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New invariants of $G_2$–structures

Diarmuid Crowley and Johannes Nordström

Geometry & Topology 19 (2015) 2949–2992

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.

$G_2$–structures, spin geometry, diffeomorphisms, $h$–principle, exceptional holonomy
Mathematical Subject Classification 2010
Primary: 53C10, 57R15
Secondary: 53C25, 53C27
Received: 12 September 2014
Revised: 27 January 2015
Accepted: 10 March 2015
Published: 20 October 2015
Proposed: Simon Donaldson
Seconded: Richard Thomas, Jesper Grodal
Diarmuid Crowley
Institute of Mathematics
University of Aberdeen
Aberdeen AB24 3UE
Johannes Nordström
Department of Mathematical Sciences
University of Bath
Bath BA2 7AY