One of the most important pieces of mathematical knowledge was on the verge of being lost, maybe forever. Now, a new book hopes to save it.
The Disc Embedding Theorem rewrites a proof completed in 1981 by Michael Freedman — about an infinite network of discs — after years of solitary toil on the California coast. Freedman’s proof answered a question that at the time was one of the most important unsolved questions in mathematics, and the defining problem in Freedman’s field, topology.
Freedman’s proof felt miraculous. Nobody at the time believed it could possibly work — until Freedman personally persuaded some of the most respected people in the field. But while he won over his contemporaries, the written proof is so full of gaps and omissions that its logic is impossible to follow unless you have Freedman, or someone who learned the proof from him, standing over your shoulder guiding you.
“I probably didn’t treat the exposition of the written material as carefully as I should have,” said Freedman, who today leads a Microsoft research group at the University of California, Santa Barbara focused on building a quantum computer.
Consequently, the miracle of Freedman’s proof has faded into myth.
Today, few mathematicians understand what he did, and those who do are aging out of the field. The result is that research involving his proof has withered. Almost no one gets the main result, and some mathematicians have even questioned whether it’s correct at all.
In a 2012 post on MathOverflow, one commenter referred to the proof as a “monstrosity of a paper” and said he had “never met a mathematician who could convince me that he or she understood Freedman’s proof.”
The new book is the best effort yet to fix the situation. It is a collaboration by five young researchers who were captivated by the beauty of Freedman’s proof and wanted to give it 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 important but inaccessible piece of mathematics into something that a motivated undergraduate could learn in a semester.
“There is nothing left to the imagination anymore,” 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 Technology and Economics, Min Hoon Kim of Chonnam National University in South Korea, and Mark Powell of Durham University in the U.K. “It’s all nailed down.”
In 1974, Michael Freedman was 23 years old, and he had his eye on one of the biggest problems in topology, a field of math which studies the basic characteristics of spaces, or manifolds, as mathematicians refer to them.
It was 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 characteristics 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 on the other while maintaining relative distances between points, so that points that are close together on the first manifold remain close together on the second.)
Poincaré was specifically thinking of three-dimensional manifolds, but mathematicians went on to consider manifolds of all dimensions. They also wondered if the conjecture held for two types of manifolds. The first type, known as a “smooth” manifold, doesn’t have any features like sharp corners, allowing you to perform calculus at every point. The second, known as a “topological” manifold, can have corners where calculus is impossible.
By the time Freedman started work on the problem, mathematicians had made a lot of progress on the conjecture, including solving the topological version of it in dimensions 5 and higher.
Freedman focused on the four-dimensional topological conjecture. It stated that any topological manifold that’s a four-dimensional “homotopy” sphere, which is loosely equivalent to a four-dimensional sphere, is in fact homeomorphic (strongly equivalent) to the four-dimensional sphere.
“The question we’re asking is, [for the four-sphere], is there a difference between these two notions of equivalence?” said Ray.
The four-dimensional version was arguably the hardest version of Poincaré’s problem. This is due in part to the fact that the tools mathematicians used to solve the conjecture in higher dimensions don’t work in the more constrained setting of four dimensions. (Another contender for the hardest version of the question is the three-dimensional Poincaré conjecture, which wasn’t solved until 2002, by Grigori Perelman.)
At the time Freedman set to work, no one had any fully developed idea for how to solve it — meaning that if he was going to succeed, he was going to have to invent wildly new mathematics.
Curves That Count
Before getting into how he proved the Poincaré conjecture, it’s worth digging a little more into what the question is really asking.
A four-dimensional homotopy sphere can be characterized by the way curves drawn inside it interact with each other: The interaction tells you something essential about the larger space in which they’re interacting.
In the four-dimensional case, these curves will be two-dimensional planes (and in general, the curves will be at most half the dimension of the larger space they’re drawn inside). To understand the basic setup, it’s easier to consider a simpler example involving one-dimensional curves intersecting inside two-dimensional space, like this:
These curves have something called an algebraic intersection number. To calculate this number, work left to right and assign a −1 to every place they intersect in which the arc is ascending and a +1 to every place they intersect where the arc is descending. In this example, the leftmost intersection gets a −1 and the rightmost intersection gets a +1. Add them together and you get the algebraic intersection number for these two curves: 0.
