Ben, Takahiro - slowly I got into this subject and agree with your points. Black hole singularity violates homeomorphism. Connection between black holes and quantum mechanics becomes more interesting if to consider virtual black holes pointed out by Hawking: en.m.wikipedia.org/wiki/Virtual_black_hole

I did not assume that. However, intuitively I see that homeomorphic transformation should preserve all conservation laws - energy, momentum, space, rotation etc.

I belong to high energy physics and, therefore, you can consider me as a vulgar mathematician. However, want to ask two questions: did we have 1) quantum mechanics and 2) black holes at the state of T=0 (BigBang) ?