From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Please Help: Poincare Sphere
Date: 25 Jul 1996 16:27:26 GMT
In article <4t4vjc$t12@nuscc.nus.sg>,
Wei-Choon Ng wrote:
> Has anyone heard of Poincare Sphere? What is it, some kind of vector
>space? Could someone please enlighten me. Thanks. Please email me direct.
Hoping to understand higher-dimensional manifolds as well as surfaces
(2-dimensional manifolds) are, Poincare asked if the homology groups were
sufficient to distinguish them in general, that is, if two manifolds
M and N are given and H_*(M) = H_*(N), are M and N homeomorphic?
(Perhaps he expected them even to be diffeomorphic; that certainly fails.)
One consequence of this conjecture is that a compact n-dimensional
manifold with all homology groups being zero ( 0 < * < n ) would have
to be the n-sphere. In particular (thanks to Poincare duality) a
3-dimensional compact manifold with trivial first homology would have
to be the 3-sphere.
Poincare himself found a counterexample, now called the Poincare sphere.
There is a group G of 120 rotations of R^4 which acts without fixed
points on the unit sphere S^3. Consequently, M = S^3 / G is a
3-dimensional compact manifold. It's not homeomorphic to S^3 since
it has a non-trivial fundamental group G. On the other hand, its
first homology group _is_ zero, since in this construction, H_1(M)
will be the abelianization G/[G,G] of the group G, and G happens
to be perfect (G = [G,G]).
In response to this example, Poincare amended his question to ask, if
M is a compact n-dimensional _simply-connected_ manifold with no
non-trivial homology, is it the n-sphere? Clearly any attempt at
classifying n-dimensional manifolds would have to be able to answer
this question, so this seems a natural focus to people's attention.
Considerable progress has been made, but for n=3, the problem is
still open. Arguably this is one of the most important open questions in
mathematics.
dave
posted and emailed as requested