RE: [obm-l] Conjectura de Poincare

["Leandro Lacorte Recova" <leandrorecova@xxxxxxx>]

Poincaré Conjecture 
  

In its original form, the Poincaré conjecture states that every simply
connected closed three-manifold is homeomorphic to the three-sphere (in a
topologist's sense) , where a three-sphere is simply a generalization of the
usual sphere to one dimension higher. More colloquially, the conjecture says
that the three-sphere is the only type of bounded three-dimensional space
possible that contains no holes. This conjecture was first proposed in 1904
by H. Poincaré  (Poincaré 1953, pp. 486 and 498), and subsequently
generalized to the conjecture that every compact n-manifold is
homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere.
The generalized statement reduces to the original conjecture for n = 3. 
Tirei do site: http://mathworld.wolfram.com/PoincareConjecture.html 

The Poincaré conjecture has proved a thorny problem ever since it was first
proposed, and its study has led not only to many false proofs, but also to a
deepening in the understanding of the topology of manifolds (Milnor). One of
the first incorrect proofs was due to Poincaré himself (1953, p. 370),
stated four years prior to formulation of his conjecture, and to which
Poincaré subsequently found a counterexample. In 1934, Whitehead (1962, pp.
21-50) proposed another incorrect proof, then discovered a counterexample
(the Whitehead link) to his own theorem. 

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is
classical (and was known to 19th century mathematicians), n = 3 (the
original conjecture) appears to have been proved by recent work by G.
Perelman (although the proof has not yet been fully verified), n = 4 was
proved by Freedman (1982) (for which he was awarded the 1986 Fields medal),
n = 5 was demonstrated by Zeeman (1961), n = 6 was established by Stallings
(1962), and  was shown by Smale in 1961 (although Smale subsequently
extended his proof to include all ). 

The Clay Mathematics Institute included the conjecture on its list of $1
million prize problems. In April 2002, M. J. Dunwoody produced a five-page
paper that purports to prove the conjecture. However, Dunwoody's manuscript
was quickly found to be fundamentally flawed (Weisstein 2002). A much more
promising result has been reported by Perelman (2002, 2003; Robinson 2003).
Perelman's work appears to establish a more general result known as the
Thurston's geometrization conjecture, from which the Poincaré conjecture
immediately follows (Weisstein 2003). Mathematicians familiar with
Perelman's work describe it as well thought-out and expect that it will be
difficult to locate any substantial mistakes (Robinson 2003, Collins 2004).
In fact, Collins (2004) goes so far as to state, "everyone expects [that]
Perelman's proof is correct."

-----Original Message-----
From: owner-obm-l@mat.puc-rio.br [mailto:owner-obm-l@mat.puc-rio.br] On
Behalf Of Nicolau C. Saldanha
Sent: Thursday, March 31, 2005 6:09 AM
To: obm-l@mat.puc-rio.br
Subject: Re: [obm-l] Conjectura de Poincare

On Mon, Mar 28, 2005 at 05:58:38PM -0300, Bruno Lima wrote:
> Pessoal, uma duvida minha, ha mais ou menos ums ano
> anunciaram por ai que um russo, acho que se chamava
> Perelman havia resolvido a Conjectura de Poincare,
> depois nao ouvi mais falar, afinal resolveu?? E o cara
> recebeu o 1 mi de dolares? Pois eu acho que esse era
> um dos problemas do intituto Clay.

Tanto quanto eu sei, a demonstração ainda está sendo verificada
pelos especialistas da área, mas a impressão geral é de que está
tudo certo. A demonstração usa análise pesada. E sim, este é
um dos problemas milionários.

[]s, N.
=========================================================================
Instruções para entrar na lista, sair da lista e usar a lista em
http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html
=========================================================================

=========================================================================
Instruções para entrar na lista, sair da lista e usar a lista em
http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html
=========================================================================