A homotopy sphere has the feature that any pair of half-dimensional curves drawn inside it has an algebraic intersection number of 0.
This is true for the regular sphere, too. But the regular sphere also has a slightly different property related to intersections: You can always draw two curves so that they don’t intersect each other at all. So while a homotopy sphere has the property that a pair of curves always has an algebraic intersection number of 0, the regular sphere has the property that any pair of curves can be separated from each other so that they have a geometric intersection number of 0. That is, they literally don’t intersect at all.
For Freedman to prove the four-dimensional Poincaré conjecture, he needed to show that it’s always possible to take particular pairs of curves with algebraic intersection 0 and “push” them off each other so that their geometric intersection number is still 0. If you have pairs of curves with algebraic intersection 0, and you prove you can always push them apart, you prove that the space they’re embedded in must be the regular sphere.
“It’s like social distancing for these half-dimensional submanifolds,” said Ray.
Previous work on higher-dimensional versions of the problem had established a method for doing this. It involved looking for objects called Whitney discs, which are flat two-dimensional spaces bounded by the curves you want to 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 you do so, the disc starts to vanish, like a setting sun. Eventually, the disc disappears completely, and the curves have been separated.
“The Whitney disc is giving you the path of the isotopy. You’re continuously moving one curve until 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 four-dimensional Poincaré conjecture, was to prove that these flat Whitney discs were present whenever you had a pair of intersecting curves with algebraic intersection 0. Establishing that it was true took Freedman to unimaginable new heights of mathematics.
As Freedman worked, he confronted a particular stumbling block that comes up in four dimensions. He needed to prove that it’s always possible to separate intersecting 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 is that in four dimensions, the two-dimensional Whitney discs can intersect themselves, rather than lying flat. The places that a disc intersects itself form obstructions to the process of sliding one curve off the other. You can think of the self-intersection as a snag that catches one of your 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 intersects itself,” said Ray.
So Freedman needed to prove that it’s always possible to undo the places the Whitney discs intersect themselves, lay them flat and then proceed with the separation. Luckily for him, he wouldn’t be starting from scratch. In the 1970s, a mathematician named Andrew Casson came up with a strategy for removing the self-intersections from discs.
The point of the discs is to establish that it’s possible to separate curves so that they don’t intersect. If a disc itself contains an intersection, the method for alleviating it is the same: Look for a second disc bounded by the intersecting parts of the first disc. If you find that second disc, you know you can iron out the intersection in the first disc.
OK, but what if the second disc — which is helping the first disc — also intersects itself? Then you look for a third disc contained in the second disc. However, that disc could intersect itself as well, so you look for a fourth disc, and the process goes on, forever, producing an infinite stack of discs inside discs — all erected in the hope of establishing that the original disc, all the way at the bottom, can be made to not intersect itself.
Casson established that these “Casson handles” are loosely equivalent to actual Whitney discs — homotopy equivalent, to put it more precisely — and he used this equivalence to investigate many important questions in four-dimensional topology. But he could not prove that Casson handles are equivalent to discs in an even stronger sense — that they’re homeomorphic to discs. This stronger equivalence is what mathematicians needed in order to use the handles to prove the biggest open question of all.
“If we show these are actual honest-to-goodness discs, we could prove the Poincaré conjecture and a whole bunch of other things in dimension four,” said Ray. “But [Casson] couldn’t do it.”
It took Freedman seven years, from 1974 to 1981, but he managed it. Most of that time he barely talked to anyone about what he was up to, save his older colleague Robert Edwards, who served as a kind of mentor.
“He locked himself up for seven years in [San Diego] to think about 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 at the University of California, Berkeley, was one of the first mathematicians to learn about Freedman’s proof. To assess the magnitude of major mathematical results, Kirby tries to imagine how long it would have taken before someone else came up with it, and by this standard Freedman’s proof is the most amazing result Kirby has seen in his long career.
