Volume 19, issue 5 (2015)

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New invariants of $G_2$–structures

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.

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