In a burst of work, mathematicians have moved closer to understanding just how many ways there are of contorting a simple sphere so that, in the end, it still resembles itself.
Much of the progress comes from results by Tadayuki Watanabe at Kyoto University. Over the last several years, he has figured out how to adapt a powerful technique for the broader purpose of studying spheres of any number of dimensions.
As a result, he resolved the last open case of an important problem known as the Smale conjecture, and along the way he created a surprising new way of looking at spheres.
The most familiar spheres are three-dimensional objects: baseballs and planets, for example. They have many obvious symmetries; you can flip or rotate them and they still look the same.
In 1959 a mathematician named Stephen Smale asked whether these basic symmetries of the three-dimensional sphere represent, in a way, all the symmetries it has. Over the subsequent decades mathematicians solved the Smale conjecture for the three-dimensional sphere (where those are, indeed, all the symmetries), as well as for spheres of nearly all dimensions (where, generally speaking, those are not all the symmetries). But there was one situation they couldn’t decipher: the four-dimensional sphere.
But then in late 2018, Watanabe solved the question there too. He proved that the four-dimensional sphere has families of symmetries that go well beyond the basic ones. In doing so, he finished the Smale conjecture and invigorated a search for those aberrant symmetries. Progress on that search has come rapidly, with contributions including a new paper by Watanabe in September.
“When you know that the Smale conjecture is not true, you shouldn’t stop there. You should try and figure out how far is it from being true,” said Alexander Kupers of the University of Toronto, Scarborough.
Mathematicians in the field of topology study the properties of shapes, called manifolds, like the sphere. They consider rigid symmetries, such as rotations and reflections, but also a slightly looser notion of symmetry that encompasses any way of rearranging points on a manifold while preserving some of its basic properties.
The sphere, for example, is smooth everywhere, meaning it’s possible to perform calculus at every point on its surface. If you rearrange the points so that the sphere you end up with is still smooth everywhere, you’ve performed a type of transformation that mathematicians call a diffeomorphism.
All rotations and reflections are diffeomorphisms of the sphere, but there are many other diffeomorphisms as well. For instance, you can keep most points where they are and restrict your rearranging to a small region of the sphere. There, let’s say you move one point and stretch the rest of the points in the region. This “point-pushing” strategy is just one of many examples of a diffeomorphism of the sphere that is not a rigid symmetry — neither a rotation nor a reflection.
Smale wanted to understand, all those years ago, how diffeomorphisms of the sphere relate to rigid symmetries of the sphere. There are more diffeomorphisms, but are they of a qualitatively different type?
Smale posed this question in a very particular way. He compared the two classes of symmetries by thinking of all rotations and reflections as forming their own space, and of all diffeomorphisms as forming a different space, and asked how similar the first space is to the second.
But what does it mean to turn symmetries into a space?
It’s possible to take each rotation or reflection of the sphere and represent it as a point in space. Rotations and reflections that are similar to each other — say, a rotation by 59 degrees and a rotation by 60 degrees — are represented by nearby points, and those that are different from one another are represented by more distant points. Altogether you end up with a space whose points index all rotations and reflections of the sphere.
You can do something equivalent for diffeomorphisms, such that the more similar two diffeomorphisms are, the closer together are the points that represent them. Again you end up with a space — different from the first one — whose points index all diffeomorphisms of the sphere.
The space representing diffeomorphisms will be larger, in a sense, because there are more diffeomorphisms than rigid symmetries. It will also contain the space representing rigid symmetries because rigid symmetries are themselves a subset of diffeomorphisms.
The two spaces are not exactly the same, but Smale wanted to know whether they were loosely equivalent in a particular way. When two spaces can be deformed to look like one another through stretching and shrinking, and without tearing, topologists say that they are “homotopy” equivalent. So, for example, a single point is homotopy equivalent to a disk, because you can shrink the disk to a point, or stretch the point to fill the disk.
This is the type of comparison Smale had in mind as he considered how rigid symmetries of the sphere compare to diffeomorphisms of the sphere. In particular, he wanted to know whether the space representing rigid symmetries was homotopy equivalent to the space representing diffeomorphisms for the three-dimensional sphere.
This way of phrasing the question captures something essential about the relationship between the two types of transformations. If the space of one can be deformed into the other, it indicates that in a sense, the types of transformations the two spaces represent are not all that different.
“Maybe the easiest way to say it is that the only symmetries of the sphere are the ones you know about that come from rigid rotations and reflections,” said Oscar Randal-Williams of the University of Cambridge.
In his 1959 paper, Smale himself proved that the conjecture is true for the two-dimensional sphere: Yes, the space of diffeomorphisms is homotopy equivalent to the space of rigid symmetries. In other words, in two dimensions, the sphere’s symmetries are pretty much limited to the obvious ones.
