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New
invariants of $G_2$–structures
Diarmuid Crowley and Johannes Nordström |
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Geometry & Topology 19 (2015) 2949–2992
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Abstract |
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We define a –valued homotopy invariant of a –structure on the tangent bundle of a closed –manifold in terms of the signature and Euler characteristic of a coboundary with a –structure. For manifolds of holonomy obtained by the twisted connected sum construction, the associated torsion-free –structure always has . Some holonomy examples constructed by Joyce by desingularising orbifolds have odd . We define a further homotopy invariant such that if is –connected then the pair determines a –structure up to homotopy and diffeomorphism. The class of a –structure is determined by on its own when the greatest divisor of modulo torsion divides 224; this sufficient condition holds for many twisted connected sum –manifolds. We also prove that the parametric –principle holds for coclosed –structures. |
Keywords
$G_2$–structures, spin geometry, diffeomorphisms,
$h$–principle, exceptional holonomy
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Mathematical Subject Classification 2010
Primary: 53C10, 57R15
Secondary: 53C25, 53C27
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References |
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
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Authors |
| Diarmuid Crowley | |
| Institute of Mathematics University of Aberdeen Aberdeen AB24 3UE UK |
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| Johannes Nordström | |
| Department of Mathematical
Sciences University of Bath Bath BA2 7AY UK |
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