1-odda most primordial pieces of mathematical knowledge was onna verge of bein’ lost, maybe forever. now, a new book hopes to save it.

*the disc embedding theorem* rewrites a proof completed in 1981 by michael freedman — bout an ∞ network of discs — after yrs of solitary toil onna california coast. freedman’s proof answered a ? that atta time was 1-odda most primordial unsolved ?s in mathematics, na defining problem in freedman’s field, topology.

freedman’s proof felt miraculous. nobody atta time believed it ‘d possibly work — til freedman personally persuaded somd' most respected pplz inna field. but while he won over his contemporaries, the written proof is so full of gaps and omissions that its logic is impossible to follo unless you ‘ve freedman, or some1 who learned the proof from him, standing over yr ‘der guiding you.

“i probably didn’t treat the exposition of the written material as carefully as i ‘d ‘ve,” said freedman, who tody leads a microsoft research group atta university of california, santa barbara focused on building a quantum computer.

consequently, the miracle of freedman’s proof has faded into myth.

tody, few mathematicians cogg wha’ he did, and those who do are aging out of the field. the result s'dat research involving his proof has withered. almost no one gets the main result, and some mathematicians ‘ve even ?ed whether it’s correct at all.

in a 2012 post on mathoverflo, one commenter referred to the proof as a “monstrosity offa paper” and said he had “never met a mathematician who ‘d convince me that he or she understood freedman’s proof.”

the new book tis best effort yet to fix the situation. tis a collaboration by 5 young researchers who were captivated by the beauty of freedman’s proof and wanted t'giv't new life. over nearly 500 pages, it spells out the steps of freedman’s argument in complete detail, using clear, consistent terminology. the goal was to turn this primordial but inaccessible piece of mathematics into something dat a' motivated undergraduate ‘d learn in a semester.

“thris nothing left to the imagination any+,” said arunima ray of the max planck institute for mathematics in bonn, co-editor of the book along with stefan behrens of bielefeld university, boldizsár kalmár of the budapest university of tek and economics, min hoon kim of chonnam national university in south korea, and mark powell of durham university inna u.k. “it’s all nailed down.”

## sorting spheres

in 1974, michael freedman was 23 yrs old, and he had his eye on 1-odda biggest problems in topology, a field of math which studies the basic toonistics of spaces, or manifolds, as mathematicians refer to them.

twas called the poincaré conjecture, after the french mathematician henri poincaré, who’d posed it in 1904. poincaré predicted that any shape, or manifold, with certain generic toonistics must be equivalent, or homeomorphic, to the sphere. (two manifolds are homeomorphic when you can take all the points on one and map them over to points onna other while maintaining relative distances tween points, so that points tha're close together onna 1st manifold remain close together onna 2nd.)

poincaré was specifically thinking of 3-dimensional manifolds, but mathematicians went onna ponder manifolds of all dimensions. they also wandaed if the conjecture held for two types of manifolds. the 1st type, known as a “smooth” manifold, doesn’t ‘ve any features like sharp corners, alloing you to perform calculus at every point. the 2nd, known as a “topological” manifold, can ‘ve corners where calculus is impossible.

by the time freedman started work onna problem, mathematicians had made a lotta progress onna conjecture, including solving the topological version o'it in dimensions 5 and higher.

freedman focused onna 4-dimensional topological conjecture. it stated that any topological manifold that’s a 4-dimensional “homotopy” sphere, which is loosely equivalent to a 4-dimensional sphere, is in fact homeomorphic (strongly equivalent) to the 4-dimensional sphere.

“the ? we’re asking is, [for the 4-sphere], is there a difference tween these two notions of equivalence?” said ray.

the 4-dimensional version was arguably the hardest version of poincaré’s problem. this is due in pt to the fact that the tulz mathematicians used to solve the conjecture in higher dimensions don’t work inna + constrained setting of 4 dimensions. (another contender for the hardest version of the ? tis 3-dimensional poincaré conjecture, which wasn’t solved til 2002, by grigori perelman.)

atta time freedman set t'work, no one had any fully developed idea for how to solve it — meaning that if he was goin to succeed, he was goin to ‘ve to invent wildly new mathematics.

