Geometry & Topology

New invariants of $G_2$–structures

Diarmuid Crowley and Johannes Nordström

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We define a 48–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.

Article information

Source
Geom. Topol., Volume 19, Number 5 (2015), 2949-2992.

Dates
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
https://projecteuclid.org/euclid.gt/1510858853

Digital Object Identifier
doi:10.2140/gt.2015.19.2949

Mathematical Reviews number (MathSciNet)
MR3416118

Zentralblatt MATH identifier
1346.53029

Subjects
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

Keywords
$G_2$–structures spin geometry diffeomorphisms $h$–principle exceptional holonomy

Citation

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


Export citation

References

  • M F Atiyah, I M Singer, The index of elliptic operators, III, Ann. of Math. 87 (1968) 546–604
  • J C Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002) 145–205
  • M Berger, Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955) 279–330
  • R L Bryant, Some remarks on $G\sb 2$–structures, from: “Proceedings of Gökova Geometry–Topology Conference 2005”, (S Akbulut, T Önder, R J Stern, editors), GGT, Gökova, Turkey (2006) 75–109
  • R L Bryant, Non-embedding and non-extension results in special holonomy, from: “The many facets of geometry”, (O García-Prada, J P Bourguignon, S Salamon, editors), Oxford Univ. Press (2010) 346–367
  • R L Bryant, S M Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989) 829–850
  • R Bryant, F Xu, Laplacian flow for closed $G_2$–structures: Short time behavior
  • M Čadek, M Crabb, J Vanžura, Obstruction theory on $8$–manifolds, Manuscripta Math. 127 (2008) 167–186
  • E Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. Sup. 12 (1979) 269–294
  • A Corti, M Haskins, J Nordstr öm, T Pacini, $\mathrm{G}_2$–manifolds and associative submanifolds via semi-Fano $3$–folds, Duke Math. J. 164 (2015) 1971–2092
  • D Crowley, J Nordström, The classification of $2$–connected $7$–manifolds
  • D Crowley, J Nordström, Exotic $G_2$–manifolds
  • H Donnelly, Spectral geometry and invariants from differential topology, Bull. London Math. Soc. 7 (1975) 147–150
  • J Eells, Jr, N Kuiper, H, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. 60 (1962) 93–110
  • Y Eliashberg, N Mishachev, Introduction to the $h$–principle, Graduate Studies in Mathematics 48, Amer. Math. Soc. (2002)
  • M Fernández, A Gray, Riemannian manifolds with structure group $G\sb{2}$, Ann. Mat. Pura Appl. 132 (1982) 19–45
  • A Gray, Vector cross products on manifolds, Trans. Amer. Math. Soc. 141 (1969) 465–504
  • A Gray, P S Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. 34 (1970) 83–96
  • S Grigorian, Short-time behaviour of a modified Laplacian coflow of $G\sb 2$–structures, Adv. Math. 248 (2013) 378–415
  • R Harvey, H B Lawson, Jr, Calibrated geometries, Acta Math. 148 (1982) 47–157
  • N Hitchin, Stable forms and special metrics, from: “Global differential geometry: The mathematical legacy of Alfred Gray”, (M Fernández, J A Wolf, editors), Contemp. Math. 288, Amer. Math. Soc. (2001) 70–89
  • D D Joyce, Compact Riemannian $7$–manifolds with holonomy $G\sb 2$, I, II, J. Differential Geom. 43 (1996) 291–328, 329–375
  • D D Joyce, Compact manifolds with special holonomy, Oxford University Press (2000)
  • A Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003) 125–160
  • A Kovalev, J Nordstr öm, Asymptotically cylindrical $7$–manifolds of holonomy $G\sb 2$ with applications to compact irreducible $G\sb 2$–manifolds, Ann. Global Anal. Geom. 38 (2010) 221–257
  • H B Lawson, Jr, M-L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton Univ. Press (1989)
  • H-V Lê, Existence of symplectic $3$–forms on $7$–manifolds, preprint (2007)
  • J Milnor, On manifolds homeomorphic to the $7$–sphere, Ann. of Math. 64 (1956) 399–405
  • J Milnor, Spin structures on manifolds, Enseignement Math. 9 (1963) 198–203
  • J Milnor, D Husemoller, Symmetric bilinear forms, Ergeb. Math. Grenzgeb. 73, Springer, New York (1973)
  • S Salamon, Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics Series 201, Longman, Harlow, UK (1989)
  • C T C Wall, Non-additivity of the signature, Invent. Math. 7 (1969) 269–274
  • H Weiss, F Witt, Energy functionals and soliton equations for $G\sb 2$–forms, Ann. Global Anal. Geom. 42 (2012) 585–610
  • F Witt, Generalised $G\sb 2$–manifolds, Comm. Math. Phys. 265 (2006) 275–303
  • F Xu, R Ye, Existence, convergence and limit map of the Laplacian flow, preprint (2009)