Dimension 126 Contains Strangely Twisted Shapes, Mathematicians Prove
Kristina Armitage/Quanta Magazine
Introduction
It can be tempting to assume that your intuitions about three-dimensional space carry over to higher-dimensional realms. After all, adding another dimension simply creates a new direction to move around in. It doesn’t change the defining features of space: its endlessness and its uniformity.
But different dimensions have decidedly different personalities. In dimensions 8 and 24, it’s possible to pack balls together especially tightly. In other dimensions, there are “exotic” spheres that look irremediably crumpled. And dimension 3 is the only one that can contain knots — in any higher dimension, you can untangle a knot even while holding its ends fast.
Now, mathematicians have put the finishing touches on a story of dimensional weirdness that has been 65 years in the making. For many decades, researchers have wanted to know which dimensions can host particularly strange shapes — ones so twisted that they cannot be converted into a sphere through a simple procedure called surgery. The existence of these shapes, mathematicians have shown, is intimately intertwined with fundamental questions in topology about the relationships between spheres of different dimensions.
Over the years, mathematicians found that the twisted shapes exist in dimensions 2, 6, 14, 30 and 62. They also showed that such shapes could not possibly exist in any other dimension — save one. No one could determine the status of dimension 126.
Three mathematicians have now settled this final problem. In a paper posted online last December, Weinan Lin and Guozhen Wang of Fudan University in Shanghai, along with Zhouli Xu of the University of California, Los Angeles, proved that 126 is indeed one of the rare dimensions that can host these strangely twisted shapes.
It’s “a very long program, finally finished,” said Ulrike Tillmann of the University of Oxford.
The proof, which uses a combination of computer calculations and theoretical insights, is “like a monumental engineering project,” said Michael Hopkins of Harvard University. “It’s just jaw-dropping how they did it.”
The Doomsday Hypothesis
In the 1950s, the mathematician John Milnor astonished the mathematical world by showing that dimension 7 is home to “exotic” spheres. An exotic sphere looks exactly like an ordinary sphere from the perspective of topology, which only considers the features of a shape that don’t change when it is stretched or deformed. But the two spheres have incompatible definitions of smoothness — a curve that’s smooth on an ordinary sphere might not be considered smooth on an exotic sphere. Milnor was eager to explore and classify these exotic spheres, which in some dimensions turned out to be rare and in others numbered in the thousands.
To do this, he introduced a technique called surgery, a controlled way to simplify a mathematical shape, or manifold, and potentially convert it into an exotic sphere. The method would become essential to the study of manifolds more generally.
As its name suggests, surgery involves slicing out a piece of a manifold and then sewing in one or more new pieces along the boundary of the cut. You must sew in these new pieces smoothly, without creating sharp corners or edges. (When it comes to questions about twisted shapes, mathematicians also require the surgery to respect the manifold’s “framing,” a technical attribute of how the manifold sits in space.)
To see this process in action, let’s surgically transform a torus (the two-dimensional surface of a doughnut) into a sphere (the two-dimensional surface of a ball):
Samuel Velasco/Quanta Magazine
The result is an ordinary sphere — in fact, there are no 2D exotic spheres. But in certain dimensions, surgery converts some manifolds into ordinary spheres and others into exotic spheres. And sometimes, there’s yet another possibility: manifolds that can’t be converted into a sphere at all.
To visualize this last scenario, we can again look at a torus, only this time we’ll give it some special twists to obstruct surgeries:
Samuel Velasco/Quanta Magazine
Mathematicians have proved that there is no surgery that can transform this twisted torus into a sphere, whether regular or exotic. It’s an entirely different class of manifold.
In 1960, the French mathematician Michel Kervaire came up with an invariant — a number you can calculate for a given smooth manifold — that equals zero when the manifold can be surgically converted into a sphere, and 1 when it cannot. So the ordinary torus has a Kervaire invariant of zero, while the twisted torus has a Kervaire invariant of 1.
Kervaire used his invariant to explore the menagerie of possible manifolds in different dimensions. He even used it to come up with a 10-dimensional manifold that has no Kervaire invariant, either zero or 1 — meaning that this manifold must be so crooked that it can have no sensible notion of smoothness at all.
