Talk:Hilbert's fifth problem

Redirect

Could someone redirect [[Hilbert's 5th problem}} here? —Preceding unsigned comment added by 90.191.102.163 (talk) 19:40, 6 November 2008 (UTC)

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 C^k-class actually leads to a real analytic (C^ω) 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.

(The second part of Hilbert's Fifth Problem is also concerned with groups, but is really quite different from the above.)Daqu (talk) 07:38, 1 April 2009 (UTC)

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 C^k gamut" is confusingly phrased.143.167.237.208 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 ):

1. 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.
2. 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.
3. 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.
4. Applications: descriptions of why the problem is important should be included.

Well, I hope this list will be of some help to someone (maybe even me, :D ) who wants to improve the entry on this interesting topic. Daniele.tampieri (talk) 20:19, 15 September 2011 (UTC)

textbook draft

http://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/

69.111.193.46 (talk) 08:41, 2 April 2012 (UTC)

Needs a much bettter description of the problem

The first sentence of the section Classical formulation is as follows:

"A formulation that was accepted for a long period was that the question was to characterize Lie groups as the topological groups that were also topological manifolds."

By describing the problem in such vague terms, this description avoids focusing on what the problem is.

The fact that the problem is restated later in the same section is no reason not to improve this first sentence.

I hope someone familiar with the problem can rewrite the sentence crisply and clearly so that the nature of the problem is clear. 2601:200:C000:1A0:6C9E:36DF:DC73:342E (talk) 01:26, 2 September 2021 (UTC)

Possible Error in the article

In the article it is said: If we consider a general locally compact group G and the connected component of the identity G₀, we have a group extension G₀ → G → G/G₀. As a totally disconnected group, G/G₀ has an open compact subgroup, and the pullback G′ of such an open compact subgroup is an open, almost connected subgroup of G. In this way, we have a smooth structure on G, since it is homeomorphic to (G′ × G′ )/G₀, where G′/G₀ is a discrete set.

And something is wrong here. If I take G to be the p-adic integers, then the connected component G₀ is just {0}. So, G/G₀ is just G. Since G/G₀ is compact, we can take G′ to be G as well. Now, the article claims that G′/G₀ is discrete, but that is not true, because that is again just G and G is not discrete (as it is infinite and compact). So, something is wrong here...

Or am I missing something? --2003:C9:7F2D:3097:D814:2968:9A79:E173 (talk) 09:58, 9 April 2022 (UTC)

I agree this cannot be correct. It is probably supposed to be a proof of the claim just above, that "any almost connected l.c. group is a projective limit of Lie groups", but the argument should use that $G/G_{0}$, being compact and totally disconnected, is a projective limit of finite groups and then construct Lie groups from $G_{0}$ and these finite groups such that $G$ is the projective limit of these. This does not seem completely obvious to me, and in any case there should be a proper reference there instead. jraimbau (talk) 12:51, 10 April 2022 (UTC)

I made some changes and added references to Tao's book. I think the article is still not in an acceptable state ; at least there should be a more complete discussion of the Montgomery--Zippin--Gleason solution in a section and more detailed comments on the Gleason--Yamabe results on lc groups (including examples) in another. jraimbau (talk) 15:49, 12 April 2022 (UTC)