A small nit. If two finitely presented groups are isomorphic, this fact can be verified in finite time. [Potentially huge time, Busy Beaver type numbers.] This is a partial decision procedure, solving one side of the problem. The unsolvable side is when the two groups are not isomorphic, but we can never confirm it. [A finitely presented group is the usual generator & relations style: [ a,b; a^2 = 1, b^3 = 1 ] with a finite number of generators and a finite number of finite relations.] Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Daniel Asimov Sent: Mon 8/21/2006 1:03 AM To: math-fun Subject: Re: [math-fun] Poincare Conjecture & Perelman Now that the Poincare conjecture appears to be solidly proven, I'd like to see the known corollaries. Over the years a number of theorems have proved "Either there's a counterexample to the Poincare conjecture, or else something very strange happens in 4 or 5 dimensions" (with the something very strange being specified). Very weird stuff is already known to happen in 4 dimensions. E.g., it's the unique dimension whose Euclidean space R^4 carries more than one differentiable structure, up to equivalence (diffeomorphism). There is a continuum of them! Also, given any finitely-presented group G, there is a 4-dimensional manifold fundamental group M whose fundamental group pi_1(M) is isomorphic to G (using topological operations that mimic the presentation using glueing of 1-handles & 2-handles, then capping it off with a conractible 4-manifold with the same boundary -- I think). Since a theorem of Markov says there is no algorithm which, given two arbitrary finite group presentations, can determine if they define isomorphic groups, it follows that there is no algorithm to classify all (even compact) 4-manifolds up to homeomorphism. (Okay, okay, this is also true in dimensions > 4, but 4 is the lowest case where it happens.) --Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun