Celebratio Mathematica

Michael H. Freedman

Michael Hartley Freedman

by Rob Kirby

Mike’s family

Mike Freed­man was born in Los Angeles on 21 April 1951, to Nancy Mars Freed­man and Be­ne­dict Freed­man. Ben’s fath­er, Dav­id, was born in Ro­mania and emig­rated to New York, where he be­came a ma­jor writer of com­edy sketches for ra­dio and Broad­way shows, in par­tic­u­lar, the Ziegfield Fol­lies, Burns and Al­len, Ed­die Can­tor, and Fanny Brice. He died early in 1936, when Ben was 16.

Ben’s moth­er, Be­atrice, was a for­mid­able wo­man, and be­lieved that Ben would be wast­ing his time in part B of a course with two parts, ex­pect­ing Ben to learn it all in part A. Con­sequently, Ben fin­ished Town­send Har­ris High School at the age of 13 and star­ted Columbia in fall 1933. Be­atrice’s de­term­in­a­tion worked even bet­ter for a young­er son, Dav­id Noel Freed­man, who fin­ished high school at 12 and went on to be a dis­tin­guished pro­fess­or in Bib­lic­al Stud­ies, but not so well for young­est son Toby Freed­man who only fin­ished high school at 16. “Toby the dope”, as his moth­er af­fec­tion­ately called him, played foot­ball at UC Berke­ley, got an MD from Stan­ford Med­ic­al School, and be­came the team phys­i­cian for the Los Angeles Rams and LA Lakers.

Dur­ing 1933–34, Kurt Gödel was lec­tur­ing on his fam­ous work in lo­gic at the In­sti­tute of Ad­vanced Study, where Alonzo Church and two of his gradu­ate stu­dents, Steph­en Kleene and Barkley Ross­er, were listen­ers. Ross­er came weekly to Columbia to re­port on Gödel’s lec­tures, and fresh­man Ben, even at his age, was a ser­i­ous par­ti­cipant, al­though he only re­turned to lo­gic dec­ades later.

In high school, Ben had learned quite a bit of math (in par­tic­u­lar, cal­cu­lus of fi­nite dif­fer­ences). When Ben’s fath­er died, he left school and star­ted work­ing for an ac­tu­ary, and writ­ing com­edy sketches. He met Nancy, a dan­cer, act­ress and writer, and they were mar­ried in 1941, a mar­riage per­haps hastened be­cause Nancy had bac­teri­al en­do­cardit­is and was giv­en only months to live.

The new­ly­weds moved to Hol­ly­wood and con­tin­ued to write scripts. At first, Ben was not draf­ted ow­ing to poor vis­ion but, as World War II grew, he took a job as drafts­man with Cur­tiss–Wright Avi­ation. When the oth­er drafts­men were draf­ted, Ben was pro­moted to teach­ing aero­naut­ic­al en­gin­eer­ing, re­ly­ing on his early math back­ground. He shortly ended up with Hughes Air­craft, in­volved in build­ing the fam­ous ply­wood air­plane, the Spruce Goose. His skill with cal­cu­lus of fi­nite dif­fer­ences was valu­able in build­ing a ply­wood plane, for ply­wood was in­her­ently dis­crete, where­as reg­u­lar cal­cu­lus was ap­pro­pri­ate for met­al planes.

An an­ec­dote from those days in­volves Ben’s boss wak­ing him up at 2 AM on the phone, telling him to take two showers and to meet him and Howard Hughes at the air­field. Ben did so, a small plane ar­rived and out stepped Hughes, who com­manded Ben to re­main at a dis­tance (Hughes was fam­ously ger­mo­phobic). Hughes said that he had a secret job for Ben, doub­ling his salary, but Ben could not tell any­one where he was, al­though mail would be de­livered to and from Ben. Ben said he couldn’t leave Nancy for the war nor for Hughes, so there ended that bizarre epis­ode.

Ben and Nancy con­tin­ued to write, and in 1947 pub­lished their first nov­el, Mrs. Mike, a best seller about a teen­age girl who went to live in north­ern Canada in the 1800s. Ben wrote scripts for the Red Skel­et­on Show for 12 years, and for Mickey Rooney movies, such as Atom­ic Kid and Hatari; he still en­joyed do­ing math in his spare time. In 1965, he went back to school at UCLA, ob­tain­ing a Ph.D. in lo­gic un­der Yi­an­nis Moschova­kis in 1970. He taught at Oc­ci­dent­al Col­lege un­til his re­tire­ment. Nancy died in 2010.

Mike has two sis­ters: Jo­hanna, who has a PhD in psy­cho­logy from Stan­ford and is a pro­fess­or at UC Irvine Med­ic­al School (where she dir­ects a pro­gram in the med­ic­al hu­man­it­ies and the arts), and De­borah, who is an op­era sing­er and teaches voice, some­times at UC Berke­ley.

Mike has four sons, three with his wife Leslie (Sam) How­land; the sons have tough acts to fol­low.

