New invariants of G_2-structures

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.

26 pages, 1 figure. v3: Defined further invariant, strengthened classification results, changed title

References in corpus (3)

• G_2-manifolds and associative submanifolds via semi-Fano 3-folds
• Short-time behaviour of a modified Laplacian coflow of G2-structures
• Obstruction theory on 8-manifolds

Cited by in corpus (17)

• Framed M-branes, corners, and topological invariants
• A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces
• The classification of 2-connected 7-manifolds
• Effective action from M-theory on twisted connected sum $G_2$-manifolds
• On the automorphism group of a closed G${}_2$-structure
• Estimates and monotonicity for a heat flow of isometric G2-structures
• An analytic invariant of G_2 manifolds
• Closed G${}_2$-structures on non-solvable Lie groups
• Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds
• Gauge theory on Aloff-Wallach spaces
• A compact $G_2$-calibrated manifold with first Betti number $b_1=1$
• Geometric Flows of $G_2$-Structures on 3-Sasakian 7-Manifolds
• Supersymmetric gauge theory on curved 7-branes
• Purely coclosed G${}_2$-structures on 2-step nilpotent Lie groups
• Special types of locally conformal closed G${}_2$-structures
• Moduli spaces of (co)closed ${\mathrm{G}}_2$-structures on nilmanifolds
• Coclosed $G_2$-structures on $\text{SU}(2{)}^2$-invariant cohomogeneity one manifolds