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Re: Re: I broke the sum into pieces

  • To: mathgroup at smc.vnet.net
  • Subject: [mg105174] Re: [mg105149] Re: I broke the sum into pieces
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 23 Nov 2009 06:52:47 -0500 (EST)
  • References: <4B0551F8.1050308@sbcglobal.net> <144987c90911190624s3f42b7e7g6545e37062631d1f@mail.gmail.com> <200911221111.GAA10575@smc.vnet.net>

On 22 Nov 2009, at 20:11, Roger Bagula wrote:

> he Poincare conjecture was proved so that all the 3d manifolds reduce
> to a circle
> ( hyperbolic included), so up to 4 dimensions 3 d manifolds Pi should
> be
> the same constant in our universe.
> I think that conjecture was already proved true in higher dimensions.


This is a fantastic mi-statement of the Poincare conjecture and its 
solutions. It is absolutely not true that all 3d manifolds "reduce to a 
circle" in any sense whatever. This is even less true in higher 
dimensions!

The original Poincare conjecture in dimensions 3 states that a simply 
connected manifold which is a homology sphere is homeomorphic to the 
standard sphere. The 3-d case is rather special since any simply 
connected closed manifold always is a homology sphere so the conjecture 
can be stated in the form: every simply connected closed 3-manifold is 
homeomorphic to a sphere. This however certainly does not mean that all 
3 manifolds "reduce" to a sphere (not a circle) as there are infinitely 
many non-simply connected ones!

The 3 dimensional Poincare conjecture was proved several years ago by G. 
Perelman, building on ideas introduced by Richard Hamilton.

In higher dimensions the conjecture is even more different from what was 
implied in this post. It states that a homotopy n-sphere is homomorphic 
to an n-sphere. The condition that something is a homotopy n-sphere is a 
very strong one - relatively few manifolds are homotopy spheres! To say 
that this somehow implies that *all* manifolds "reduce" to spheres is a 
comical travesty.

Just for completeness: the Poincare conjecture in dimensions >=5  was 
proved in 1961 by Stephen Smale. The Poincare conjecture in dimension 4 
was proved in 1982 by Michael Freedman. He actually only proved the so 
called topological Poincare conjecture - the smooth version still 
remains unproved. (Dimenion 4 is thought to be very special. The four 
dimensional Euclidean space is the only one in which the topological 
structure does not determine the smooth structure - in other words, 
there exist exotic smooth structures on R^4).

Smale, Freedman and Perelman received Fields medals for their solutions 
but Perelman refused to accept his.

Andrzej Kozlowski



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