Mike’s youth

Mike’s fath­er (in 2011) re­calls with a scriptwriter’s gift this next pic­ture of Mike as a teen­ager, which nicely cap­tures the Mike we know as an adult:

Mike and Doug tackle El Capitan

Mike was fif­teen and in Pali High (Pa­cific Pal­is­ades, CA, a Los Angeles sub­urb over­look­ing the Pa­cific). We had a pool, and he learned to swim at the age of 2, be­com­ing a strong swim­mer with su­per­i­or en­dur­ance. He nev­er liked team sports, but got his ex­er­cise hik­ing through the hills of the Santa Mon­ica Moun­tains with his dog Rex. Be­ing already a math­em­atician, his route was gen­er­ally a geodes­ic, which gave him some train­ing in rock climb­ing.

At Pali High, he tried out for and made the swim­ming team, but al­ways came down with a cold just be­fore the meet. He quit, and re­sumed his sol­it­ary hikes with Rex. Then, he found an­oth­er loner of his age, Doug McK­en­zie, and they teamed up.

After a few weeks of trudging through the hills, fall­ing off boulders, and be­ing cut to rib­bons by the brush, they de­cided that this was their true vo­ca­tion. They bought ropes, pitons, ca­ra­bin­ers — the whole works. They went back in­to the hills and looked around for cliffs to con­quer. They came home hours after din­ner, and ig­nored their home­work. It had no ef­fect on their grades; they already knew everything high school had to teach them.

When they thought they were ready, they ap­proached their par­ents with a plan. They would have their drivers’ li­censes in a few weeks, drive north to Yosemite, and after a few prac­tice as­cents climb El Cap­it­an by one of the easi­er routes.

Mrs. McK­en­zie, who had done some rock climb­ing her­self, dis­missed the whole pro­ject as ab­surd, im­possible, and def­in­itely cal­cu­lated to get them killed. Nancy and I, who knew noth­ing of rock climb­ing and had nev­er been to Yosemite, thought it a reas­on­able plan that might suc­ceed a couple of years down the road.

Mrs. McK­en­zie called on her hus­band for re­in­force­ment, and they laid down the law: for­get the whole thing.

I don’t know how Doug re­spon­ded, but Nancy and I real­ized it was a year too late to take that line with Mike. He’d spent the sum­mer be­fore at UC Berke­ley un­der a spe­cial pro­gram in math­em­at­ics for bril­liant high-school stu­dents. When he came home, he an­nounced that he was now an adult, and would no longer check in be­fore mid­night or re­veal whom he was dat­ing. In fact, next time he was at Berke­ley, he might nev­er come back.

To which Nancy replied with a straight face, “If that’s what you want, Mike. We’ll miss you. Any­time you’re in the neigh­bor­hood, drop in.”

Mean­while, Mrs. McK­en­zie had come up with a com­prom­ise solu­tion. She had loc­ated a young man with Yosemite-climb­ing ex­per­i­ence, who had gone in­to busi­ness tu­tor­ing am­a­teurs on the ba­sics. We hired him, and he went along with the boys and the dog in­to the hills, for an overnight­er. When they re­turned, Mike re­por­ted, “He’s not a bad guy. He knows a few things, and really loves the Si­er­ras. But he keeps telling the same joke over and over, and like Dad says, ‘If it ain’t funny the first time, it ain’t funny the fifth time.’ Also, he’s on drugs and lost his edge. Doug and I had to save his neck a couple of times.”

We gave in. But ex­trac­ted a prom­ise they would study the the­ory of climb­ing the way you learn any sci­entif­ic dis­cip­line, and test it in prac­tice the way sci­ent­ists test their the­or­ies. I bought a couple of books to get star­ted on.

They took me ser­i­ously and camped out in our big eu­ca­lyptus to hone their skills. They con­ceived the idea of haul­ing Rex up in­to the tree and im­prov­ing his edu­ca­tion, thereby im­prov­ing theirs. It says something about hu­man vs. an­im­al cog­nit­ive abil­it­ies that Rex made the fast­est pro­gress, at least in eu­ca­lyptus as­cents. It was fas­cin­at­ing to watch the ex­pres­sion on the face of a cat who thought he had es­caped scot-free, and then saw this su­per-dog com­ing up the tree right after him.

To cut the story short, with­in the year they did climb El Cap. Nancy and I didn’t get par­tic­u­larly ex­cited — Mike had said they would do it, and they did it. But, sev­er­al years later, en route to a writers’ con­fer­ence in Bish­op, we took the long way through Yosemite and came un­ex­pec­tedly on a little sign on the side of the road. I stopped the car. “Look, Nancy, that’s El Cap.” She nearly fain­ted.

Mike be­came, and still is, an ex­cel­lent rock climber, but stor­ies of his ex­ploits ap­pear else­where.

Dur­ing high school, Mike had a real in­terest in paint­ing, pick­ing up an ex­pres­sion­ist per­spect­ive from his moth­er. He took a batch of paint­ings (mostly wa­ter­col­ors) to col­lege with him, but the idea of be­com­ing a paint­er soon dis­ap­peared.