“If he hadn’t done it, I can’t imagine who would have 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 have a Casson handle, you have a Whitney disc, and if you have 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 objects — the Casson handle and the 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 could build almost all of the Casson handle and almost all of the disc with the same components.
But there were places where he couldn’t quite complete the picture — as if he were creating a portrait and there were some aspects of his subject’s face he couldn’t see. His last move, then, was to prove that those gaps in his picture — the places he couldn’t see — didn’t matter from the standpoint of the type of equivalence he was after. That is, the gaps in the picture could not possibly prevent the Casson handle from being homeomorphic to the disc, no matter what they contained.
“I have 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 techniques from an area of math called Bing topology, after the mathematician R.H. Bing, who developed it in the 1940s and ’50s. But he applied them in a completely novel setting to generate a conclusion that seemed nearly preposterous — that in the end, the gaps didn’t matter.
“That’s what made the proof so remarkable and made it so unlikely that anybody else would have found it,” said Kirby.
Freedman completed his outline of the proof in the summer of 1981. The factors that would ultimately place it at risk of being lost to mathematical memory became apparent soon after.
Spreading the News
Freedman announced his proof at a small conference at the University of California, San Diego, that August. About 10 of the most respected mathematicians, with the best chance of understanding Freedman’s work, attended.
Ahead of the event he sent out copies of a 20-page handwritten manuscript outlining his proof. On the conference’s second evening, Freedman began presenting his work. He couldn’t finish in one sitting, so his talk carried over to the next night. When he finished, his small audience was bewildered — Freedman’s mentor, Edwards, among them. In a 2019 interview about 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 in the audience found his presentations to be both mind-boggling and incomprehensible, thinking that his ideas were harebrained and crazy,” Edwards said.
Freedman’s proof seemed improbable in large part because it wasn’t really fleshed out. He had an idea for how the proof should go and a strong, almost preternatural intuition that the approach would work. But he hadn’t actually carried it out all the way.
“I couldn’t imagine how Mike had the nerve to announce a proof when he was so shaky on the 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 more days of conversation, Edwards had enough of a sense of what Freedman was trying to do to evaluate whether it really worked. And on the first Saturday morning after the conference, he realized that it did.
“[Edwards] said, ‘I’m the first person who really knows this is true,’” said Kirby.
Once Edwards was convinced, he helped convince others. And in a way, that was enough. There is no high commission of mathematics that officially certifies results as correct. The actual process by which a new statement is accepted is more informal, relying on the assent of the members of the mathematical community who are supposed to know best.
“Truth in mathematics means you convince the experts that your proof is correct. Then it becomes true,” said Teichner. “Freedman convinced all the experts that his proof is correct.”
But 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 people who had never met him could read and learn on their own. And that is what he never produced.
Freedman submitted the outline of his proof — which was all he really had — to the Journal of Differential Geometry. The journal’s editor, Shing-Tung Yau, assigned it to an outside expert for review before deciding whether to publish it — a standard safeguard in all academic publishing. But the 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, and the editors of the journal grew restless. Edwards received regular calls from the journal’s secretary asking if he had a verdict on the legitimacy of the proof. In that same 2019 interview Edwards explained that finally, he told the journal the proof was right, even though he knew he hadn’t had time to fully check it out.
“The next time the secretary called I said ‘Yes, the paper is correct, I assure you. But I can’t generate a proper referee’s report any time soon.’ So they decided to accept and published it as it was,” he said.
The paper appeared in 1982. It contained typos and misspellings and was still effectively the same outline Freedman had circulated right after he’d finished the work. Anyone trying to read it would need to fill in many steps of the wholly novel argument on their own.
The limitations of the published article were evident right away, but no one stepped forward to address them. Freedman moved on to other work and stopped lecturing on his Poincaré proof. Almost a decade later, in 1990, a book appeared that tried to present a more accessible version of the proof. It was by Freedman and Frank Quinn, now at the Virginia Polytechnic Institute and State University, though it was primarily written by Quinn.
