The world problem: on the computability of the topology of 4-manifolds

Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing machine with an arbitrary input can be encoded into the topology of a 4-manifold, such that the 4-manifold is homeomorphic to a certain other 4-manifold if and only if the corresponding Turing machine halts on the associated input. Physical implications are briefly discussed.