Mike star­ted col­lege at UC Berke­ley in 1968. He thought, per­haps in­cor­rectly, that he knew enough Cal­cu­lus already, and skipped the lower-di­vi­sion math courses. He en­rolled in an ab­stract al­gebra course, a course in lo­gic giv­en by Michel Mendès from France, the son of the one-time French Prime Min­is­ter, and courses in bot­any and French. His dif­fi­culty with the lat­ter con­vinced him to try for gradu­ate school at Prin­ceton after just one year.

Ed Nel­son tells this story about Mike’s ad­mis­sion to Prin­ceton gradu­ate school:

Here’s the un­var­nished tale. As al­ways, Fox was chair­man of the ad­mis­sions com­mit­tee. We had fin­ished mak­ing our list of ac­cept­ances, when a slightly late ap­plic­a­tion came in from Mike. I looked through the folder and it looked very good, so I took it to Fox to show it to him. Mike had in­cluded a photo of him­self stand­ing on the top of a peak. That was a no-no at that time, since a photo would re­veal the ap­plic­ant’s race. I star­ted to show the folder to Fox, but he saw the photo and said “ad­mit him.” Just to be com­pletely ex­pli­cit, the photo is the only thing in the folder Fox looked at. Hon­est, that’s what happened.

Mike drove across coun­try to Prin­ceton with a copy of Hoff­man and Kun­ze propped up on the steer­ing wheel — he figured he ought to know some lin­ear al­gebra be­fore ar­riv­ing. At Prin­ceton, at the open­ing lec­ture of Don­ald Spen­cer’s class on el­lipt­ic PDEs, he heard Spen­cer’s first sen­tence: “Let \( S \) be the sheaf of germs of sec­tions of a vec­tor bundle…” Well, he thought, at least I know what a vec­tor is!

Des­pite the gaps in his know­ledge, Mike fin­ished a good thes­is with Bill Browder in just four years, and moved on to a postdoc at UC Berke­ley in 1973.

1973–75 in Berkeley

Mike spent the aca­dem­ic years 1973–75 at UC Berke­ley as a Lec­turer (the name for our postdocs in those days). I knew him slightly from climb­ing with him near Prin­ceton when he was a gradu­ate stu­dent, and we con­tin­ued to climb to­geth­er when he was here. The next story is from that peri­od.

A bouldering championship

Many of us have watched Mike climb­ing build­ings (see pho­tos), called “buil­d­er­ing,” or climb­ing boulders (boul­der­ing), or joined him on trips of great­er mag­nitude. I had the priv­ilege of join­ing him and Den­nis John­son on a four-day as­cent of the clas­sic route on the north­w­est face of the Half Dome in Yosemite Val­ley. This was in Ju­ly 1975, only 18 years after the first as­cent (in five days) by Roy­al Rob­bins, Mike Sher­rick and Jerry Gall­was in 1957, the first grade VI climb in the world.

It’s clear that Mike was a star climber among math­em­aticians, but how did he rank among the pros? This fol­low­ing story of­fers some evid­ence.

The Great West­ern Boul­der­ing Cham­pi­on­ship was held on Mike’s birth­day, 21 April 1979, just south of Mount Wood­son in South­ern Cali­for­nia. The area is covered with many gran­ite boulders, up to 100 feet high. A com­mit­tee chose a few dozen routes on these boulders, none more than 17-feet high, for ropes were not used. The com­mit­tee graded them, with the most dif­fi­cult worth 22 points, the next 18 points, and then sev­er­al in the 17s, 16s and 15s. Each climber in the com­pet­i­tion had to climb ten routes, and re­ceived either zero or the grade of the climb if they were suc­cess­ful. Once both feet left the ground, it was all or noth­ing.

Among the con­test­ants was John Bachar, already one of the most fam­ous climbers in the world, with le­gendary feats of free, solo climb­ing in Yosemite (see his ob­it in the Eco­nom­ist). As the day went by, the best climbers, in­clud­ing Bachar, had done nine climbs with sim­il­ar total scores, but the two hard­est climbs were still un­tried. As day­light waned and the or­gan­izers got antsy, Mike at­temp­ted to or­gan­ize a lot­tery, with the odd-man out hav­ing to try a hard climb first, thereby per­haps giv­ing away in­form­a­tion about the dif­fi­cult parts. Mike was not suc­cess­ful, and gradu­ally the oth­er climbers drif­ted away and fin­ished with easi­er climbs. But not Mike, who gave the 18-point­er a try, and of course suc­ceeded and won the Cham­pi­on­ship. Why didn’t Bachar try? Per­haps he was not as in­clined as Mike to try “hard prob­lems”!

Now, it was up to Mike to see if he could be­come a bet­ter math­em­atician than a climber, and he did so in late Au­gust 1981.