No one had imagined that such a manifold could exist. Faced with the power of the new invariant, mathematicians rushed to determine the Kervaire invariants of manifolds in different dimensions.
Within a few years, they’d proved that twisted manifolds of Kervaire invariant 1 exist in dimensions 2, 6, 14 and 30. These dimensions fit a pattern: Each number is 2 less than a power of 2 (for example, 30 is 25 − 2). In 1969, the mathematician William Browder proved that dimensions of this form are the only ones that might host shapes with a Kervaire invariant of 1.
It was natural to assume that twisted manifolds would exist in all dimensions of this form: 62, 126, 254 and so forth. Based on this assumption, one mathematician even built an entire edifice of conjectures about exotic spheres and other shapes. But the possibility that the original assumption might be false still loomed. It came to be known as the doomsday hypothesis, since it would falsify all these other conjectures.
Zhouli Xu was warned against trying to resolve the Kervaire conjecture while in graduate school. But the problem remained “a constant source of fascination,” he said.
Archives of the Mathematisches Forschungsinstitut Oberwolfach
And indeed, while mathematicians showed in 1984 that twisted manifolds exist in dimension 62, no one could prove that such manifolds exist in any of the remaining dimensions. As search after search came up dry, mathematicians eventually ran out of steam, and the problem became a backwater.
In 2009, “to stem the tide of oblivion,” the mathematician Victor Snaith wrote a book exploring the implications of having manifolds with Kervaire invariant 1 in all the dimensions on Browder’s list. But, Snaith warned in the preface, “this might turn out to be a book about things which do not exist.”
If Snaith had published his book a year later, it might have read very differently. For within weeks of the book’s publication, Hopkins and two other researchers confounded mathematicians by announcing that Snaith’s warning to readers had been all too apt: The doomsday hypothesis was true. Manifolds of Kervaire invariant 1, they had proved, cannot exist in dimensions 254 and higher.
The result left mathematicians in a curious position. Out of the entire infinitude of possible shapes in possible dimensions, the shapes in precisely one dimension still resisted classification. In the words of Douglas Ravenel, a mathematician at the University of Rochester and one of the co-authors of the doomsday-hypothesis proof, there was “a big loose end.” It took the form of dimension 126.
Surviving to Infinity
In 2011, Zhouli Xu arrived at the University of Chicago as a new graduate student planning to study computational aspects of manifolds. His adviser, Peter May, proposed the problem of dimension 126, which mathematicians thought was likely to involve massive computations. May sent Xu to Mark Mahowald of Northwestern University, an expert on the Kervaire invariant problem who had even named his sailboat after one of its key symbols — θj, or “Thetajay.”
But Mahowald, who died in 2013, immediately vetoed the proposal. The dimension 126 problem was too hard, he told Xu — “a lifelong problem.” Instead, he guided the young mathematician toward related problems in lower dimensions.
But for Xu, the dimension 126 problem remained “a constant source of fascination,” he said.
Guozhen Wang studies complicated objects to gain insights into high-dimensional shapes that defy imagination.
Courtesy of Guozhen Wang
The potential strategy to settle the problem was no mystery. Mathematicians had long known that crucial secrets about exotic spheres and other manifolds are encoded in objects known as stable homotopy groups of spheres. These are collections of functions, or “mappings,” that send points from a high-dimensional sphere to a lower-dimensional one.
Imagine, for instance, a mapping that sends every point in a 44-dimensional sphere to a point in a 33-dimensional sphere. This mapping essentially crushes down 11 dimensions’ worth of the bigger sphere. If you choose some point on the smaller sphere and find all the points on the bigger sphere that got mapped to it, these points will typically form an 11-dimensional manifold.
Now consider all the different points on your smaller sphere. Roughly speaking, each point gives you another 11-dimensional manifold. So your mapping doesn’t just create a single 11-dimensional manifold. It creates a bucket of 11-dimensional manifolds.
A stable homotopy group is a set in which each element is a collection of mappings like this one.
Mathematicians knew that to settle the Kervaire invariant question in a given dimension, they only needed to understand the stable homotopy group for that dimension. There’s just one difficulty: Understanding stable homotopy groups is one of the most challenging and foundational problems in topology. “I do not expect it to be solved within the lifetime of my granddaughters,” Ravenel said.