## curves that count

b4 gettin into how he proved the poincaré conjecture, it’s worth digging a lil + into wha’ the ? is really asking.

a 4-dimensional homotopy sphere can be toonized btw curves drawn inside it interact with each other: the interaction tells you something primordial bout the larger space in which they’re interacting.

inna 4-dimensional case, these curves ll'be two-dimensional planes (and in general, the curves ll'be at most ½ the dimension of the larger space they’re drawn inside). to cogg the basic setup, it’s easier to ponder a simpler ex involving one-dimensional curves ∩ing inside two-dimensional space, like this:

these curves ‘ve something called an algebraic ∩ion №. to calcul8 this №, work left to rite and assign a −1 to every place they ∩ in which the arc is ascending and a +1 to every place they ∩ where the arc is descending. in this ex, the leftmost ∩ion gets a −1 na ritemost ∩ion gets a +1. add them together and you get the algebraic ∩ion № for these two curves: 0.

a homotopy sphere has the feature that any pair of ½-dimensional curves drawn inside t'has an algebraic ∩ion № of 0.

this is true for the regular sphere, too. but'a regular sphere also has a slitely ≠ property rel8d to ∩ions: you can always draw two curves so t'they don’t ∩ each other at all. so while a homotopy sphere has the property dat a' pair of curves always has an algebraic ∩ion № of 0, the regular sphere has the property that any pair of curves can be separated from each other so t'they ‘ve a geometric ∩ion № of 0. that is, they literally don’t ∩ at all.

for freedman to prove the 4-dimensional poincaré conjecture, he needed to show that it’s always possible to take pticular pairs of curves with algebraic ∩ion 0 and “push” them off each other so that their geometric ∩ion № is still 0. if you ‘ve pairs of curves with algebraic ∩ion 0, and you prove you can always push them apt, you prove that the space they’re embedded in must be the regular sphere.

“it’s like social distancing for these ½-dimensional submanifolds,” said ray.

previous work on higher-dimensional versions of the problem had established a method for doin’ this. it involved looking for essentialisms called whitney discs, which are flat two-dimensional spaces bounded by the curves you wanna separate.

these discs become a kind of guide for a mathematical process called isotopy in which you move two curves away from each other. the presence of these flat whitney discs ensures that it’s possible to gradually shift the arcing curve down. as ye do so, the disc starts to vanish, like a setting sun. eventually, the disc disappears completely, na curves ‘ve been separated.

“the whitney disc is giving you the path of the isotopy. you’re continuously movin one curve til the two curves are separate. the disc is like a road map for this process,” said ray.

freedman’s main task, as he confronted the 4-dimensional poincaré conjecture, was to prove that these flat whitney discs were present whenever you had a pair of ∩ing curves with algebraic ∩ion 0. establishing that twas true took freedman to unimaginable new h8s of mathematics.

## unknotting discs

as freedman worked, he confronted a pticular stumbling block that comes up in 4 dimensions. he needed to prove that it’s always possible to separate ∩ing two-dimensional curves — to push them off each other — and to do that he had to establish the presence of whitney discs, which ensure the separation is possible.

the trouble s'dat in 4 dimensions, the two-dimensional whitney discs can ∩ themselves, rather than lying flat. the places dat a' disc ∩s itself form obstructions to the process of sliding one curve off the other. you can think of the self-∩ion as a snag that catches one of yr curves as you’re trying to pull it off the other.

