From: baez@math.ucr.edu (John Baez)
Subject: Re: Poincare Conjecture
Date: 9 Nov 1999 08:00:06 -0600
Newsgroups: sci.math.research
Keywords: Classification algorithms in topology -- Poincare conj. undecidable?

In article <VJgkOHZh7Yeze0ail4J+UCbN64gl@4ax.com>,
Stephen Paul King  <stephenk1@home.com> wrote:

>I have a question about the Poincare Conjecture.  Is it possible that
>the classification of 3-manifolds is undecidable in a Goedel's
>Incompleteness sense?

As far as I know this is an open question, bracketed by the following
results: 

1) it was shown by Markov that there is no algorithm to classify 
compact 4-manifolds

2) it was shown by Thurston and Haken that there is an algorithm
to classify knots.

Using the ideas behind the proof of 1), I think you can cook up pairs 
of 4-manifolds for which it is undecidable in ZF set theory whether
they are homeomorphic or not.  

I don't know if there is an algorithm to classify links.  

Since the 4-dimensional version of Poincare's conjecture was only
proved by Freedman in the 1980s, I don't think it's time to give up 
hope on the 3d version just yet.  (All the dimensions above 4 were
done earlier by Smale.)