Celebratio Mathematica

As Hil­bert stated it in his lec­ture de­livered be­fore the In­ter­na­tion­al Con­gress of Math­em­aticians in Par­is in 1900 [e2], the Fifth Prob­lem is linked to Sophus Lie’s the­ory of trans­form­a­tion groups [e1], i.e., Lie groups act­ing as groups of trans­form­a­tions on man­i­folds. The “groups” that Lie dealt with were really just neigh­bor­hoods of the iden­tity in what we now call a Lie group, and his group ac­tions were defined only loc­ally, but we will ig­nore such loc­al versus glob­al con­sid­er­a­tions in what fol­lows. However, it was cru­cial to the tech­niques that Lie used that his man­i­folds should be ana­lyt­ic and that both the group law and the func­tions de­fin­ing the ac­tion of the group on the man­i­fold should be ana­lyt­ic, that is, giv­en by con­ver­gent power series. For Lie, who ap­plied his the­ory to such things as study­ing the sym­met­ries of dif­fer­en­tial equa­tions, the ana­lyti­city as­sump­tions were nat­ur­al enough. But Hil­bert wanted to use Lie’s the­ory as part of his lo­gic­al found­a­tions of geo­metry, and for this pur­pose Hil­bert felt that ana­lyti­city was un­nat­ur­al and per­haps su­per­flu­ous. So Hil­bert asked if ana­lyti­city could be dropped in fa­vor of mere con­tinu­ity. More pre­cisely, if one only as­sumed a pri­ori that the group $G$ was a loc­ally Eu­c­lidean to­po­lo­gic­al group, that the man­i­fold $M$ was a to­po­lo­gic­al man­i­fold, and that the ac­tion of $G$ on $M$ was con­tinu­ous, could one nev­er­the­less al­ways choose loc­al co­ordin­ates in $G$ and $M$ so that both the group op­er­a­tions and the ac­tion be­came ana­lyt­ic when ex­pressed in these co­ordin­ates? We shall speak of the prob­lem in this gen­er­al­ity as the un­res­tric­ted Hil­bert Fifth Prob­lem. The re­stric­ted prob­lem is the im­port­ant spe­cial case in which $G=M$ and the ac­tion is left trans­la­tion. Ask­ing wheth­er we can al­ways find ana­lyt­ic co­ordin­ates in the re­stric­ted prob­lem is clearly the same as ask­ing wheth­er a loc­ally Eu­c­lidean group is ne­ces­sar­ily a Lie group.

It turned out that there are many — and in fact many dif­fer­ent kinds of — counter­examples to the un­res­tric­ted Hil­bert Fifth Prob­lem. Per­haps the first pub­lished counter­example, due to R. H. Bing [e18], is an ac­tion of ${\mathbf{Z}}^2$ on ${\mathbf{S}}^3$ whose fixed-point set is the Al­ex­an­der Horned Sphere $\mathrm{\Sigma }$. Now $\mathrm{\Sigma }$ is not “tamely em­bed­ded” in ${\mathbf{S}}^3$, mean­ing that there are points where it is im­possible to choose co­ordin­ates so that loc­ally $\mathrm{\Sigma }$ looks like the usu­al em­bed­ding of ${\mathbf{R}}^2$ in ${\mathbf{R}}^3$. If the ac­tion were even dif­fer­en­ti­able in some suit­able co­ordin­ates, then it is easy to see that the fixed-point set would in fact be tamely em­bed­ded. (For an even more bizarre type of counter­example, re­call that in 1960 M. Ker­vaire [e24] con­struc­ted a to­po­lo­gic­al man­i­fold that did not ad­mit any dif­fer­en­ti­able struc­ture, provid­ing what can be con­sidered a counter­example even for the case when $G$ is the trivi­al group.)