I had heard An­drew Cas­son give a talk about an­oth­er, sim­pler proof of Rokh­lin’s The­or­em, that a spin, closed, smooth 4-man­i­fold had to have sig­na­ture (in­dex) di­vis­ible by 16, not just 8. I re­coun­ted Cas­son’s proof in a sem­in­ar in Berke­ley, and af­ter­ward Mike and I talked about the proof. My re­col­lec­tion is that he was the driv­ing force in com­ing up with the ar­gu­ment that ap­pears in our joint pa­per [1], and that he did most of the writ­ing (Mike gets math writ­ten up quickly, the op­pos­ite of me). The ar­gu­ments in that pa­per led me to fur­ther work, and even­tu­ally to the proof of Rokh­lin’s The­or­em (and its ex­ten­sions) in my book, The to­po­logy of 4-man­i­folds [e12], which I con­sider the right way to prove the the­or­em. This was a fine col­lab­or­a­tion in all re­spects, from my point of view!

In spring 1974, I spent 3 months at IHES, and again heard fas­cin­at­ing lec­tures from Cas­son, both on the Cas­son–Gor­don work on knots, and more im­port­antly on what are now called Cas­son handles (Cas­son called them flex­ible handles). These are smooth 4-man­i­folds that are prop­er-ho­mo­topy equi­val­ent to open 2-handles, \( \mathbb{B}^2\times\mathbb{R}^2 \), and can be smoothly em­bed­ded wherever such handles ought to ex­ist were the Whit­ney trick to work in di­men­sion 4 the way it does in high­er di­men­sions.

I brought back my hand­writ­ten notes on Cas­son handles (even­tu­ally pub­lished in [e8]) and showed them to Mike; that may have been the start of his sev­en-year odys­sey, end­ing with his re­mark­able work in 1981. Fur­ther­more, his know­ledge of the ex­ten­sions of Cas­son’s work, done with Bob Ed­wards and Larry Sieben­mann and ap­pear­ing in [e8], en­abled him to be the first to real­ize that there ex­is­ted exot­ic smooth struc­tures on or­din­ary 4-space, \( \mathbb{R}^4 \).

Topological 4-manifolds

Mike be­came an as­sist­ant pro­fess­or at UC San Diego and pub­lished enough to get ten­ure, but his main fo­cus was on at­tempt­ing to prove that Cas­son handles were smoothly stand­ard. This in­volved try­ing to show that, for ex­ample, the \( n \)-th White­head double of a knot \( K \) was smoothly con­cord­ant to the \( n+1 \)-st White­head double of \( K \) for some \( n \). These ef­forts led to his Re­imbed­ding The­or­ems, which are a tech­nic­al but im­port­ant part of his fu­ture work. After sev­er­al years, Mike shif­ted to try­ing at least to show that Cas­son handles were to­po­lo­gic­ally stand­ard, and this he fi­nally achieved with an amaz­ing use of Bing shrink­ing tech­niques.

I re­call when Mike an­nounced his res­ults in late Au­gust 1981, at a con­fer­ence at UC San Diego (Den­nis Sul­li­van was giv­ing a ten-lec­ture CBMS series). His talk left me very skep­tic­al, and won­der­ing how he had the nerve to an­nounce such a bomb­shell with such shaky ar­gu­ments.

But the ex­perts — Rick An­cel, Bob Ed­wards, Larry Sieben­mann et al. — stayed around after the con­fer­ence, and with­in a week the proof seemed nailed down. It was writ­ten up re­mark­ably quickly, by the end of the year.

To­geth­er with Cas­son’s earli­er work, what Mike had done was to show that to­po­lo­gic­al sur­gery worked in di­men­sion 4 es­sen­tially the same way it worked in high­er di­men­sions, at least for 4-man­i­folds that were simply con­nec­ted or had fun­da­ment­al groups that did not grow too fast (these were labeled good groups).

His clas­si­fic­a­tion of closed, simply con­nec­ted 4-man­i­folds is just beau­ti­ful, and is ana­log­ous to the 2-di­men­sion­al case:

For each even in­teger \( \leq 2 \), there ex­ists a unique, ori­ent­able, closed sur­face with that Euler char­ac­ter­ist­ic.

For each in­teg­ral, un­im­od­u­lar, sym­met­ric bi­lin­ear form, there ex­ists ex­actly one (two, re­spect­ively) closed, simply con­nec­ted 4-di­men­sion­al man­i­fold if the form is even (odd, re­spect­ively) with that in­ter­sec­tion form.

For ex­ample, \( \mathbb{C}P^2 \), by hav­ing an odd form \( \langle1\rangle \), has a sis­ter, called the Chern man­i­fold by Mike in hon­or of Chern’s 70th birth­day that year. The pair of sis­ters, in gen­er­al, are dis­tin­guished in that one has non-van­ish­ing tri­an­gu­la­tion (Kirby–Sieben­mann) in­vari­ant, while the oth­er, van­ish­ing. However, both man­i­folds may not be PL.

These man­i­folds are easy to build, for a form can be used to build a 4-man­i­fold with bound­ary a ho­mo­logy 3-sphere, and one of Mike’s the­or­ems is that every ho­mo­logy 3-sphere bounds a con­tract­ible 4-man­i­fold.