Over the next decade, mathematicians proved that the same statement is false for spheres of five or more dimensions. Then, in 1983, Allen Hatcher proved the conjecture is true in dimension three, settling Smale’s original question. With that, the final open case of the Smale conjecture was for the four-dimensional sphere. (For various technical reasons, the four-dimensional case is the hardest one to settle for many foundational questions in topology.)
Solving it required coming up with some kind of test, called an invariant, that could be applied to the two spaces in question to determine whether they are homotopy equivalent. Mathematicians had used invariants to prove the Smale conjecture in other dimensions, but none of those invariants worked in dimension four.
Often invariants involve building some kind of geometric construction on a manifold. One of the most basic invariants in topology involves cutting up a surface into uniform triangular tiles. Then you count the number of tile faces, add that to the number of vertices (corners where tiles meet), and subtract the number of tile edges. This invariant is called the Euler characteristic of a surface. If it comes out to be 2, you know the surface you started with is a sphere.
So when Watanabe approached the problem, he knew that he’d need a new kind of invariant capable of providing him with characteristic information about the spaces of symmetries of the four-dimensional sphere. He needed a test of sorts to tell them apart (or confirm they were the same). Luckily for him, another mathematician had been developing just such a tool.
Follow the Links
In the 1990s Maxim Kontsevich was on his way to becoming one of the most renowned mathematicians of his generation. Among many interests, he was studying topological objects called links, which are entangled loops — like two closed loops of string that wind around each other.
Given a link made of two loops, a basic question is: Can you pull the loops apart without pulling one through the other or ripping either of them? The answer isn’t always obvious, but invariants can tease it out.
“Your example may look very complicated, but you have to find an invariant that exactly detects that,” said Danica Kosanović of the Swiss Federal Institute of Technology Zurich.
In the early 1800s Carl Friedrich Gauss came up with an invariant that allows you to test whether the separation is possible. To calculate it, first designate your two loops as A and B. Now you’re going to inventory all the times one loop appears to pass in front of another using these rules:
- Each time Loop A passes over Loop B going left to right, assign a +1.
- Each time Loop A passes under Loop B going right to left, assign a +1.
- Each time Loop A passes under Loop B going left to right, assign a −1.
- Each time Loop A passes over Loop B going right to left, assign a −1.
Rather than hand-counting these interactions, Gauss devised a formula, based in calculus, for computing them. He considered two points, each constrained to move along one of the different loops, like trains on a track. Each time the loops cross, the corresponding +1 or −1 gets added.
The sum is called the linking number of two loops. If it is nonzero, you’ll know straightaway that the loops can’t be pulled apart. But if it’s zero you’re in an ambiguous situation: Some entangled loops with linking number zero can be pulled apart, while others can’t. In other words, Gauss’ invariant wasn’t powerful enough to completely characterize which links can be disentangled and which can’t.
In the 1990s, Kontsevich (inspired by the work of physicist Edward Witten) developed a more powerful version of Gauss’ linking number. While Gauss’ invariant was based on interactions between two points moving along loops in a link, Kontsevich realized he could say more about the structure of a link by considering the interactions between many points.
“If you combine these kinds of numbers of configurations in a clever way, then you obtain invariants,” said Christine Lescop of the French National Center for Scientific Research.
Kontsevich introduced this method in a lecture at the 1992 European Congress of Mathematics. He explained how to calculate complicated integral formulas which counted crossings between many points moving on links — adding plus or minus 1 each time a particular crossing occurred. His formula worked as a more precise invariant of links than Gauss’ had. Partly due to this work, Kontsevich won math’s most prestigious award, the Fields Medal, in 1998.
In the same lecture where he introduced the Kontsevich invariant, he suggested that the technique should be useful for more than detecting properties of one-dimensional links in three-dimensional space. He proposed that with the right ingenuity, a mathematician might get it to serve as an invariant of higher-dimensional spaces, including the spaces at the heart of the Smale conjecture.
Years later, Watanabe figured out how to make it work.
Into the Fourth Dimension
In 2006 Watanabe successfully reformulated Kontsevich’s invariant to apply to Smale’s conjecture. He gained a feel for it by first re-proving the fact that the Smale conjecture is false for odd dimensions 5 and higher. He also went beyond earlier work on those dimensions by identifying specific examples of diffeomorphisms that can’t be reduced to rigid symmetries.
The work provided a new way of thinking about the Smale conjecture and might have emboldened him to tackle the one unsolved dimension. But he didn’t give much thought to extending his work to dimension four.
“I stupidly was satisfied with the odd-dimensional result,” he said.
It was only in 2013, during a conversation with the mathematician Ryan Budney at a conference in Sapporo, Japan, that Watanabe began to consider whether it might be possible to do more. At first, he wasn’t optimistic.
