Geometry & Topology
- Geom. Topol.
- Volume 19, Number 5 (2015), 2949-2992.
New invariants of $G_2$–structures
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.
Geom. Topol., Volume 19, Number 5 (2015), 2949-2992.
Received: 12 September 2014
Revised: 27 January 2015
Accepted: 10 March 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 53C10: $G$-structures 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C27: Spin and Spin$^c$ geometry
Crowley, Diarmuid; Nordström, Johannes. New invariants of $G_2$–structures. Geom. Topol. 19 (2015), no. 5, 2949--2992. doi:10.2140/gt.2015.19.2949. https://projecteuclid.org/euclid.gt/1510858853