One could make a case that these ex­amples are “mon­sters” that could have been ruled out if Hil­bert had phrased his state­ment of the Fifth Prob­lem more care­fully. But there is a more ser­i­ous kind of counter­example that is so ele­ment­ary that it makes one won­der how much thought Hil­bert had giv­en to the Fifth Prob­lem be­fore pro­pos­ing it. Here is a par­tic­u­larly ele­ment­ary ex­ample, due to Mont­gomery and Zip­pin [e22], with $G=\mathbf{R}$, the ad­dit­ive group of the real num­bers, and $M=\mathbf{C}$, the com­plex plane. Let $f$ be a con­tinu­ous real-val­ued func­tion defined on the pos­it­ive real ax­is, and define the ac­tion $\phi :\mathbf{R}\times \mathbf{C}\to \mathbf{C}$ by $\phi (t,r{e}^{i\theta }):=r{e}^{i(\theta +f(r)t)}$ . (In words, $\phi$ is a one-para­met­er group of homeo­morph­isms of the plane that ro­tates each circle centered at the ori­gin in­to it­self, the circle of ra­di­us $r$ be­ing ro­tated with an­gu­lar ve­lo­city $f(r)$.) If we choose $f(r)$ to equal 1 for $r\le 1$ and 0 for $r\ge 2$, the ac­tion is the stand­ard one-para­met­er group of ro­ta­tions of $\mathbf{C}$ in­side the unit disk and is trivi­al out­side the disk of ra­di­us 2, so by the Prin­ciple of Ana­lyt­ic Con­tinu­ation, this ac­tion can­not be made ana­lyt­ic in any co­ordin­ate sys­tem. What is worse, we can choose $f$ to have these prop­er­ties and also be smooth (mean­ing $C^{\mathrm{\infty }}$), so we see that even if we as­sume a pri­ori that the ac­tion of a Lie Group on a man­i­fold is smooth, it does not fol­low that it can be made ana­lyt­ic!

After these counter­examples to the un­res­tric­ted Hil­bert Fifth Prob­lem be­came known, a ta­cit un­der­stand­ing grew up to in­ter­pret “the Fifth Prob­lem” as re­fer­ring to the re­stric­ted ver­sion: Is every loc­ally Eu­c­lidean group a Lie group? and we shall fol­low this con­ven­tion be­low.

It was fairly easy to settle the one-di­men­sion­al case. The only (para­com­pact) con­nec­ted man­i­folds of di­men­sion one are the real line, $\mathbf{R}$, and the circle, ${\mathbf{S}}^1$, and both of course are Lie groups. In 1909 L. E. J. Brouwer [e3] showed that a to­po­lo­gic­al group that is homeo­morph­ic to either of these is in fact iso­morph­ic to it as a to­po­lo­gic­al group. Us­ing res­ults from Brouwer’s pa­per, B. Kerékjártó [e6] settled the two-di­men­sion­al case in 1931. There seems to have been little if any pub­lished work on the Fifth Prob­lem between the pa­pers of Brouwer and Kerékjártó, but that is not too sur­pris­ing con­sid­er­ing that much of the mod­ern math­em­at­ic­al in­fra­struc­ture re­quired for a rig­or­ous dis­cus­sion of to­po­lo­gic­al groups and the Fifth Prob­lem be­came avail­able only after a 1926 pa­per by O. Schreier [e4]. The three-di­men­sion­al and four-di­men­sion­al cases of the Fifth Prob­lem were settled much later, by Mont­gomery [e12] in 1948 and by Mont­gomery and Zip­pin [e16] in 1952.

The first ma­jor break­through in the gen­er­al the­ory came in 1933, when J. von Neu­mann [e8], us­ing the re­cently dis­covered Haar [e7] meas­ure, ex­ten­ded the Peter–Weyl The­or­em [e5] to gen­er­al com­pact groups and used it to settle the Fifth Prob­lem in the af­firm­at­ive for com­pact groups. We will sketch a proof of von Neu­mann’s the­or­em be­low. Sev­er­al years later, build­ing on von Neu­mann’s work, Pontry­agin [e9] settled the abeli­an case, and Che­val­ley [e10] the solv­able case.

