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Celebratio Mathematica

Morton Brown

Morton Brown

by Rob Kirby

Mor­ton Brown was born in the Bronx on 12 Au­gust 1931. His fath­er, Irving Brown came from Ro­mania, and owned and op­er­ated a fruit and ve­get­able store in Man­hat­tan. His moth­er, Shir­ley Lehr­man, was of Pol­ish-Rus­si­an des­cent. Mort had an older broth­er, Jack, who went to en­gin­eer­ing at NYU and the Navy.

Mort fin­ished high school in June 1948, hav­ing skipped 7th grade. He went to the Uni­versity of Wis­con­sin and first met R H Bing in spring 1950 in an ele­ment­ary cal­cu­lus class. Bing urged Mort to take his gradu­ate to­po­logy course, which Mort did, but then he dropped out and re­took it the fol­low­ing year, find­ing it much easi­er the second time. Mort fin­ished his BS in 1953, after tak­ing a one-semester break in or­der to, as he put it, fig­ure out how to have a so­cial life and also be a math­em­atician.

While still an un­der­gradu­ate in 1952, Mort heard that count­able Haus­dorff spaces were nev­er con­nec­ted, but he pro­duced an ex­ample of one any­way, put­ting an in­ter­est­ing to­po­logy on the pos­it­ive in­tegers. It ap­peared as an ab­stract in Bull. AMS in 1953 [1], but the de­tails were nev­er pub­lished (this is the first ex­ample of Mort’s re­luct­ance to write up his work, and hence his debt to his col­lab­or­at­ors who usu­ally did the li­on’s share of the writ­ing).

Bing en­cour­aged Mort to give a short talk on his ex­ample at an AMS meet­ing at NYU in April 1953. Mort relates that he sat through the day’s talks, watch­ing the audi­ence grow un­til it was quite large. He was sur­prised there was so much in­terest in his ex­ample, but then the pre­vi­ous speak­er, Sammy Ei­len­berg, walked out, after which the audi­ence dis­ap­peared! Many years later, Ei­len­berg wrote to Mort ask­ing for the de­tails of his ex­ample (see [◊]; the pre­cise date of this let­ter is not known).

Mort stayed on at Wis­con­sin for gradu­ate work with Bing. In 1956 he mar­ried Kaar­en Strauch, and they later had three boys, Aaron (1958), Alan (1959) and Carl (1961). Kaar­en even­tu­ally be­came a Pro­fess­or of So­cial Work at East­ern Michigan Uni­versity.

Mort had the res­ults of his PhD thes­is by 1957, and took a job at Ohio State for 1957–58. Michigan then offered him a ten­ure-track job, but this re­quired a PhD, so Mort had to ac­tu­ally write up his thes­is. He partly chose Michigan be­cause he wanted to work with El­don Dyer who was be­ing wooed by Michigan but did not in fact come.

In 1958 Barry Mazur found [e1], [e2] his strik­ing proof of the Schoen­flies Con­jec­ture (mod­ulo a small hy­po­thes­is, later re­moved by Mar­ston Morse). Ed Moise said, ac­cord­ing to Mort, that “Mazur went in­to the jungle with a string and caught the only li­on that could be cap­tured with a string.” Mort had, un­til then, nev­er heard of the con­jec­ture but star­ted to think about it.

Mort found his proof of the Schoen­flies Con­jec­ture in early fall 1959, and showed it to Hans Samel­son and Moise (who said that it “used mir­rors”).

At an AMS meet­ing in De­troit around Thanks­giv­ing, Mort met Henry White­head who had been work­ing with M. H. A. New­man on the Con­jec­ture. Mort ex­plained his proof quickly in four steps and Henry said “won­der­ful.” Mort guessed Henry had not really un­der­stood the proof, un­til shortly af­ter­ward he heard Henry ex­plain the proof with pre­ci­sion to someone else. Henry wrote to New­man to con­vey his ex­cite­ment [private com­mu­nic­a­tion]: “Here is news which dwarfs everything else.” New­man dis­cusses the Con­jec­ture at length in [e3].

Mort pub­lished what he con­siders his two hard­est pa­pers in 1960: “On the in­verse lim­it of Eu­c­lidean \( N \)-spheres” [3] and “Some ap­plic­a­tions of an ap­prox­im­a­tion the­or­em for in­verse lim­its” [2].

The bound­ary of a smooth or PL man­i­fold \( M \) nat­ur­ally has a “col­lar” neigh­bor­hood of the form \( \partial M \times [0,1] \). This is loc­ally true for a to­po­lo­gic­al man­i­fold by defin­i­tion, but when Mort raised the ques­tion of the ex­ist­ence of a to­po­lo­gic­al col­lar, White­head said that ques­tion was “very deep.” Non­ethe­less, Mort found a proof [4] that in hind­sight is quite easy. Add a col­lar to \( M \), and then push the col­lar in­to \( M \) us­ing a par­ti­tion of unity and the loc­al col­lars.

