Question

The common knowledge in this regard seems to be that it was completely solved in the 1950s by a few Americans. About a decade ago, Olver contested that in one of his books. Recently, Palais wrote about it in the Notices but he only treats the old story from the 1950s and seems not to be aware of Olver’s facts.

Answer 1

Recent work of Pardon appears to show (See http://arxiv.org/abs/1112.2324) that that $\mathbb{Z}_p$ does not act faithfully on a connected 3-manifold. His work uses "tools from low dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group"

Answer 2

The OP says:

" ...Recently, Palais wrote about it in the Notices but he only treats the old story from the 1950s and seems not to be aware of Olver’s facts."

Actually, I am aware of Olver's work and also Sören Illman's contribution.

Sören is an old friend and wrote to me somewhat miffed that I did not mention his work on the problem. What he proved was a very nice fact---that if a proper Lie group action is differentiable, then it can be made real analytic. As I pointed out in my article, there are very simple examples that Hilbert should have noticed (see my article---linked below---if you think I am being hard on Hilbert) that show that without properness this is false.

As for Olver, his contribution is also nice but a little off the beaten track. Here is a quick version. One facet of what Hilbert asked was whether a "local Lie group" (i.e., an open set in $R^n$ with a continuous group operation and inverse defined near the identity) could always be embedded in a global Lie group. Cartan anwered that in a way that suffices for all practical purposes; he showed that after cutting back the original neighborhood to a smaller one it could be embedded. However a number of people (including Malcev and Douady) showed that without cutting back the answer could be "no". Their examples were infinite dimensional however, and Olver in his paper "Non-Associative Local Lie Groups" provided a class of finite dimensional examples.

OK, so why didn't I mention the work of Illman, Olver and a host of others who worked on the Fifth Proble after the 1950s. If you look at my article, available for download here:

http://www.ams.org/notices/200910/rtx091001236p.pdf

the answer is clear. My article was part of a larger memorial artcle for Andy Gleason (my PhD advisor) and it was titled "Gleason's Contribution to the Solution of the Hilbert Fifth Problem". There was plenty to talk about there, and a discussion of other contributions to the Fifth Problem that happened decades later would have been out of place.

By the way, in regard to what is called "Route B" in an answer above, the first section of my article is titled "What IS Hilbert's Fifth Problem" in which I try to explain aat least a little bit about how and why Hilbert's original statement of the Fifth Problem morphed over time.

Answer 3

As mentioned by BS and wikipedia, there are two different formulations of Hilbert's fifth problem. One was solved by Gleason, Montgomery, and Zippin. The second more general conjecture, which is also called the Hilbert-Smith conjecture, is still open. My knowledge of this is nil, but my rather hazy recollection is that the remaining step is to prove that a p-adic group (or something like that) cannot act continuously and nontrivially on a manifold.

I do not know for sure but I suspect that Montgomery and my father (C. T. Yang) worked on this for a long time without success. And indeed in the late 70's and early 80's, McAuley claimed to have a proof and distributed a manuscript. Actually, he distributed many manuscripts, because he would find gaps in his own proof and distribute corrected versions. As far as I know, McAuley's claim was never taken seriously.

Answer 4

http://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem is a decent survey. In general in the discussion of "status" of the Hilbert problems, there are at least two recognisable routes.

Route A is the more natural for contemporary mathematicians. Roughly speaking it equates with asking first for the version of Hilbert Problem N that has entered mathematical folklore (the tea-room version if you like, "Hilbert N asked if [add translation into contemporary jargon]"), and then giving the update on that $folklore$ version.

Route B involves reading what Hilbert actually wrote in German, comparing with accepted English translations, discussing ambiguities and parsing out the issues where Hilbert deliberately made open-ended remarks. In other words Route B treats the problem set as a historical document, and allows for a degree of quibbling and/or interpretative queries.

The reason these routes don't always give the same answer should be relatively obvious once they are formulated this way. But it is worth making the further point, given the tone of various MO discussions, that 1900 is quite close to the cusp at which "discursive mathematics" gives way to "formal axiomatic mathematics". Also (no one express shock, please) Hilbert did not have a definition of topological space, let alone Lie group. The issues here cannot be resolved by saying that without definitions he had no right to pose problems!

There are quite a number of the problems where the accepted Route A box-ticking answer has been queried. Some of these are worth further questions on MO.

Edit: I think problems 12, 15 and 21 are others among those problems where there is a worthwhile and clarifying debate about the status.