“There is a technical difficulty to extend my previous result to other dimensions,” he said. “Now I know that this was not really a difficult one, but at that time I did not understand my proof well enough to get over it.”
It took another paper, in 2017, by other mathematicians (including Kupers) to begin to suggest how Watanabe might surmount it. Finally, on December 6, 2018, he posted a paper disproving the Smale conjecture in dimension four, finishing off the last open dimension of the problem. The result startled mathematicians, who aren’t accustomed to techniques that work in higher dimensions adapting well to dimension four.
“He’s got a method for dealing with spheres of any dimension. What’s surprising is that it doesn’t fail in dimension four,” said Randal-Williams.
Flows on Fibers
In its original form, Kontsevich’s invariant captured the degree of entanglement of a link. Watanabe’s remarkable insight was that the same type of invariant could be used to say something about the complexity of the space of diffeomorphisms of the four-dimensional sphere.
To take a simpler example than the one Watanabe used, your base might be a circle. To every point on your circle you associate a vertical line. Collect all those vertical lines (the fibers) into a single object and you get a fiber bundle over the circle. This bundle has the shape of either a cylinder (this is called the trivial bundle) or a Möbius band.
In Watanabe’s case, imagine a unique two-dimensional disk living over each point on the space of diffeomorphisms. Those disks are the fibers. Combine them and you have a fiber bundle, which encodes properties of the underlying space, but in a richer way.
Watanabe considered how points flow on this fiber bundle, just as Gauss and Kontsevich had studied the way points flow, or move, on links. The ways in which flowing points cross convey information about the complexity of the fiber bundle they’re flowing on.
“We see configurations of points that go around the fibers of our fiber bundle and kind of detect what’s going on. It’s like you’re doing sampling of your geometric space,” said Kosanović.
The Kontsevich invariant of the space of rigid symmetries of the sphere (or the fiber bundle based on it) is zero. This is easy enough to calculate and indicates the underlying simplicity of that space. To solve the Smale conjecture in four dimensions, Watanabe had to figure out how to calculate the Kontsevich invariant of the fiber bundle based on the space of diffeomorphisms. If that came out to be nonzero, the Smale conjecture would false.
But figuring out how to count interactions between flowing points on the fiber bundle was no simple task. “These invariants are very difficult to compute,” said Lescop. “His constructions are very ingenious.”
Ultimately, Watanabe used insights he had developed studying an area of math called Morse theory to devise a formula for counting the number of specific crossings. He proved that the invariant comes out to be nonzero, meaning the space in which the flows occur is complicated — nontrivial, in the parlance of mathematicians. The fiber bundle is complicated because the diffeomorphisms are complicated — in the same way that the fiber bundle associated to the space of rigid symmetries is simple because those symmetries are simple.
“If the diffeomorphisms were trivial, then the bundle you make would be trivial,” said Randal-Williams.
Because Watanabe proved that the Kontsevich invariant of fiber bundles based on diffeomorphisms of the sphere is nonzero, he proved that in dimension four the Smale conjecture is false. There are fundamentally more diffeomorphisms, and different types of them, than there are rigid symmetries.
The Mystery Continues
Watanabe’s result closed one front of mathematical research. It also opened another.
After his work, mathematicians knew the Smale conjecture was false in all dimensions four and higher. That is, those spheres have diffeomorphisms that are too complicated to be reduced to mere rotations and reflections. Since then, mathematicians have been intensifying their efforts to identify those diffeomorphisms.
“[You want to know] what weird families live there, what crazy creatures live there that you haven’t thought of,” said Kupers.
Some of the first results in that direction came prior to Watanabe’s work. In 2015 Michael Weiss of the University of Münster identified a few sporadic examples of outlier diffeomorphisms for spheres of even dimensions greater than four.
But more recently Watanabe and others have begun to understand entire swaths of diffeomorphisms for even-dimensional spheres, starting in dimension six. In 2020 Kupers and Randal-Williams proved that there are ranges of diffeomorphisms in which there are either no families of diffeomorphisms that are responsible for making the Smale conjecture false, or only Weiss’ families — meaning, for these higher-dimensional spheres, diffeomorphisms in those ranges are not ones that depart radically from the rigid symmetries.
Then, this past September, Watanabe proved a complementary result, showing that there are large families of diffeomorphisms that certifiably account for the outlier symmetries. His result establishes that a main reason the Smale conjecture is false in those dimensions is the behavior of those families of diffeomorphisms.
“It is very interesting that [Kupers and Randal-Williams] have also found some phenomenon that is compatible with mine,” said Watanabe.
None of these recent papers relate to dimension four, though. There, mathematicians now know the Smale conjecture is false, but there’s currently little hope of identifying the diffeomorphisms that make it so.
Having resolved one mystery in topology’s toughest dimension, mathematicians must now stare down another.