Geometry & Topology

New invariants of $G_2$–structures

Diarmuid Crowley and Johannes Nordström

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We define a 48–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.

Article information

Geom. Topol., Volume 19, Number 5 (2015), 2949-2992.

Received: 12 September 2014
Revised: 27 January 2015
Accepted: 10 March 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C10: $G$-structures 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C27: Spin and Spin$^c$ geometry

$G_2$–structures spin geometry diffeomorphisms $h$–principle exceptional holonomy


Crowley, Diarmuid; Nordström, Johannes. New invariants of $G_2$–structures. Geom. Topol. 19 (2015), no. 5, 2949--2992. doi:10.2140/gt.2015.19.2949.

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