New invariants of G2–structures

Diarmuid Crowley, Johannes Nordström

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29 Citations (Scopus)
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We define a Z48–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.
Original languageEnglish
Pages (from-to)2949-2992
Number of pages43
JournalGeometry & Topology
Issue number5
Publication statusPublished - 20 Oct 2015


  • G2–structures
  • spin geometry
  • diffeomorphisms
  • h–principle
  • exceptional holonomy


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