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."

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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 space-time. -- ---------------------------------------------------------------------------- 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 ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.