Morse functions on the moduli space of $G_2$ structures

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 $\mathfrak{M}_1$.