“In a particular approach to quantum gravity, Robert Geroch and James Hartle (1986) found themselves confronted with a computationally unsolvable problem, namely the topological equivalence problem for 4-manifolds. Basically, their approach involved the questions of deciding when two four-dimensional spaces are 'the same', from the topological point of view (i.e. when it is possible to deform one of them continuously until it coincides with the other, where the deformation does not allow tearing or gluing the spaces in any way). In Fig. 7.14, this is illustrated in the two-dimensional case, where we see that the surface of a ball is different. In two dimensions, the topological equivalence problem is computationally solvable, but it was shown by A.A. Markov in 1958 that there is not algorithm for solving this problem in the four-dimensional case. In fact, what is shown effectively demonstrates that if there were such an algorithm, then one could convert that algorithm into another algorithm which could solve the halting problem, i.e. it could decide whether or not a Turing-machine action will stop. Since, as we have seen in 2.5, there is no such algorithm, it follows that there cannot be any algorithm for solving the equivalence problem for 4-manifolds either.”