  Motivated by the description of $\mathcal{N}=1$ M-theory compactifications to
four-dimensions given by Exceptional Generalized Geometry, we propose a way to
geometrize the M-theory fluxes by appropriately relating the compactification
space to a higher-dimensional manifold equipped with a torsion-free structure.
As a non-trivial example of this proposal, we construct a bijection from the
set of $Spin(7)$-structures on an eight-dimensional $S^{1}$-bundle to the set
of $G_{2}$-structures on the base space, fully characterizing the
$G_{2}$-torsion clases when the total space is equipped with a torsion-free
$Spin(7)$-structure. Finally, we elaborate on how the higher-dimensional
manifold and its moduli space of torsion-free structures can be used to obtain
information about the moduli space of M-theory compactifications.