The first time I en­countered the phrase “group without small sub­groups” I wondered what kind of sub­group a “small” one could pos­sibly be. Of course, what the phrase means is a to­po­lo­gic­al group without ar­bit­rar­ily small sub­groups, i.e., one hav­ing a neigh­bor­hood of the iden­tity that in­cludes no sub­group ex­cept the trivi­al group. We shall fol­low Ka­plansky [e25] and call this the NSS Con­di­tion and a group sat­is­fy­ing it an NSS group. Since NSS may seem a little con­trived, here is a brief dis­cus­sion of the “why and how” of its use in solv­ing the Fifth Prob­lem.

It turns out to be dif­fi­cult to draw use­ful con­clu­sions about a to­po­lo­gic­al group from the as­sump­tion that it is loc­ally Eu­c­lidean. So the strategy used for set­tling the Fifth Prob­lem was to look for a more group-ori­ented “bridge con­di­tion” and use it in a two-pronged at­tack: on the one hand show that a to­po­lo­gic­al group that sat­is­fies this con­di­tion is a Lie group, and on the oth­er show that a loc­ally Eu­c­lidean group sat­is­fies the con­di­tion. If these two pro­pos­i­tions can be proved, then the pos­it­ive solu­tion of the Fifth Prob­lem fol­lows — and even a little more.

As you may have guessed, NSS turned out to be ideally suited to play the role of the bridge. In ret­ro­spect this is not en­tirely sur­pris­ing. A power­ful but re­l­at­ively ele­ment­ary prop­erty of Lie groups is the ex­ist­ence of so-called ca­non­ic­al co­ordin­ates, or equi­val­ently the fact that the ex­po­nen­tial map is a dif­feo­morph­ism of a neigh­bor­hood of zero in the Lie al­gebra onto a neigh­bor­hood $U$ of the iden­tity in the group (see be­low). Since a line through the ori­gin in the Lie al­gebra maps to a one-para­met­er sub­group of the group, it fol­lows that such a $U$ con­tains no non­trivi­al sub­group and hence that Lie groups sat­is­fy NSS.

Start­ing in the late 1940s Gleason [1],1 Mont­gomery [e15], and Iwas­awa [e13] made sev­er­al sol­id ad­vances re­lated to the Fifth Prob­lem. This led in 1952 to a sat­is­fy­ing de­noue­ment to the story of the Fifth Prob­lem, with Gleason and Mont­gomery–Zip­pin car­ry­ing out the above two-pronged at­tack. First Gleason [3] proved that a loc­ally com­pact group sat­is­fy­ing NSS is a Lie group, and then im­me­di­ately af­ter­wards Mont­gomery and Zip­pin [e16] used Gleason’s res­ult to prove in­duct­ively that loc­ally Eu­c­lidean groups of any di­men­sion sat­is­fy NSS. Their two pa­pers ap­peared to­geth­er in the same is­sue of the An­nals of Math­em­at­ics, and at that point one knew that for loc­ally com­pact to­po­lo­gic­al groups: $$\text{Locally Euclidean }⟺\text{NSS }⟺\text{Lie}.$$ (Ac­tu­ally, the above is not quite the full story; Gleason as­sumed a weak form of fi­nite di­men­sion­al­ity in his ori­gin­al ar­gu­ment that NSS im­plies Lie, but shortly there­after Yamabe [e20] showed that fi­nite di­men­sion­al­ity was not needed in the proof.)

