Classifying manifolds is impossible.

 February 12, 2007 mathematics

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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.

There is a paper [1] discussing some of these issues. In particular, that paper discusses Novikov’s proof that S^n cannot be recognized for n \geq 5 .

[1] A.V. Chernavsky, V.P. Leksine, Unrecognizability of manifolds, Ann. Pure Appl. Logic. 141 (2006) 325–335.