# Classifying manifolds is impossible.

##### February 12, 2007 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 , 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.

There is a paper [1] discussing some of these issues. In particular, that paper discusses Novikov’s proof that cannot be recognized for .

[1] A.V. Chernavsky, V.P. Leksine, Unrecognizability of manifolds, Ann. Pure Appl. Logic. 141 (2006) 325–335. https://doi.org/10.1016/j.apal.2005.12.011.