Here is a quick sketch of how the proof of the Fifth Prob­lem for a com­pact NSS group $G$ fol­lows. Let $\mathcal{H}$ de­note the Hil­bert space $L^2(G)$ of square-in­teg­rable func­tions on $G$ with re­spect to Haar meas­ure. Left trans­la­tion in­duces an or­tho­gon­al rep­res­ent­a­tion of $G$ on $\mathcal{H}$, the so-called reg­u­lar rep­res­ent­a­tion, and, ac­cord­ing to the Peter–Weyl The­or­em, $\mathcal{H}$ is the or­tho­gon­al dir­ect sum of fi­nite-di­men­sion­al sub­rep­res­ent­a­tions, ${\mathcal{H}}_i$, i.e., $\mathcal{H}=\underset{i=1}{\overset{\mathrm{\infty }}{⨁}}{\mathcal{H}}_i$. Define $W_N:=\underset{i=1}{\overset{N}{⨁}}{\mathcal{H}}_i$. We will show that for $N$ suf­fi­ciently large, the fi­nite-di­men­sion­al rep­res­ent­a­tion of $G$ on $W_N$ is faith­ful or, equi­val­ently, that for some $N$ the ker­nel $K_N$ of the reg­u­lar rep­res­ent­a­tion re­stric­ted to $W_N$ is the trivi­al group $\{e\}$. Since the reg­u­lar rep­res­ent­a­tion it­self is clearly faith­ful, $K_N$ is a de­creas­ing se­quence of com­pact sub­groups of $G$ whose in­ter­sec­tion is $\{e\}$. Thus if $U$ is an open neigh­bor­hood of $e$ that con­tains no non­trivi­al sub­group, $K_N\mathrm{\setminus }U$ is a de­creas­ing se­quence of com­pact sets with empty in­ter­sec­tion and, by the defin­i­tion of com­pact­ness in terms of closed sets, some $K_N\mathrm{\setminus }U$ must be empty. Hence $K_N\subseteq U$, and since $K_N$ is a sub­group of $G$, $K_N=\{e\}$.

Let’s start with a Lie group $G$, and let $\mathfrak{g}$ de­note its Lie al­gebra. There are (at least!) three equi­val­ent ways to think of an ele­ment of the vec­tor space $\mathfrak{g}$. First as a vec­tor $v$ in $T{G}_e$, the tan­gent space to $G$ at $e$; second as the left-in­vari­ant vec­tor field $X$ on $G$ ob­tained by left trans­lat­ing $v$ over the group; and third as the one-para­met­er sub­group $\phi$ of $G$ ob­tained as the in­teg­ral curve of $X$ start­ing at the iden­tity. The ex­po­nen­tial map $\exp :\mathfrak{g}\to G$ is defined by $\exp (v)=\phi (1)$. It fol­lows im­me­di­ately from this defin­i­tion that $\exp (0)=e$ and that the dif­fer­en­tial of $\exp$ at 0 is the iden­tity map of $T{G}_e$, so by the in­verse func­tion the­or­em, $\exp$ maps a neigh­bor­hood of 0 in $\mathfrak{g}$ dif­feo­morph­ic­ally onto a neigh­bor­hood of $e$ in $G$. Such a chart for $G$ is called a ca­non­ic­al co­ordin­ate sys­tem (of the first kind).

Now, sup­pose we some­how “lost” the dif­fer­en­ti­able struc­ture of $G$ but re­tained our know­ledge of $G$ as a to­po­lo­gic­al group. Is there some way we could use the lat­ter know­ledge to re­cov­er the dif­fer­en­ti­able struc­ture? That is, can we re­con­struct $\mathfrak{g}$ and the ex­po­nen­tial map? If so, then we are clearly close to a solu­tion of the Fifth Prob­lem. Let’s listen in as Andy pon­ders this ques­tion.