Mort was at the In­sti­tute for Ad­vanced Study in 1960 when he began col­lab­or­at­ing with a Prin­ceton gradu­ate stu­dent, Her­man Gluck, on the An­nu­lus Con­jec­ture and re­lated is­sues. The An­nu­lus Con­jec­ture was well known to to­po­lo­gists like White­head, but it was Mort who coined the name after a con­ver­sa­tion with Lee Ru­bel who em­phas­ized the im­port­ance of a good name that people would eas­ily re­mem­ber.

The col­lab­or­a­tion with Gluck led to a series of three An­nals pa­pers on stable homeo­morph­isms [6], [7], [5], a name coined by Gluck for homeo­morph­isms which can be factored in­to \( h_1\circ h_2\circ \ldots \circ h_k \), where each \( h_i \) equals the iden­tity on an open set \( U_i \). They showed among oth­er things that the An­nu­lus Con­jec­ture held in all di­men­sions if and only if all homeo­morph­isms of Eu­c­lidean space are stable.

These no­tions turned out to be very im­port­ant in fu­ture work on tri­an­gu­la­tions of to­po­lo­gic­al man­i­folds by Kirby and Sieben­mann [e6] [e7]. In fact, these pa­pers of Mort, to­geth­er with Mazur [e1], Mil­nor’s pa­per on mi­crobundles [e5], and Kister’s proof that to­po­lo­gic­al man­i­folds have tan­gent bundles [e4], laid the found­a­tions for later work on the struc­ture of to­po­lo­gic­al man­i­folds.

Two later pa­pers, [8] and [9], con­cerned fixed points of homeo­morph­isms of the plane, \( \mathbb{R}^2 \). Con­sider an area pre­serving homeo­morph­ism of \( \mathbb{S}^1 \times [0,1] \); Poin­caré con­jec­tured that it has two fixed points and proved some spe­cial cases. Birk­hoff pub­lished a proof for the ex­ist­ence of one fixed point and in a later pa­per cor­rec­ted an er­ror and claimed a proof for two fixed points. But there was doubt in some quar­ters as to the ac­cur­acy of the second pa­per. Brown and W. Neu­mann gave an ele­ment­ary, de­tailed proof for two fixed points which ba­sic­ally fol­lows the out­line of Birk­hoff’s work, and the au­thors de­scribe their pa­per as mostly ex­pos­it­ory. A well needed ex­pos­i­tion.

Cartwright and Lit­tle­wood showed that an ori­ent­a­tion pre­serving homeo­morph­ism of \( \mathbb{R}^2 \) which leaves a com­pact, non­sep­ar­at­ing con­tinuum \( C \) in­vari­ant also leaves a point of \( C \) fixed. Mort gives a one-para­graph proof of this the­or­em [9] us­ing a res­ult of L. E. J. Brouwer.

Works

[1]M. Brown: “A count­able con­nec­ted Haus­dorff space,” Bull. Amer. Math. Soc. 59 (1953), pp. 367. Ab­stract only (in­side The April meet­ing in New York); un­pub­lished. article

[2]M. Brown: “Some ap­plic­a­tions of an ap­prox­im­a­tion the­or­em for in­verse lim­its,” Proc. Amer. Math. Soc. 11 (1960), pp. 478–​483. MR 0115157 Zbl 0113.​37705

[3]M. Brown: “On the in­verse lim­it of Eu­c­lidean \( N \)-spheres,” Trans. Amer. Math. Soc. 96 (1960), pp. 129–​134. MR 0119182 Zbl 0136.​19805

[4]M. Brown: “Loc­ally flat im­bed­dings of to­po­lo­gic­al man­i­folds,” Ann. of Math. (2) 75 (1962), pp. 331–​341. MR 0133812 Zbl 0201.​56202

[5]M. Brown and H. Gluck: “Stable struc­tures on man­i­folds, III: Ap­plic­a­tions,” Ann. of Math. (2) 79 (1964), pp. 45–​58. MR 0158385

[6]M. Brown and H. Gluck: “Stable struc­tures on man­i­folds, I: Homeo­morph­isms of \( S^{n} \),” Ann. of Math. (2) 79 (1964), pp. 1–​17. MR 0158383

[7]M. Brown and H. Gluck: “Stable struc­tures on man­i­folds, II: Stable man­i­folds,” Ann. of Math. (2) 79 (1964), pp. 18–​44. MR 0158384

[8]M. Brown and W. D. Neu­mann: “Proof of the Poin­caré–Birk­hoff fixed point the­or­em,” Michigan Math. J. 24 : 1 (1977), pp. 21–​31. MR 0448339 Zbl 0402.​55001

[9]M. Brown: “A short short proof of the Cartwright–Lit­tle­wood the­or­em,” Proc. Amer. Math. Soc. 65 : 2 (1977), pp. 372. MR 0461491 Zbl 0369.​57001