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New invariants of G<sub>2</sub>-structures
OAI: oai:purehost.bath.ac.uk:openaire_cris_publications/6f2a3924-4238-45c1-94b1-fc62e9a20d9c • DOI: https://doi.org/10.2140/gt.2015.19.2949
Abstract
We define a Z/48-valued homotopy invariant nu of a G_2-structure 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 G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples constructed by Joyce by desingularising orbifolds have odd nu.
We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) 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(M) 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.
We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) 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(M) 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.