“Well, if I think of $\mathfrak{g}$ as be­ing the one-para­met­er groups, that’s a group the­or­et­ic concept. Let’s see — is there some way I can in­vert $\exp$? That is, giv­en $g$ in $G$ close to $e$, can I find the one-para­met­er group $\phi$ such that $\phi (1)=\exp (\phi )=g$? Now I know square roots are unique near $e$ and in fact $\phi (1/2)$ is the square root of $g$. By in­duc­tion, I can find $\phi (1/2^n)$ by start­ing with $g$ and tak­ing the square root $n$ times. And once I have $\phi (1/2^n)$, by simply tak­ing its $m$-th power I can find $\phi (m/2^n)$ for all $m$. So, if I know how to take square roots near $e$, then I can com­pute $\phi$ at all the dy­ad­ic ra­tion­als $m/2^n$, and since they are dense in $\mathbf{R}$, I can ex­tend $\phi$ by con­tinu­ity to find it on all of $\mathbf{R}!$”

This was the stated mo­tiv­a­tion for Gleason’s pa­per “Square roots in loc­ally Eu­c­lidean groups” [1], and in it he goes on to take the first step and show that in any NSS loc­ally Eu­c­lidean group $G$, there are neigh­bor­hoods $U$ and $V$ of the iden­tity such that every ele­ment in $U$ has a unique square root in $V$. Al­most im­me­di­ately after this art­icle ap­peared, in a pa­per called “On a the­or­em of Gleason”, Che­val­ley [e14] went on to com­plete the pro­gram Andy out­lined. That is, Che­val­ley used Gleason’s ex­ist­ence of unique square roots to con­struct a neigh­bor­hood $U$ of the iden­tity in $G$ and a con­tinu­ous map­ping $(g,t)\mapsto {\phi }^g(t)$ of $U\times \mathbf{R}$ in­to $G$ such that each ${\phi }^g$ is a one-para­met­er sub­group of $G,{\phi }^g(t)\in U$ for $|t|\le 1$, and ${\phi }^g(1)=g$.

In his key 1952 An­nals pa­per “Groups without small sub­groups” [3], Gleason de­cided not to fol­low up this ap­proach to the solu­tion of the Fifth Prob­lem and in­stead used a vari­ant of von Neu­mann’s meth­od. His ap­proach was based on the con­struc­tion of one-para­met­er sub­groups, but these were used as a tool to find a cer­tain fi­nite-di­men­sion­al in­vari­ant lin­ear sub­space $Z$ of the reg­u­lar rep­res­ent­a­tion of $G$ on which $G$ ac­ted faith­fully and ap­pealed to Cartan’s The­or­em to com­plete the proof. The con­struc­tion of $Z$ is a tech­nic­al tour de force, but it is too com­plic­ated to out­line here, and we refer in­stead to the ori­gin­al pa­per [2] or the re­view by Iwas­awa.

Look­ing back at how it happened, it seems al­most ac­ci­dent­al that I be­came Andy Gleason’s first Ph.D. stu­dent — and Dav­id Hil­bert was partly re­spons­ible.

As an un­der­gradu­ate at Har­vard I had de­veloped a very close ment­or­ing re­la­tion­ship with George Mackey, then a res­id­ent tu­tor in my dorm, Kirk­land House. We had meals to­geth­er sev­er­al times each week, and I took many of his courses. So, when I re­turned in 1953 as a gradu­ate stu­dent, it was nat­ur­al for me to ask Mackey to be my thes­is ad­visor. When he in­quired what I would like to work on for my thes­is re­search, my first sug­ges­tion turned out to be something he had thought about him­self, and he was able to con­vince me quickly that it was un­suit­ably dif­fi­cult for a thes­is top­ic. A few days later I came back and told him I would like to work on re­for­mu­lat­ing the clas­sic­al Lie the­ory of germs of Lie groups act­ing loc­ally on man­i­folds as a rig­or­ous mod­ern the­ory of full Lie groups act­ing glob­ally. Fine, he said, but then ex­plained that the loc­al ex­pert on such mat­ters was a bril­liant young former Ju­ni­or Fel­low named Andy Gleason who had just joined the Har­vard math de­part­ment. Only a year be­fore he had played a ma­jor role in solv­ing Hil­bert’s Fifth Prob­lem, which was closely re­lated to what I wanted to work on, so he would be an ideal per­son to dir­ect my re­search.

