## Geometry & Topology

### New invariants of $G_2$–structures

#### 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.

#### Article information

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

Dates
Revised: 27 January 2015
Accepted: 10 March 2015
First available in Project Euclid: 16 November 2017

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

#### 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

#### 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)