  Hard disks systems are often considered as prototypes for simple fluids. In a
statistical mechanics context, the hard disk configuration space is generally
quotiented by the action of various symmetry groups. The changes in the
topological and geometric properties of the configuration spaces effected by
such quotient maps are studied for small numbers of disks on a square and
hexagonal torus. A metric is defined on the configuration space and the various
quotient spaces that respects the desired symmetries. This is used to construct
explicit triangulations of the configuration spaces as $\alpha$-complexes.
Critical points in a configuration space are associated with changes in the
topology as a function of disk radius, are conjectured to be related to the
configurational entropy of glassy systems, and could reveal the origins of
phase transitions in other systems. The number and topological and geometric
properties of the critical points are found to depend on the symmetries by
which the configuration space is quotiented.
