The following considerations relate to the quantum gravity programs currently being developed.

0.1 Conjectures on Black Hole Symmetry and Structure

• 1. A black hole–spinning or otherwise– can be considered as a tightly coupled boson system, and thus a quantum spin group system; therefore, it is unlikely that it would be chaotic. instead, it is predicted to be organized as some kind of spin foam that would exhibit ’extremely slow’ fluctuations related to the energy radiation leaks near the black hole horizon.
• 2. A graded Lie groupoid, G_Lg, may provide a mathematical representation of the black hole gravitational, quantized field symmetry. (Both the precise concept of a G_Lg and that of quantized gravitational fields are available at PlanetPhysics).
• 3. Instead of ”clouds” of probability one may wish to consider transition probability distributions for tightly coupled spin foams within the black hole.
• 4. Whereas space-time point topology is indeed a problem for black holes, a CW-complex non-discrete topology remains a possibility nicely represented by the spin foams that do form a CW-complex associated with the black hole. The CW-complex topology is consistent with both the G_Lgsymmetry of the quantized gravitational fields and the associated spin foams.
• 5. Instead of the ’standard’ time in QM, one would may introduce for the region inside the horizon of the black hole a quantum superoperator associated with the time observable (as our quantum group has done in a few recent publications, echoing Prigogine’s published work on quantum superoperators).

Remarks.

The black hole structure and symmetry are difficult, challenging problems that are at the cutting edge and intersections of both physics and algebraic topology.

Apparently, the question of the dimensions of a black hole is unanswered so far by M-theory.