New invariants of G2–structures

Diarmuid Crowley, Johannes Nordström

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11 Citations (Scopus)
4 Downloads (Pure)

Abstract

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
Volume19
Issue number5
DOIs
Publication statusPublished - 20 Oct 2015

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Homotopy
Connected Sum
Holonomy
Invariant
Tangent Bundle
Euler Characteristic
Orbifold
Torsion-free
Diffeomorphism
Divisor
Torsion
Divides
Modulo
Signature
Odd
Closed
Sufficient Conditions
Class

Keywords

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

Cite this

Crowley, D., & Nordström , J. (2015). New invariants of G2–structures. Geometry & Topology, 19(5), 2949-2992. https://doi.org/10.2140/gt.2015.19.2949

New invariants of G2–structures. / Crowley, Diarmuid; Nordström , Johannes.

In: Geometry & Topology, Vol. 19, No. 5, 20.10.2015, p. 2949-2992.

Research output: Contribution to journalArticle

Crowley, D & Nordström , J 2015, 'New invariants of G2–structures', Geometry & Topology, vol. 19, no. 5, pp. 2949-2992. https://doi.org/10.2140/gt.2015.19.2949
Crowley, Diarmuid ; Nordström , Johannes. / New invariants of G2–structures. In: Geometry & Topology. 2015 ; Vol. 19, No. 5. pp. 2949-2992.
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