At a recent Pizza Seminar, Matt Day gave a lovely talk explaining why it isn’t possible to classify 4-manifolds.

An algorithm for deciding whether two closed 4-manifolds are homeomorphic gives an algorithm for deciding whether a closed 4-manifold is simply connected, and therefore (since every finitely presented group is the fundamental group of a 4-manifold), and algorithm for deciding when a group is trivial. Here’s the reduction: we are given a 4-manifold , and we compute its signature . By Freedman, there are no more than two closed simply connected 4-manifolds, and , having the same signature as ; we construct and , and we use the homeomorphism decision procedure to test if or .

Since there is no algorithm for deciding when a group is trivial, there can not be an algorithm for deciding when two closed 4-manifolds are homeomorphic.

Here is a paper discussing some of these issues:

Chernavsky, A. V.; Leksine, V. P. Unrecognizability of manifolds. Ann. Pure Appl. Logic 141 (2006), no. 3, 325—335.

In particular, that paper discusses Novikov’s proof that cannot be recognized for .