“the disc was supposed to help me, but it turns out the disc also ∩s itself,” said ray.

so freedman needed to prove that it’s always possible to undo the places the whitney discs ∩ themselves, lay them flat and then proceed w'da separation. ♣ily for him, he ‘dn’t be starting from scratch. inna 1970s, a mathematician named andrew casson came up witha strategy for removin the self-∩ions from discs.

the point of the discs is to establish that it’s possible to separate curves so t'they don’t ∩. if a disc itself contains an ∩ion, the method for alleviating tis the same: look for a 2nd disc bounded by the ∩ing pts of the 1st disc. if you find that 2nd disc, you know you can iron out the ∩ion inna 1st disc.

ok, but wha’ if the 2nd disc — which is helping the 1st disc — also ∩s itself? then you look for a third disc contained inna 2nd disc. however, that disc ‘d ∩ itself swell, so u look for a 4th disc, na process goes on, forever, producing an ∞ stack of discs inside discs — all erected inna hope of establishing that the original disc, all the way atta bottom, can be made to not ∩ itself.

casson established that these “casson handles” are loosely equivalent to actual whitney discs — homotopy equivalent, to put it + precisely — and he used this equivalence to investigate many primordial ?s in 4-dimensional topology. but he ‘d not prove that casson handles are equivalent to discs in an even stronger sense — t'they’re homeomorphic to discs. this stronger equivalence is wha’ mathematicians needed in order to use the handles to prove the biggest open ? of all.

“if we show these are actual honest-to-goodness discs, we ‘d prove the poincaré conjecture and a whole bunch of other things in dimension 4,” said ray. “but [casson] ‘dn’t dweet.”

## freedman’s insite

it took freedman 7 yrs, from 1974 to 1981, but he managed it. most odat time he barely talked to any-1 bout wha’ he was up to, save his older colleague robert edwards, who served as a kind of mentor.

“he locked himself up for 7 yrs in [san diego] to think bout this. he didn’t interact much with anybody else while he was figuring it out,” said peter teichner of the max planck institute for mathematics.

robion kirby, now atta university of california, berkeley, was 1-odda 1st mathematicians to learn bout freedman’s proof. to assess the magnitude of major mathematical results, kirby tries to imagine how long it ‘d ‘ve taken b4 some1 else came up with it, and by this standard freedman’s proof tis most amazing result kirby has seen onnis long career.

“if he hadn’t done it, i can’t imagine who ‘d ‘ve for i don’t know how long,” said kirby.

freedman needed to prove that casson handles were strongly equivalent to flat whitney discs: if you ‘ve a casson handle, you ‘ve a whitney disc, and if you ‘ve a whitney disc, you can separate curves, and if you can separate curves, you’ve established that the homotopy sphere is homeomorphic to the actual sphere.

his strategy was to show that you can build both essentialisms — the casson handle na flat whitney disc — out of the same set of pieces. the idea was that if you can build two things out of the same pieces, they must be equivalent in some sense. freedman began the construction process and got pretty far with it: he ‘d build almost all odda casson handle and almost all odda disc w'da same components.

but there were places where he ‘dn’t quite complete the picture — as if he were creating a portrait and there were some aspects of his subject’s face he ‘dn’t see. his last move, then, was to prove that those gaps onnis picture — the places he ‘dn’t see — didn’t matter from the standpoint of the type of equivalence he was after. that is, the gaps inna picture ‘d not possibly prevent the casson handle from bein’ homeomorphic to the disc, no matter wha’ they contained.

“i ‘ve two jigsaw puzzles and 99 out of 100 pieces match. are these leftover bits actually changing my space? freedman showed they’re not,” said ray.

to perform this final move, freedman drew on tek knicks from an zone of math called bing topology, after the mathematician r.h. bing, who developed it inna 1940s and ’50s. but he applied them in a completely novel setting to generate a conclusion that seemed nearly preposterous — that inna end, the gaps didn’t matter.

“that’s wha’ made the proof so remarkable and made it so unlikely that anybody else ‘d ‘ve found it,” said kirby.

freedman completed his outline of the proof inna summer of 1981. the factors that ‘d ultimately place it at risk of bein’ lost to mathematical memory became apparent soon after.

## spreading the news

freedman anncd his proof at a lil conference atta university of california, san diego, that aug. bout 10 of the most respected mathematicians, w'da best chance of cogging freedman’s work, attended.

ahead of the event he sent out copies offa 20-page handwritten manuscript outlining his proof. onna conference’s 2nd evening, freedman began presenting his work. he ‘dn’t finish in one sitting, so his talk carried over to the nxt nite. when he finished, his lil audience was bewildered — freedman’s mentor, edwards, among them. in a 2019 interview bout the proceedings, edwards recalled the sense of shock — and skepticism — with which freedman’s talk was received.

