Talk:Hilbert's fifth problem
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This page doesn't actually state what the final resolution was (that group objects in the category of topological manifolds and actually Lie groups in a unique way). Is it worth mentioning that the assumption of any Ck-class actually leads to a real analytic (Cω) structure. I assume this was known prior to Hilbert's question (at least for k ≥ 2). -- Fropuff 16:19, 2 November 2005 (UTC)
- I have gone down with repetitive strain injury in one hand - the Devil's way of telling you about the amount of time you spend on Wikipedia. Yes; but I've got the big Soviet encyclopedia open now, and it says the same things, really. A sharper statement: any locally compact group and any neighbourhood of e in it contains an open set K × L where K is a compact subgroup and L a local Lie group. This gives Hilbert 5 when combined with 'no small subgroups'. Charles Matthews 16:45, 2 November 2005 (UTC)
- The first part of Hilbert's Fifth Problem is entirely concerned with the existence of that real analytic structure!!! As the article completely neglects to mention. (It is a truly terrible article.)
- The first part of Hilbert's Fifth Problem is the following: Given a locally Euclidean topological group G and a locally Euclidean topological space M, with a continuous group action f: G x M -> M (Note: a "group action" is a mapping taking (g,x) in G x M to an element f(g,x) we'll call gx of M, with the property that for any g and h of G, and any x of M, then we have g(hx) = (gh)x.) . . . then the question is whether one can always choose local coordinates for G and M such that G and M are real analytic manifolds, and such that f is a real analytic mapping.
- This is not true in full generality, and the details of exactly when it is true are not yet fully known.
- But the part of this question that was solved by the work of Andrew Gleason combined with the joint work of Deane Montgomery and Leo Zippin is this: Suppose that, in the above, the space M is the same as G -- and also let the group action f: G x G -> G be simply the group multiplication in G. Then the same question: Can we always choose real analytic coordinates, for G, such that f is also real analytic?
- Or put more simply: If G is a topological manifold that is a topological group, then does it have a real analytic structure so that the manifold G and its group multiplication are real analytic?
- The answer, according to the combined work of Gleason, and Montgomery & Zippin, is Yes. And as Fropuff said, the real analytic structure is essentially unique. Both contributions were published in 1952.
recent addition of WAREL
Well finally WAREL has added something with a reference that is capable of being followed up. However I note that his/her text is copied directly from here, and the reference looks to be copied character-for-character from here. I am not knowledgeable in this area, perhaps someone with more background can clarify the relation of Yamabe's work with the material already in the article. Dmharvey 04:13, 8 March 2006 (UTC)
That statement "the group axioms collapse the whole Ck gamut" is confusingly phrased.126.96.36.199 10:41, 21 June 2006 (UTC)
On the state of this entry.
This entry's content is really messy, reflecting perhaps the present state of the theory around the topic dealt. However, since I cannot work on it now, I decided to collect here some advice for the ones who can (and eventually for me, in the not so next future. :D ):
- A more elementary definition of the problem in the introduction: without emphasizing its importance in theoretical physics, geometry and other branches of mathematical and natural sciences, the essence of the problem should be exposed in the most simple (however confusing) terms, closely to Hilber's original formulation.
- The historical approach: a chronology of all contribution and claims should be created, with precise references and notes, starting from Hilbert, Von Neumann, Pontryagin, Montgomery and Zippin, Yamabe and Rosinger. A differences in the formulation of the problem should be sketched in this section: also it would be important to consider survey papers, such as the ones of C.T. Chang in the noted book on Hilbert's problems published by the American Mathematical Society, and the one of J. Hirschfeld in the Transactions of the AMS.
- The "Formal definition" section: this section should be as precise as possible, presenting all the approaches sketched in the historical section in a formally precise Definition-Theorem way. Obviously, proofs should be avoided as lengthy and not trivial, but maybe a sketch of them should be included.
- Applications: descriptions of why the problem is important should be included.