### Abstract

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 |
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Pages (from-to) | 2949-2992 |

Number of pages | 43 |

Journal | Geometry & Topology |

Volume | 19 |

Issue number | 5 |

DOIs | |

Publication status | Published - 20 Oct 2015 |

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### Keywords

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

### Cite this

*Geometry & Topology*,

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

**New invariants of G2–structures.** / Crowley, Diarmuid; Nordström , Johannes.

Research output: Contribution to journal › Article

*Geometry & Topology*, vol. 19, no. 5, pp. 2949-2992. https://doi.org/10.2140/gt.2015.19.2949

}

TY - JOUR

T1 - New invariants of G2–structures

AU - Crowley, Diarmuid

AU - Nordström , Johannes

PY - 2015/10/20

Y1 - 2015/10/20

N2 - 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.

AB - 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.

KW - G2–structures

KW - spin geometry

KW - diffeomorphisms

KW - h–principle

KW - exceptional holonomy

U2 - 10.2140/gt.2015.19.2949

DO - 10.2140/gt.2015.19.2949

M3 - Article

VL - 19

SP - 2949

EP - 2992

JO - Geometry & Topology

JF - Geometry & Topology

SN - 1364-0380

IS - 5

ER -