“i think it’s fair to say that everyone inna audience found his presentations to be both Ψ-boggling and incomprehensible, thinking that his ideas were harebrained and crazy,” edwards said.

freedman’s proof seemed improbable in large pt cause twasn’t really fleshed out. he had an idea for how the proof ‘d go and a strong, almost preternatural intuition that the approach ‘d work. but he hadn’t actually carried it out all the way.

“i ‘dn’t imagine how mike had the nerve to announce a proof when he was so shaky onna details,” said kirby, who also attended the conference.

but afterward, several mathematicians stayed to talk with freedman. the magnitude of the potential result seemed to merit that, at least. after two + dys of conversation, edwards had enough offa sense of wha’ freedman was trying to do to cogitate whether it really worked. and onna 1st sat morning after the conference, he realized that it did.

“[edwards] said, ‘i’m the 1st person who really knows this is true,’” said kirby.

once edwards was convinced, he helped convince others. and in a way, twas' enough. thris no high commission of mathematics that officially certifies results as correct. the actual process by which a new statement is accepted is + informal, relying onna assent of the members of the mathematical community who are supposed to know best.

“truth in mathematics means you convince the experts that yr proof is correct. then it becomes true,” said teichner. “freedman convinced all the experts that his proof is correct.”

b'that by itself was not enough to promulgate the result through the field. to do that, freedman needed a written statement of the proof that pplz who had never met him ‘d read and learn on their own. and that is wha’ he never produced.

## movin on

freedman submitted the outline of his proof — which was all he really had — to the *journal of ≠ial geometry*. the journal’s editor, shing-tung yau, assigned it to an outside expert for review b4 deciding whether to publish it — a standard safeguard in all academic publishing. but'a person he assigned it to was hardly an objective expert: robert edwards.

the review still took time. the proof itself was 50 pages long, and edwards found he was writing a page of dense mathematical notes for each page of the proof. weeks passed, na editors of the journal grew restless. edwards received regular calls from the journal’s secretary asking if he had a verdict onna legitimacy of the proof. in that same 2019 interview edwards explained that finally, he told the journal the proof was rite, even though he knew he hadn’t had time to fully check it out.

“the nxt time the secretary called i said ‘yes, the paper is correct, i assure you. but i can’t generate a proper referee’s reprt any time soon.’ so they decided to accept and published it as twas,” he said.

the paper appeared in 1982. it contained typos and misspellings and was still effectively the same outline freedman had circul8d rite after he’d finished the work. any-1 trying to read it ‘d nd'2 fill in many steps of the wholly novel argument on their own.

the limitations of the published article were evident rite away, but no one stepped forward to address them. freedman moved onna other work and stopped lecturing on his poincaré proof. almost a decade l8r, in 1990, a book appeared that tried to present a + accessible version of the proof. twas by freedman and frank quinn, now atta virginia polyteknic institute and state university, though twas primarily written by quinn.

the book version was hardly + readable. it assumed readers brought a certain amount of background knowledge to the book that almost no one actually had. there was no way to read it and learn the proof from the ground up.

“if you were fortunate enough to be round those pplz who understood the proof, you ‘d still learn it,” said teichner. “but pplz who went back to the [written] srcs realized they ‘dn’t.”

and for decades, that is where things remained: 1-odda most amazing results inna history of mathematics was known by a few pplz and inaccessible to everyone else.

the rest of the math realm mite ‘ve moved on as freedman had, but his proof was too monumental to fully ignore. so the community adapted to the strange set of circumstances. many researchers adopted freedman’s proof as a black box. if you assume his proof is correct you can prove lotso' other theorems bout 4-dimensional manifolds, and plenty of mathematicians did.

