To: WSFAlist at keithlynch.net
Date: Tue, 20 Jan 2004 01:28:29 -0500
Subject: [WSFA] a 100-year old math problem
From: ronkean at juno.com
Reply-To: WSFA members <WSFAlist at keithlynch.net>

It has been reported that a publicity-shy Russian mathematician may have
now proven the 1904 Poincaré Conjecture.

The question I have is that, since the Conjecture concerns a 'sphere'
with a 3 dimensional surface embedded in 4 dimensional space (and higher
dimensional analogs), how could the Conjecture have any practical value,
considering that the real world seems to be one of only 3 dimensional
space?  Might the Conjecture have some application in cosmology?

More information:

http://mathworld.wolfram.com/news/2003-04-15/poincare/

Excerpt:
In the form originally proposed by Henri Poincaré in 1904 (Poincaré 1953,
pp. 486 and 498), Poincaré's conjecture stated that every closed simply
connected three-manifold is homeomorphic to the three-sphere. Here, the
three-sphere (in a topologist's sense) is simply a generalization of the
familiar two-dimensional sphere (i.e., the sphere embedded in usual
three-dimensional space and having a two-dimensional surface) to one
dimension higher. More colloquially, Poincaré conjectured that the
three-sphere is the only possible type of bounded three-dimensional space
that contains no holes. This conjecture was subsequently generalized to
the conjecture that every compact n-manifold is homotopy-equivalent to
the n-sphere if and only if it is homeomorphic to the n-sphere. The
generalized statement is now known as the Poincaré conjecture, and it
reduces to the original conjecture for n = 3.
The n = 1 case of the generalized conjecture is trivial, the n = 2 case
is classical (and was known even to 19th century mathematicians), n = 3
has remained open up until now, n = 4 was proved by Freedman in 1982 (for
which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman
in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was
established by Smale in 1961 (although Smale subsequently extended his
proof to include all n >= 5).

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