Given two manifolds — smooth, well-defined geometric objects — can you always determine whether they're topologically the same? In dimensions 1 and 2, yes: curves and surfaces can be classified completely. In dimension 3, Perelman's proof of the Poincaré conjecture implies that the homeomorphism problem is decidable, though the algorithm is impractical. The question is what happens above dimension 3.
Friedl, Hirsch, and Kegel (arXiv:2603.23630) give a detailed proof of Markov's theorem: in all dimensions greater than 3, the homeomorphism problem is unsolvable. No algorithm can determine, for arbitrary pairs of manifolds, whether they are topologically equivalent. Moreover, unrecognizable manifolds exist in every dimension above 3 — manifolds for which no algorithm can determine whether a given manifold is homeomorphic to them.
The proof works by encoding the word problem for groups into the topology of manifolds. Every finitely presented group can be realized as the fundamental group of a manifold in dimension 4 or higher. Since the word problem is undecidable for certain groups (by the Adian-Rabin theorem), and since homeomorphic manifolds have isomorphic fundamental groups, distinguishing manifolds requires solving the word problem — which can't always be done.
The result means topology in high dimensions is fundamentally different from topology in low dimensions, not just harder but logically impossible to complete. In dimensions 1-3, the classification of shapes is finite and achievable. Above 3, no finite procedure suffices. The boundary isn't gradual — it's a sharp threshold at dimension 4 where algebra becomes undecidable and topology inherits the impossibility.
The structural point is that manifolds in high dimensions carry enough algebraic complexity to encode computation, and any classification problem that can encode arbitrary computation is vulnerable to undecidability. Low-dimensional topology is special not because it's simpler, but because it can't encode enough algebra to become undecidable.