Celebratio Mathematica

Andrew Mattei Gleason

Gleason’s contribution to the solution of
Hilbert’s Fifth Problem

by Richard Palais

What is Hilbert’s Fifth Problem?

Andy Gleason is prob­ably best known for his work con­trib­ut­ing to the solu­tion of Hil­bert’s Fifth Prob­lem. We shall dis­cuss this work be­low, but first we need to know just what the “Fifth Prob­lem” is. In its ori­gin­al form it asked, roughly speak­ing, wheth­er a con­tinu­ous group ac­tion is ana­lyt­ic in suit­able co­ordin­ates. But as we shall see, the mean­ing has changed over time.

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 [e15], is an ac­tion of \( \mathbf{Z}^2 \) on \( \mathbf{S}^3 \) whose fixed-point set is the Al­ex­an­der Horned Sphere \( \Sigma \). Now \( \Sigma \) is not “tamely em­bed­ded” in \( \mathbf{S}^3 \), mean­ing that there are points where it is im­possible to choose co­ordin­ates so that loc­ally \( \Sigma \) 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 \( r \leq 1 \) and 0 for \( r \geq 2 \), 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 \( C^{\infty} \)), 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.

Early history of the Fifth Problem

Excerpt from Gleason’s vacation journal, July 1947, in which he mentions working on the “Hilbert fifth”.
Courtesy of Jean Berko Gleason.

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 [e17] 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 no small subgroups (NSS) condition

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 [e16], 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 [e17] 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: \[ \text{Locally Euclidean } \Longleftrightarrow \text{NSS } \Longleftrightarrow \text{Lie}. \] (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.)

Cartan’s theorem

Start­ing with von Neu­mann, all proofs of cases of the Fifth Prob­lem, in­clud­ing Gleason’s, were ul­ti­mately based on the fol­low­ing clas­sic res­ult that goes back to É. Cartan. (For a mod­ern proof, see Che­val­ley ([e11], page 130).)
The­or­em (Cartan): If a loc­ally com­pact group has a con­tinu­ous, in­ject­ive ho­mo­morph­ism in­to a Lie group and, in par­tic­u­lar, if it has a faith­ful fi­nite-di­men­sion­al rep­res­ent­a­tion, then it is a Lie group.

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., \( \mathcal{H} =\bigoplus^{\infty}_{i=1} \mathcal{H}_i \). Define \( W_N := \bigoplus^N_{i=1}\mathcal{H}_i \). 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, \( K_N \backslash U \) is a de­creas­ing se­quence of com­pact sets with empty in­ter­sec­tion and, by the defin­i­tion of com­pact­ness in terms of closed sets, some \( K_N \backslash U \) must be empty. Hence \( K_N \subseteq U \), and since \( K_N \) is a sub­group of \( G \), \( K_N = \{e\} \).

Following in Gleason’s footsteps

Andy Gleason put lots of re­marks and clues in his pa­pers about his mo­tiv­a­tions and trains of thought, and it is an en­joy­able ex­er­cise to read these chro­no­lo­gic­ally and use them to guess how he de­veloped his strategy for at­tack­ing the Fifth Prob­lem.

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 \( |t| \leq 1 \), 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.

Andy Gleason as mentor

Andy (left) with George Mackey (2000). Although Andy never earned a Ph.D., he thought of George as his mentor and advisor and lists himself as George’s student on the Mathematics Genealogy Project website.
Photo courtesy of Jean Berko Gleason.

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.


[1]A. M. Gleason: “Square roots in loc­ally Eu­c­lidean groups,” Bull. Am. Math. Soc. 55 : 4 (1949), pp. 446–​449. MR 0028841 Zbl 0041.​16002 article

[2]A. M. Gleason: “The struc­ture of loc­ally com­pact groups,” Duke Math. J. 18 : 1 (1951), pp. 85–​104. MR 0039730 Zbl 0044.​01901 article

[3]A. M. Gleason: “Groups without small sub­groups,” Ann. Math. (2) 56 : 2 (September 1952), pp. 193–​212. MR 0049203 Zbl 0049.​30105 article