Morse functions on the moduli space of $G_2$ structures

Abstract

Let $ \mathfrak{M}$ be the moduli space of torsion free $ G_2$ structures on a compact 7-manifold $ M$, and let $ \mathfrak{M}_1 \subset \mathfrak{M}$ be the $ G_2$ structures with volume($M$) $=1$. The cohomology map $ \pi^3: \mathfrak{M} \to H^3(M, R)$ is known to be a local diffeomorphism. It is proved that every nonzero element of $ H^4(M, R) = H^3(M, R)^*$ is a Morse function on $ \mathfrak{M}_1 $ when composed with $ \pi^3$. When dim $H^3(M, R) = 2$, the result in particular implies $ \pi^3$ is one to one on each connected component of $ \mathfrak{M}$. Considering the first Pontryagin class $ p_1(M) \in H^4(M, R)$, we formulate a compactness conjecture on the set of $ G_2$ structures of volume($M$) $=1$ with bounded $L^2$ norm of curvature, which would imply that every connected component of $ \mathfrak{M}$ is contractible. We also observe the locus $ \pi^3(\mathfrak{M}_1) \subset H^3(M, R)$ is a hyperbolic affine sphere if the volume of the torus $ H^3(M, R) / H^3(M, Z)$ is constant on $ \mathfrak{M}_1$.