Mike proved an \( h \)-cobor­d­ism the­or­em, and from this showed that any 4-man­i­fold prop­er-ho­mo­topy equi­val­ent to \( \mathbb{R}^4 \) was in­deed \( \mathbb{R}^4 \).

Exotic \( \mathbb R^4 \)’s

It was March 1982, and Mike was on the phone ex­plain­ing this amaz­ing the­or­em of Si­mon Don­ald­son [e6] and how it im­plied the ex­ist­ence of an exot­ic smooth struc­ture on \( \mathbb{R}^4 \). Mike had been vis­it­ing IHES and had heard a lec­ture about Si­mon Don­ald­son’s Ph.D. thes­is by his ad­viser, Mi­chael Atiyah. Mike told me that there were these things called “in­stan­tons,” cer­tain nice sec­tions of cer­tain com­plex 2-plane bundles over a 4-man­i­fold \( X \), and that, when the in­ter­sec­tion form of \( X \) was pos­it­ive def­in­ite, these in­stan­tons formed a mod­uli space of di­men­sion 5, which was smooth ex­cept for cones on \( \mathbb{C}P^2 \), and whose bound­ary was \( X \). This im­plied that the in­ter­sec­tion form had to be a dir­ect sum of \( \langle1\rangle \)’s, and that oth­er pos­it­ive-def­in­ite in­ter­sec­tion forms were not real­ized as in­ter­sec­tion forms of smooth 4-man­i­folds.

My im­me­di­ate re­ac­tion was that, c’mon, we don’t make 5-man­i­folds in that way! But then Mike fol­lowed this up by stat­ing that an exot­ic smooth struc­ture on \( \mathbb{R}^4 \) must ex­ist. He sketched an ar­gu­ment on the phone which used Don­ald­son’s res­ult, the work of Cas­son on “Cas­son” handles, Mike’s proof that Cas­son handles were in fact to­po­lo­gic­ally stand­ard, and a fur­ther res­ult of Bob Ed­wards that these handles lived in \( \mathbb{S}^2\times\mathbb{S}^2 \).

It was most re­mark­able that \( \mathbb{R}^4 \) had an exot­ic smooth struc­ture, for it had been shown by Stallings [e3] that in di­men­sions \( > 4 \) there were unique smooth struc­tures, and in di­men­sions \( < 4 \) unique­ness was well known [e1], [e2], [e4]. That di­men­sion four was dif­fer­ent was a com­plete sur­prise.

I filled in some de­tails to Mike’s sketch and gave some talks about it. Later, prob­ably in sum­mer 1983 dur­ing a dis­cus­sion at tea in Berke­ley, Mike re­marked that the exot­ic \( \mathbb{R}^4 \) smoothly em­bed­ded in \( \mathbb{S}^4 \). I said, it didn’t. After a few rounds of “Yes, it does,” “No, it doesn’t,” we com­pared con­struc­tions. It turned out that I was us­ing the nonex­ist­ence of a smooth, closed 4-man­i­fold with neg­at­ive-def­in­ite in­ter­sec­tion form \( E_8 \oplus\langle-1\rangle \), where­as Mike needed a com­pact counter­example to the \( h \)-cobor­d­ism the­or­em, which Don­ald­son was to pro­duce a year later [e9]. Both ex­amples are writ­ten up in [e12]; see also [e15] and [e16]. For com­plete­ness, I’ll give an out­line of my ar­gu­ment here:

Constructing an exotic \( \mathbb{R}^4 \)

Start with \( \mathbb{C}P^2 \) with ten points blown up, that is, \( X^4 = \mathbb{C}P^2 \mathbin{\#} 10(-\mathbb{C}P^2) \). The in­ter­sec­tion form on \( X \) is \[ \langle1\rangle \oplus10\langle-1\rangle = \bigl(E_8\oplus\langle-1\rangle\bigr)\oplus \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \] where \( E_8 \) is the neg­at­ive-def­in­ite sig­na­ture -8 bi­lin­ear form. Sup­pose \( X^4 \) is dif­feo­morph­ic to \( Y^4 \mathbin{\#}\mathbb{S}^2 \times\mathbb{S}^2 \), where \( E_8\oplus\langle-1\rangle \) is the neg­at­ive-def­in­ite in­ter­sec­tion form of the simply con­nec­ted smooth 4-man­i­fold \( Y \). But Don­ald­son’s thes­is [e6] (which Atiyah had de­scribed at IHES) ruled out the ex­ist­ence of any such \( Y \). For he had proved that a smooth, neg­at­ive-def­in­ite, simply con­nec­ted, closed 4-man­i­fold must have in­ter­sec­tion form equal to \( \bigoplus n\langle-1\rangle \). Thus, \( X \) does not smoothly split off an \( \mathbb{S}^2 \times \mathbb{S}^2 \).

