The authors of the Rainich-Misner-Wheeler theory no longer believe that everything physical can be accounted for in terms of the topological-geometrical structure of ordinary spacetime. However, many physicists and philosophers entertain the possibility that a geometrodynamics (a theory which accounts for sources as well as fields in terms of topological-geometrical structure) may be feasible in the context of a more general topology. In this dissertation I consider two topological-geometrical models (based upon a single suggestive formalism) in which a geometrodynamics is both feasible and pedagogically advantageous. Specifically I consider the topology which is constituted by the real domains of the two broad classes of rotation groups: those characterized by the commutator and anti-commutator algebras. I then adopt a Riemannian geometric structure and show that the monistically geometric interpretation of this formalism restricts displacements on the proposed manifold to integral multiples of a universal constant. Secondly I demonstrate that in the context under consideration, this constraint affects a very interesting ontological reduction: the unification of quantum mechanics with a discrete, multidimensional extension of general relativity. A particularly interesting feature of this unification is that it includes and (for the world which is characterized by energy levels which range in magnitude from low to intermediately high) requires the choice cf an SL(2,R)xSU(3)-symmetric realization of the proposed, generic formalism which is a lattice of spins π and π/2. (This is in the context of the same universally constant scale factor as that which yields the quantization conditions described above.) If the vertices of this lattice are associated with the fundamental particles, then the resulting theory predicts and precludes the same interactions as the standard supersymmetry theory. In addition to the ontological reduction which is provided, and the restriction to supersymmetry, the proposed theory may also represent a scientifically useful extension of conventional theory in that it suggests a means of understanding the apparently large energy productions of the quasars and relates Planck's constant to the size of the universe.