Abstract
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.
Citation
Diarmuid Crowley. Johannes Nordström. "New invariants of $G_2$–structures." Geom. Topol. 19 (5) 2949 - 2992, 2015. https://doi.org/10.2140/gt.2015.19.2949
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