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 $H^4(M,R)=H^3(M,R{)}^{\ast }$ 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)$,weformulateacompactnessconjectureonthesetof$ 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 $H^3(M,R)\mathrm{/}{H}^3(M,Z)$ is constant on ${\mathfrak{M}}_1$.