Topological Quantum Gravity through Harmonic S${}^2$ Maps

By virtue of harmonic maps on two-dimensional spheres (S${}^2$), a topological quantization in spacetime is proposed. The discrete character of all physical quantities follows naturally. A Schwarzschild black hole, non-black hole and wormhole based geometries are considered in which a quantum hair becomes effective. A thermometer or curvature-detecting device can record the macroscopic quantumness of spacetime.