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It is the goal of this article to extend the notion of quantization from thestandard interpretation focused on non-commuting observables defined startingfrom classical analogues, to the topological equivalents defined in terms ofcoefficient groups in (co)homology. It is shown that the commutation relationsbetween quantum observables become (non)compatibility relations betweencoefficient groups. The main result is the construction of a new, higher-levelform of quantization, as seen from the perspective of the universal coefficienttheorem. This idea brings us closer to a consistent quantization of gravity,allows for a systematic description of topology changing string interactionsbut also gives new, quantum-topological degrees of freedom in discussionsinvolving quantum information. On the practical side, a possible connection tothe fractional quantum Hall effect is explored.

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