotopy invariant ξ(φ) such that if M is 2–connected then the pair (ν,ξ) determines a G2–structure up to homotopy and diffeomorphism. The class of a G2–structure is determined by ν on its own when the greatest divisor of p1(M) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G2–manifolds.

We also prove that the parametric h–principle holds for coclosed G2–structures.

Original language | English |
---|---|

Pages (from-to) | 2949-2992 |

Number of pages | 43 |

Journal | Geometry & Topology |

Volume | 19 |

Issue number | 5 |

DOIs | |

Publication status | Published - 20 Oct 2015 |

### Keywords

- G2–structures
- spin geometry
- diffeomorphisms
- h–principle
- exceptional holonomy

## Fingerprint Dive into the research topics of 'New invariants of G2–structures'. Together they form a unique fingerprint.

## Cite this

Crowley, D., & Nordström , J. (2015). New invariants of G2–structures.

*Geometry & Topology*,*19*(5), 2949-2992. https://doi.org/10.2140/gt.2015.19.2949