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