The quantization of topology, from quantum Hall effect to quantum gravity

It is the goal of this article to extend the notion of quantization from the standard interpretation focused on non-commuting observables defined starting from classical analogues, to the topological equivalents defined in terms of coefficient groups in (co)homology. It is shown that the commutation relations between quantum observables become (non)compatibility relations between coefficient groups. The main result is the construction of a new, higher-level form of quantization, as seen from the perspective of the universal coefficient theorem. This idea brings us closer to a consistent quantization of gravity, allows for a systematic description of topology changing string interactions but also gives new, quantum-topological degrees of freedom in discussions involving quantum information. On the practical side, a possible connection to the fractional quantum Hall effect is explored.