The moduli space of torsion-free G2 structures

Let M be the moduli space of torsion-free G(2) structures on a compact oriented G(2) manifold M. The natural cohomology map pi(3) : M --> H-3(M, R) is known to be a local diffeomorphism [Compact Manifolds with Special Holonomy, Oxford University Press, 2000]. Let M-1, subset of M be the subset of G(2) structures with volume (M) = 1. We show every nonzero element of H-4 (M, R) = H-3(M, R)* is a Morse function on M-1 when composed with pi(3), and we compute its Hessian. The result implies a special case of Torelli's theorem: if H-1(M, R) = 0 and dim H-3 (M, R) = 2, the cohomology map pi(3) : M --> H-3 (M, R) is one to one on each connected component of M. We formulate a compactness conjecture on the set of G(2) structures of volume (M) = 1 with bounded L-2 norm of curvature. If this conjecture were true, it would imply that every connected component of M is contractible, and that every compact G(2) manifold supports a G(2) structure whose fundamental 4-form represents the negative of the (nonzero) first Pontryagin class of M. We also observe that when H-1(M, R) = 0, and the volume of the torus H-3(M, R) / H-3(M, Z) is constant along M-1,M- the locus pi(3)(M-1) subset of H-3(M, R) is a hyperbolic affine sphere. (C) 2004 Elsevier BX All rights reserved.