So instead, mathematicians approach this problem by degrees. Since 1958, they’ve been organizing information about the structure of stable homotopy groups into an enormous but unfinished atlas of dots called the Adams spectral sequence.
Weinan Lin wrote a computer program that helped him and his colleagues resolve a decades-old conjecture about the strange properties of shapes in high dimensions.
Zhonglin Wu
Imagine a book with an infinite number of pages, each of which consists of infinitely many columns of dots. Let’s open the book and examine a single page. Each column on this page represents a dimension. Each dot in a given column represents a different potential “flavor” for sphere mappings in that dimension. The flavor always comes in two types, like “regular” or “extra-crispy” (or, say, Kervaire invariant zero or 1).
In some ways, the book is exceedingly repetitive — each page has the same list of columns, and many of the same dots in each column. But as you turn the pages, you’ll notice a key difference: Each page captures successively finer details of sphere mappings and manifolds. The atlas’s early pages are only approximations of the truth. The approximations get better as you turn the pages, until on the final page of the atlas — known as the “infinity” page — the representation is perfect.
Navigating the atlas is like exploring the universe of manifolds with increasingly powerful telescopes. On page 1, the details of each manifold are hazy, and many manifolds that don’t really belong have been erroneously included. But if you make a better telescope, you may be able to detect that a manifold has some “flaw” that would exclude it from the atlas. In that case, you would delete the relevant dot on all subsequent pages of the atlas. If your telescope doesn’t find any flaws, then the dot survives to the next page, where you might hope to turn an even stronger telescope on it.
In 1969, Browder showed that one specific dot in the 126th column of the atlas holds the key to the Kervaire invariant problem in that dimension. If this dot survives to the infinity page, then buckets of 126-dimensional manifolds must come in both flavors: Half of the buckets will consist of manifolds with Kervaire invariant zero, and half will consist of manifolds with Kervaire invariant 1. If the dot doesn’t survive, then 126-dimensional manifolds come in just one flavor, Kervaire invariant zero.
For the special dot in column 126, there were 105 different hypothetical ways that it might vanish before the infinity page. To get a handle on these possibilities, Xu partnered with Guozhen Wang, his longtime collaborator and former college roommate. As they developed new computational techniques, they passed them along to Weinan Lin, a mathematician Xu had known since graduate school. Lin wrote a program that was able to rule out 101 of the possibilities. Then, over the course of another year, the researchers painstakingly developed new methods to rule out the last four. Browder’s special dot, they concluded, does survive to the infinity page — meaning that dimension 126 has manifolds with Kervaire invariant 1.
Prior to the team’s announcement, mathematicians thought such a computation was way out of reach, Hopkins said. The new work is “heroically computational,” he added. Its methods may eventually help mathematicians chart even more of the giant atlas.
While the new paper proves that strangely twisted shapes exist in dimension 126, it gives no hint of how to construct them. Researchers have identified specific twisted shapes in the first four special Kervaire dimensions: 2, 6, 14 and 30. But no one has found one yet in dimension 62 or 126, even though in each dimension where these shapes exist, they make up fully half of all possible shapes. Despite their abundance, “we can’t actually point to one,” Tillmann said.
If mathematicians do figure out how to construct twisted shapes in dimensions 62 and 126, it might give them some clue about what makes these six dimensions special — why it’s possible to build such twisty shapes in those and only those dimensions. “Usually when [something like] that happens, there’s some very beautiful construction,” Hopkins said. “It’s very ephemeral, because it’s got to only work five or six times, and not infinitely many times.” The new work “makes it more inspiring to really try to find the special construction of these six things.”
And the Kervaire problem is just one type of dimensional weirdness encoded in the Adams spectral sequence. The special Kervaire dimensions correspond to six special dots in the second row of the atlas. Recently, Xu and Robert Burklund of the University of Copenhagen figured out that a handful of special dimensions seem to exhibit another kind of oddball behavior in the third row of the atlas. No one knows yet what kind of strange manifolds correspond to the special dots in these dimensions — but mathematicians hope to find out.
Discoveries about subsequent rows will likely also be forthcoming, Xu said. “There should be a lot more stories later on, waiting for us to explore.”