“if you just accept that it’s true, you can go and use it in many ways,” said powell. “b'that doesn’t mean you wanna take everything on faith.”

n'oer time, as younger researchers entered mathematics and ‘d choose t'work in any zone they wanted, fewer chose t'work w'da proof at all.

freedman understood. “it’s not so satisfying t'work in an zone where you don’t cogg the primordial theorem,” he said. “basically, the situation arose where no one under 40 yrs old knew the proof, and twas a lil friteening that this bit of information mite eventually be lost.”

twas at this point that teichner — who’d learned the proof inna early 1990s from freedman himself — decided to launch a rescue mission. he wanted to create a text that ‘d allo any qualified person to learn the proof on their own.

“i decided it’s bout time we write something you can cogg,” he said.

## future-proofing freedman

teichner began by goin straite back to the src. in 2013 he asked freedman t'give a series of lectures ‘oer the course offa semester atta max planck institute describing the proof — a modern-dy version of the talks he’d delivered 30 yrs earlier to announce the result. freedman agreed eagerly.

“he was definitely worried it ‘d be lost. that’s why he was so supportive,” said teichner.

backin 1981, freedman had lectured to a handful of senior figs inna field — the experts he needed to win over. this time his audience was a group of 50 young mathematicians teichner had brought together to receive the baton. the lectures, which freedman delivered by video feed from his office in santa barbara, were an event unto themselves inna topology realm.

“in my institution we used to ‘ve fri afternoon freedman lectures, where we’d get a beer and watch him talk bout his proof,” said ray, who was a graduate student at rice university in houston atta time.

after the lectures the mathematician stefan behrens led an effort to turn freedman’s remarks into + formal lecture notes. several yrs l8r, in 2016, powell nother mathematicians, including behrens, delivered a new series of lectures based on those notes, continuing the process of transforming freedman’s work into something + durable.

“mark gave lectures and we started filling in + and + details to those lecture notes and then it sort of went from there,” said ray.

‘oer the nxt 5 yrs, powell, ray and their 3 co-editors organized a team of mathematicians to turn freedman’s proof into a book. the final product, released in jul, is almost 500 pages and includes contributions from 20 ≠ authors. freedman hopes the book will revitalize research inna zone of math he revolutionized.

“i think the book comes at a good moment. pplz are looking at 4-manifolds with fresh eyes,” he said.

the book improves onna written presentation of freedman’s proof in several ways. while writing the book, the authors discovered a handful of errors inna arguments freedman used to prove ≠ theorems inna original journal article. the book fixes those. it also provides a thorough introduction to bing topology, the zone of math freedman used to prove that the gaps onnis constructions of the casson handle and whitney disc don’t matter. and altogether, the book is designed to be pedagogical and easy to approach. early chapters provide a broad outline of the proof that l8r chapters then fill in.

“having summaries, and then + detailed summaries, then full details, is supposed to make it readable,” said powell. “you can get the big picture of wha’’s goin to happen b4 you get all the details. b'we still ‘ve all the details.”

the editors hope to propel freedman’s uber tek knicks back inna'da mainstream of mathematical thinking. the third pt of the book details the biggest open problems in 4-dimensional topology that researchers mite approach once equipped with knowledge of freedman’s proof.

“this pt of the book has absolutely nothing to do w'da proof of freedman’s original work,” said ray. “it talks bout how to use this to do wha’ comes nxt.”

and already, several of the mathematicians involved w'da book ‘ve produced new research that builds on freedman’s ideas. one paper, posted in 2013 atta very beginning of the book process, finds some new uses for previously dormant tek knicks in bing topology. another, from last yr, uses ideas that the editors learned assembling the book to address a ? bout “surgery” on knots in 4-dimensional manifolds.

“it’s now movin forward cause they’re comfortable using the disc-embedding theorem,” said teichner.

the book serves an instrumental purpose within the field of mathematics, maybe even an primordial one. but'a editors say t'they were motivated by + than practical ends to see the long project through. when they started the work, freedman’s proof was presh, but hidden. now, at last, it’s on full display.

*correction: sep 10, 2021**freedman anncd his proof atta university of california, san diego, not atta university of san diego. the article s'been revised accordingly. a fig depicting ∩ing curves has also been revised to + accurately cogitate the contents of the article.*

original content at: www.quantamagazine.org…

authors: kevin hartnett

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