I felt a little un­happy at be­ing cast off like that by Mackey, but of course I knew per­fectly well who Gleason was and I had to ad­mit that George had a point. Andy was already fam­ous for be­ing able to think com­plic­ated prob­lems through to a solu­tion in­cred­ibly fast. “Johnny” von Neu­mann had a sim­il­ar repu­ta­tion, and since this was the year that High Noon came out, I re­call jokes about hav­ing a math­em­at­ic­al shootout — Andy vs. Johnny solv­ing math prob­lems with blaz­ing speed at the OK Cor­ral. In any case, it was with con­sid­er­able trep­id­a­tion that I went to see Andy for the first time.

Totally un­ne­ces­sary! In our ses­sions to­geth­er I nev­er felt put down. It is true that oc­ca­sion­ally when I was telling him about some pro­gress I had made since our pre­vi­ous dis­cus­sion, part­way through my ex­plan­a­tion Andy would see the crux of what I had done and say something like, “Oh! I see. Very nice! and then…,” and in a mat­ter of minutes he would re­con­struct (of­ten with im­prove­ments) what had taken me hours to fig­ure out. But it nev­er felt like he was act­ing su­per­i­or. On the con­trary, he al­ways made me feel that we were col­leagues, col­lab­or­at­ing to dis­cov­er the way for­ward. It was just that when he saw his way to a solu­tion of one prob­lem, he liked to work quickly through it and then go on to the next prob­lem. Work­ing to­geth­er with such a math­em­at­ic­al power­house put pres­sure on me to per­form at top level — and it was sure a good way to learn hu­mil­ity!

My ap­pren­tice­ship wasn’t over when my thes­is was done. I re­mem­ber that shortly after I had fin­ished, Andy said to me, “You know, some of the ideas in your thes­is are re­lated to some ideas I had a few years back. Let me tell you about them, and per­haps we can write a joint pa­per.” The ideas in that pa­per were in large part his, but on the oth­er hand, I did most of the writ­ing, and in the pro­cess of cor­rect­ing my at­tempts he taught me a lot about how to write a good journ­al art­icle.

But it was only years later that I fully ap­pre­ci­ated just how much I had taken away from those years work­ing un­der Andy. I was very for­tu­nate to have many ex­cel­lent stu­dents do their gradu­ate re­search with me over the years, and of­ten as I worked to­geth­er with them I could see my­self be­hav­ing in some way that I had learned to ad­mire from my own ex­per­i­ence work­ing to­geth­er with Andy.

Let me fin­ish with one more an­ec­dote. It con­cerns my fa­vor­ite of all Andy’s the­or­ems, his el­eg­ant clas­si­fic­a­tion of the meas­ures on the lat­tice of sub­spaces of a Hil­bert space. Andy was writ­ing up his res­ults dur­ing the 1955–56 aca­dem­ic year, as I was writ­ing up my thes­is, and he gave me a draft copy of his pa­per to read. I found the res­ult fas­cin­at­ing, and even con­trib­uted a minor im­prove­ment to the proof, as Andy was kind enough to foot­note in the pub­lished art­icle. When I ar­rived at the Uni­versity of Chica­go for my first po­s­i­tion the next year, Andy’s pa­per was not yet pub­lished, but word of it had got­ten around, and there was a lot of in­terest in hear­ing the de­tails. So when I let on that I was fa­mil­i­ar with the proof, Ka­plansky asked me to give a talk on it in his ana­lys­is sem­in­ar. I’ll nev­er for­get walk­ing in­to the room where I was to lec­ture and see­ing Ed Span­i­er, Mar­shall Stone, Saun­ders Mac Lane, An­dré Weil, Ka­plansky, and Chern all look­ing up at me. It was pretty in­tim­id­at­ing, and I was suit­ably nervous!

But the pa­per was so el­eg­ant and clear that it was an ab­so­lute breeze to lec­ture on it, so all went well, and this “in­aug­ur­al lec­ture” helped me get off to a good start in my aca­dem­ic ca­reer.