Classifying manifolds is impossible.
2007-02-12 16:35:45 +0000 mathematics
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 $M$, and we compute its signature $\sigma(M)$. By Freedman, there are no more than two closed simply connected 4-manifolds, $M_1$ and $M_2$, having the same signature as $M$; we construct $M_1$ and $M_2$, and we use the homeomorphism decision procedure to test if $M \cong M_1$ or $M \cong M_2$.
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:
MR2234702
In particular, that paper discusses Novikov’s proof that $S^n$ cannot be recognized for $n \geq 5$.