  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$.