However, in spring 1974 at IHES, Cas­son had lec­tured on his de­vel­op­ment of Cas­son handles, namely, smooth 4-man­i­folds pairs \( (H,\partial H) \), which were prop­er-ho­mo­topy equi­val­ent to \( (\mathbb{B}^2 \times \mathbb{B}^2, \,\mathbb{S}^1 \times\mathbb{B}^2) \), with \( \partial H \) dif­feo­morph­ic to \( \mathbb{S}^1 \times\mathbb{R}^2 \). Cas­son showed [e8] that whenev­er the ho­mo­topy-the­or­et­ic con­di­tions im­plied (if we were in high­er di­men­sions) that there would ex­ist an em­bed­ded smooth 2-handle, then, in di­men­sion 4, a smooth Cas­son handle could be em­bed­ded. In par­tic­u­lar, two Cas­son handles could be at­tached to a 0-handle such that the res­ult­ing in­teri­or is an open 4-man­i­fold \( M \) prop­er-ho­mo­topy equi­val­ent to \( \mathbb{S}^2 \times\mathbb{S}^2-\{q\} \) for some point \( q \). Then, \( M \) could be smoothly em­bed­ded in \( X \) to rep­res­ent the hy­per­bol­ic sum­mand of the in­ter­sec­tion form on \( X \).

After hear­ing Cas­son’s lec­tures, Bob Ed­wards showed an im­port­ant ad­di­tion­al fact about Cas­son handles, namely that \( H \) was also dif­feo­morph­ic, rel bound­ary, to a stand­ard 2-handle \( \mathbb{B}^2 \times \mathbb{B}^2 \) minus the cone (in­to the in­teri­or of \( \mathbb{B}^2 \times \mathbb{B}^2) \) on a gen­er­al­ized White­head con­tinuum \( \mathit{Wh} \) em­bed­ded in \( \mathbb{B}^2 \times\mathbb{S}^1 \). Thus, \( M^4 \) is a smooth sub­man­i­fold of \( \mathbb{S}^2 \times \mathbb{S}^2 \), as well as of \( X \).

Then, in 1981 Freed­man showed [2] that Cas­son handles were to­po­lo­gic­ally stand­ard, mean­ing that the dif­feo­morph­ism from \( \partial H \) to \( \mathbb{S}^1 \times \mathbb{B}^2 \) ex­ten­ded to a homeo­morph­ism from \( H \) to \( \mathbb{B}^2\times\mathbb{B}^2 \). It fol­lows that \( M \) is homeo­morph­ic to \( \mathbb{S}^2 \times \mathbb{S}^2 - \{q\} \), and that \( X \) to­po­lo­gic­ally splits off an \( \mathbb{S}^2 \times\mathbb{S}^2 \), leav­ing a closed 4-man­i­fold \( Y^{\prime} \) that can­not have a smooth struc­ture.

The exot­ic \( \mathbb{R}^4 \) arises as the com­ple­ment of the to­po­lo­gic­ally em­bed­ded \( \mathbb{S}^2 \wedge\mathbb{S}^2 \) giv­en by the cores of the 0-handle and two Cas­son handles in \( \mathbb{S}^2 \times\mathbb{S}^2 \) (as real­ized by Ed­wards in the para­graph above). This com­ple­ment is smooth (as an open sub­set of \( \mathbb{S}^2 \times \mathbb{S}^2 \)), is eas­ily seen to be prop­er-ho­mo­topy equi­val­ent to \( \mathbb{R}^4 \), and is then homeo­morph­ic to \( \mathbb{R}^4 \) by Freed­man. Call this com­ple­ment \( \mathbb{R}^4_{\Sigma} \), with \( \Sigma \) de­not­ing the smooth struc­ture

Sup­pose that \( \mathbb{R}^4_{\Sigma} \) were dif­feo­morph­ic to \( \mathbb{R}^4 \). Then, it would have smoothly em­bed­ded \( \mathbb{S}^3 \)’s out­side any com­pact sub­set. In par­tic­u­lar, there would be a smoothly em­bed­ded \( \mathbb{S}^3 \) close enough to the wedge \( \mathbb{S}^2 \wedge \mathbb{S}^2 \) so that the \( \mathbb{S}^3 \) lay smoothly in­side \( M \), and thus smoothly in­side \( M \) — now thought of as a smooth sub­man­i­fold of \( X \) and rep­res­ent­ing the hy­per­bol­ic pair in the in­ter­sec­tion form.

Now, cut \( X \) along this smooth \( \mathbb{S}^3 \), throw away the part ly­ing in \( M \), and smoothly glue-in \( \mathbb{B}^4 \). This smoothly splits off the hy­per­bol­ic pair, leav­ing a simply con­nec­ted 4-man­i­fold with neg­at­ive-def­in­ite in­ter­sec­tion form, con­tra­dict­ing Don­ald­son’s the­or­em. Thus, \( \mathbb{R}^4_{\Sigma} \) is not dif­feo­morph­ic to \( \mathbb{R}^4 \).

For a while, some the­or­et­ic­al phys­i­cists wondered wheth­er an exot­ic \( \mathbb{R}^4 \) might un­der­lie our uni­verse, but then it was seen that these exot­ic \( \mathbb{R}^4 \)’s were really strange 4-man­i­folds; what was weird was that they were homeo­morph­ic to stand­ard \( \mathbb{R}^4 \).

