Empirical evidence of quantization is found in experiments demonstrating the Aharonov-Bohm and integer and fractional Quantum Hall effects. In the associated ongoing open areas of research there have been numerous attempts to explain the observed nature of such quantization. Of particular note, and one motivation for the topological concepts of space-time addressed here, is the occurrence of certain sequences of plateaus in fractional Quantum Hall results, represented by positive integer multiples of quantum units where nature selects certain integers as multipliers of fundamental quantum measures of electrical charge and magnetic flux. The micro-origins of such selections are unknown. Our recently deceased colleague, Evert Jan Post, espoused a universal view of integer and fractional QH impedance characterized by the ratio of period integrals for flux and charge, leading to a ratio of the corresponding quantum integers, often referred to as filling factors. Our main purpose in the present article is to build upon previous topological results toward the ultimate goal of accommodating singularities in a space-time Riemannian manifold, representing the experimentally observed specific sequences of integers and fractions as an extension of the familiar manoeuvres such as the residue theorem of Cauchy in complex analysis, or, in a more general topological setting of exterior calculus, Hodge-de Rham cohomology, and the Mittag-Leffler theorems. It is our ultimate intention to shed light on the nature of particles and space by examining such singular features through extensions of classical singularity theory to the space-time pseudo-Riemannian manifold.

quantization, Hodge-de Rham theory, generalized Mittag-Leffler’s theorems.