On Sat, May 19, 2012 at 01:17:18PM -0400, Stephen P. King wrote:
>  Dear Bruno,
>     I finally found a good and accessible paper 
> <http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&sqi=2&ved=0CEoQFjAA&url=http%3A%2F%2Fntrs.nasa.gov%2Farchive%2Fnasa%2Fcasi.ntrs.nasa.gov%2F20050243612_2005246604.pdf&ei=8NO3T9LmFu-d6AHAq_3uCg&usg=AFQjCNHMBmwAi1K7yJY3oRBnJrYRC2H9RA&sig2=yb-YNcKWR6LNPSVy8bQquA>
> that discusses my bone of contention. To quote from it:
> "A  theorem  proved by Markov  on  the  non-classifiability  of  the
> 4-manifolds  implies
> that, given  some comprehensive specification  for  the  topology
> of  a manifold  (such  as
> its triangulation,  a  la  Regge  calculus,  or  instructions  for
> constructing  it  via  cutting
> and  gluing  simpler  spaces) _there  exists  no  general  algorithm
> to  decide  whether  the
> manifold is homeomorphic to some other manifold _ [l].  The
> impossibility of  classifying
> the  4-manifolds is  a well-known  topological result,  the proof of
> which,  however,  may
> not  be  well known  in  the  physics  community.  It  is
> potentially  a  result  of  profound
> physical  implications,  as  the  universe  certainly  appears  to
> be  a manifold  of  at  least
> four  dimensions."

Funnily enough, I remember from the dim-distant undergraduate days,
that the classifiability of 3 and 4-manifolds were open problems. 1 &
2-manifolds had known classifications (2-manifolds are classified by
the number of "holes" (aka genus), for instance). Manifolds of
dimension higher than 4 are known to be unclassifiable. So a result
that 4-manifolds are unclassifiable would be a significant topological
result. What's suspicious is the claim that this was proved in
1960. Also suspicious in light of the Wikipedia entry claiming the
problem is still open: http://en.wikipedia.org/wiki/4-manifold

Conversely, as for the 3-manifold problem, this looks it might have been
solved by Perelman's work that also solved the more famous Poincare
conjecture in 2003. If there's anybody about that more knowledgeable on these
matters, please comment.

I remember there was something peculiar about 4-dimensional space that
wasn't true of any other dimension - unfortunately, the sands of time
have erased the details from my memory. But I remember people were
speculating that it was a possible reason for why we lived in 4D


Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au

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