Fairly soon, Bob Gom­pf [e7] had pro­duced a few more exot­ic \( \mathbb{R}^4 \)’s. Even­tu­ally, there were un­count­ably many [e10] (these are most eas­ily thought of as all the open 4-balls of any ra­di­us big­ger than a giv­en con­stant, with the smooth struc­tures in­her­ited as open sets of one big exot­ic \( \mathbb{R}^4 \)).

It is pos­sible that all com­pact to­po­lo­gic­al 4-man­i­folds have count­ably many smooth struc­tures, and that all non-com­pact ones have un­count­ably many smooth struc­tures.


For a while, Mike may have been the most highly dec­or­ated young math­em­atician.

  • 1984: Elec­ted to US Na­tion­al Academy of Sci­ences
  • 1984: Cali­for­nia Sci­ent­ist of the Year
  • 1984–1989: Ma­cAr­thur Found­a­tion Fel­low
  • 1986: Veblen Prize in Geo­metry, AMS
  • 1986: Fields Medal
  • 1987: Na­tion­al Medal of Sci­ence (US)

These awards go to, at most, a few math­em­aticians per year.

There are vari­ous ways to meas­ure the qual­ity of a pa­per or body of work. One way is to as­sume the in­di­vidu­al had not done the work and try to guess how long af­ter­wards the work would have been done by someone else. In Mike’s case, I could guess that his the­or­ems on to­po­lo­gic­al 4-man­i­folds would still not have been car­ried out, even thirty years later, as this is writ­ten.

There is no photo of Mike re­ceiv­ing the Na­tion­al Medal of Sci­ence from the Pres­id­ent, for, as Mike’s fath­er, Ben, re­col­lects:

When Mike re­ceived the no­ti­fic­a­tion from the White House that he had won the Na­tion­al Medal of Sci­ence, he showed it to us. Nancy said we’d fly there with him; it was a high point in our lives. Mike said he was think­ing of not go­ing. His first son was just born, and he didn’t want to leave him. Also, he was close to a break­through on a big prob­lem. Third, he didn’t really en­joy hon­ors and medals and shak­ing hands with pres­id­ents. So we went, Re­agan shook my hand, and we cel­eb­rated with Raoul Bott, who was also a Medal­ist.

The in-between years, 1982–1997

In the mid 1980s I re­mem­ber Mike say­ing that, with all those awards, es­pe­cially the Fields Medal, he needed to not only pro­duce re­search but look like a real math­em­atician, so he in­ten­ded to fill in some of the gaps in his know­ledge of math­em­at­ics due to his ab­bre­vi­ated years in school and his fo­cus on 4-man­i­folds. He did so re­mark­ably well, in my opin­ion, but at the same time con­tin­ued his work in low-di­men­sion­al to­po­logy.

Wheth­er or not Mike’s work on 4-man­i­folds covered all fun­da­ment­al groups, or just the good ones, led to what is called the A-B-slice prob­lem, which Slava Krushkal sum­mar­izes in his com­ment­ary on those pa­pers. Mike and Slava are cur­rently skep­tic­al that all groups are good.

A quantum computer

In the fall of 1988, Mike vis­ited Har­vard on sab­bat­ic­al. At that time, Wit­ten’s pre­print [e11] on quantum field the­ory and the Jones poly­no­mi­al had ap­peared, and Raoul Bott and Cliff Taubes or­gan­ized a sem­in­ar centered on the pre­print. Al­though the pre­print had no men­tion of fram­ings on links, which was sor­ted out soon, Mike thought the pa­per was re­volu­tion­ary and that quantum field the­ory was a power­ful tool.

It had also been shown [e13] that com­put­ing the val­ues of the Jones poly­no­mi­al is \#P-hard, that is, it be­longs to the class of NP prob­lems where, in­stead of ask­ing wheth­er there is a solu­tion, it asks how many solu­tions. This led Mike to think about build­ing a quantum com­puter to solve these prob­lems; as he wrote later [3],

Non-abeli­an to­po­lo­gic­al quantum field the­or­ies ex­hib­it the math­em­at­ic­al fea­tures ne­ces­sary to sup­port a mod­el cap­able of solv­ing all #P prob­lems, a com­pu­ta­tion­ally in­tract­able class, in poly­no­mi­al time. Spe­cific­ally, Wit­ten [e11] has iden­ti­fied ex­pect­a­tion val­ues in a cer­tain \( \mathit{SU}(2) \) field the­ory with val­ues of the Jones poly­no­mi­al that are #P-hard. This sug­gests that some phys­ic­al sys­tem whose ef­fect­ive Lag­rangi­an con­tains a non-abeli­an to­po­lo­gic­al term might be ma­nip­u­lated to serve as an ana­log com­puter cap­able of solv­ing NP- or even #P-hard prob­lems in poly­no­mi­al time. De­fin­ing such a sys­tem and ad­dress­ing the ac­cur­acy is­sues in­her­ent in pre­par­a­tion and meas­ure­ment is a ma­jor un­solved prob­lem.