The book version was hardly more 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 around those people who understood the proof, you could still learn it,” said Teichner. “But people who went back to the [written] sources realized they couldn’t.”
And for decades, that is where things remained: One of the most amazing results in the history of mathematics was known by a few people and inaccessible to everyone else.
The rest of the math world might have 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 lots of other theorems about four-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. “But that doesn’t mean you want to take everything on faith.”
And over time, as younger researchers entered mathematics and could choose to work in any area they wanted, fewer chose to work with the proof at all.
Freedman understood. “It’s not so satisfying to work in an area where you don’t understand the fundamental theorem,” he said. “Basically, the situation arose where no one under 40 years old knew the proof, and it was a little frightening that this bit of information might eventually be lost.”
It was at this point that Teichner — who’d learned the proof in the early 1990s from Freedman himself — decided to launch a rescue mission. He wanted to create a text that would allow any qualified person to learn the proof on their own.
“I decided it’s about time we write something you can understand,” he said.
Teichner began by going straight back to the source. In 2013 he asked Freedman to give a series of lectures over the course of a semester at the Max Planck Institute describing the proof — a modern-day version of the talks he’d delivered 30 years earlier to announce the result. Freedman agreed eagerly.
“He was definitely worried it would be lost. That’s why he was so supportive,” said Teichner.
Back in 1981, Freedman had lectured to a handful of senior figures in the 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 in the topology world.
“In my institution we used to have Friday afternoon Freedman lectures, where we’d get a beer and watch him talk about his proof,” said Ray, who was a graduate student at Rice University in Houston at the time.
After the lectures the mathematician Stefan Behrens led an effort to turn Freedman’s remarks into more formal lecture notes. Several years later, in 2016, Powell and other mathematicians, including Behrens, delivered a new series of lectures based on those notes, continuing the process of transforming Freedman’s work into something more durable.
“Mark gave lectures and we started filling in more and more details to those lecture notes and then it sort of went from there,” said Ray.
Over the next five years, Powell, Ray and their three co-editors organized a team of mathematicians to turn Freedman’s proof into a book. The final product, released in July, is almost 500 pages and includes contributions from 20 different authors. Freedman hopes the book will revitalize research in the area of math he revolutionized.
“I think the book comes at a good moment. People are looking at four-manifolds with fresh eyes,” he said.
The book improves on the written presentation of Freedman’s proof in several ways. While writing the book, the authors discovered a handful of errors in the arguments Freedman used to prove different theorems in the original journal article. The book fixes those. It also provides a thorough introduction to Bing topology, the area of math Freedman used to prove that the gaps in his 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 later chapters then fill in.
“Having summaries, and then more detailed summaries, then full details, is supposed to make it readable,” said Powell. “You can get the big picture of what’s going to happen before you get all the details. But we still have all the details.”
The editors hope to propel Freedman’s powerful techniques back into the mainstream of mathematical thinking. The third part of the book details the biggest open problems in four-dimensional topology that researchers might approach once equipped with knowledge of Freedman’s proof.
“This part of the book has absolutely nothing to do with the proof of Freedman’s original work,” said Ray. “It talks about how to use this to do what comes next.”
And already, several of the mathematicians involved with the book have produced new research that builds on Freedman’s ideas. One paper, posted in 2013 at the very beginning of the book process, finds some new uses for previously dormant techniques in Bing topology. Another, from last year, uses ideas that the editors learned assembling the book to address a question about “surgery” on knots in four-dimensional manifolds.
“It’s now moving forward because they’re comfortable using the disc-embedding theorem,” said Teichner.
The book serves an instrumental purpose within the field of mathematics, maybe even an essential one. But the editors say that they were motivated by more than practical ends to see the long project through. When they started the work, Freedman’s proof was beautiful, but hidden. Now, at last, it’s on full display.
Correction: September 10, 2021
Freedman announced his proof at the University of California, San Diego, not at the University of San Diego. The article has been revised accordingly. A figure depicting intersecting curves has also been revised to more accurately reflect the contents of the article.