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.