Mike was bark­ing up the wrong phys­ics tree for a while. Know­ing \( \mathit{SU}(2) \) was as­so­ci­ated to the weak force, he pre­sumed some kind of nuc­le­ar phys­ics would be re­quired to build a quantum com­puter. He only learned around 1995–96 about a pa­per of Greg Moore and Nich­olas Read [e14] in which “the mod­ern tech­niques of two-di­men­sion­al con­form­al field the­ory are ap­plied to the the­ory of the frac­tion­al quantum Hall ef­fect.” This pa­per shows that the “ef­fect­ive” low-en­ergy field the­ory might be an \( \mathit{SU}(2) \) gauge the­ory. Since 2005, Mike’s re­search unit has been sup­port­ing ex­per­i­ment­al work at nu­mer­ous loc­a­tions around the world, to ex­plore to­po­lo­gic­al phases of mat­ter. Much of this re­search has been on the frac­tion­al quantum Hall ef­fect, and on Ma­jor­ana sys­tems built from semi­con­duct­or/su­per­con­duct­or in­ter­faces.

Microsoft and Station Q

Mike felt his re­search fo­cus was chan­ging, and that it was not a good idea for him to train gradu­ate stu­dents in areas where he was a be­gin­ner him­self. When he vis­ited Mi­crosoft in 1996 to give a talk, Nath­an Myhr­vold, Chief Tech­no­logy Of­ficer at Mi­crosoft, offered him a job on the spot. This seemed a good fit for Mike’s new in­terests, and he left UC San Diego for Mi­crosoft in Ju­ly 1997.

He con­vinced Mi­crosoft to open a re­search group, called Sta­tion Q, on the UC Santa Bar­bara cam­pus on 15 March 2005. The ori­gin­al group in­cluded Chet­an Nayak, Alexei Kit­aev, Kev­in Walk­er, and Zhenghan Wang, and has ex­pan­ded since.

When I asked Mike about fa­vor­ite pa­pers out­side 4-man­i­fold the­ory, his im­me­di­ate re­sponse was his pa­per with Danny Calegari and Kev­in Walk­er [4] on the pos­it­iv­ity of uni­ver­sal pair­ings in di­men­sion 3. Kev­in gives his per­spect­ive on this work in his com­ment­ary.


For a while in the 1980s, Mike worked sum­mers for ‘JASON,’ an in­de­pend­ent group of young­er sci­ent­ists chosen to ad­vise the US gov­ern­ment in out-of-the-or­din­ary ways. It was es­tab­lished in 1960, fun­ded through the MITRE Cor­por­a­tion, and in­cluded dis­tin­guished sci­ent­ists (e.g., Free­man Dys­on, Sid­ney Drell, Wal­ter Munk, Richard Muller, and Mar­shall Rosen­bluth).

While there, Mike had an idea, per­haps too out of the or­din­ary even for JASON, which was to of­fer US state­hood to Panama, Kuwait and Is­rael; they would get se­cur­ity and vari­ous be­ne­fits of be­ing a part of the US, and the US would get in re­turn the Panama Canal, oil, and brains. Well, it would nev­er ac­tu­ally hap­pen, but the story is that Sid Drell men­tioned it to then-Sec­ret­ary of State George Shultz over din­ner, but without men­tion­ing Mike’s name so as to save his repu­ta­tion! A ver­sion of this idea, titled “A pro­pos­al on Panama”, can be read from Mike’s bib­li­o­graphy on this site.

In ad­di­tion, Mike has tried his hand at writ­ing vign­ettes from his early “ad­ven­tures”, also lis­ted as “Night climb­ing” and “Jon Loni’s Stoney Point mas­sacre: A story twice re­told”. Both are avail­able here as well.


[1] incollection M. Freed­man and R. Kirby: “A geo­met­ric proof of Roch­lin’s the­or­em,” pp. 85–​97 in Al­geb­ra­ic and geo­met­ric to­po­logy (Stan­ford Uni­versity, CA, Au­gust 2–21, 1976), part 2. Edi­ted by R. J. Mil­gram. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics XXXII. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1978. MR 0520525 Zbl 0392.​57018

[2] article M. H. Freed­man: “The to­po­logy of four-di­men­sion­al man­i­folds,” J. Dif­fer­en­tial Geom. 17 : 3 (1982), pp. 357–​453. MR 0679066 Zbl 0528.​57011

[3] article M. H. Freed­man: “P/NP, and the quantum field com­puter,” Proc. Natl. Acad. Sci. USA 95 : 1 (1998), pp. 98–​101. MR 1612425 Zbl 0895.​68053

[4] article D. Calegari, M. H. Freed­man, and K. Walk­er: “Pos­it­iv­ity of the uni­ver­sal pair­ing in 3 di­men­sions,” J. Amer. Math. Soc. 23 : 1 (2010), pp. 107–​188. MR 2552250 Zbl 1201.​57024 ArXiv 0802.​3208