# In the ‘Wild West’ of Geometry, Mathematicians Redefine the Sphere

## Introduction

If you’ve ever been stuck in traffic on a rainy afternoon, you’ve probably watched raindrops racing each other down the car window. When pairs of droplets collide, they merge into a new droplet, losing their separate identities.

That merging is possible because the water droplets are just about spherical. When shapes are flexible — as raindrops are — attaching a sphere doesn’t change anything. In certain areas of mathematics, a sphere attached to a sphere is still a sphere, though perhaps a bigger or lumpier one. And if a sphere gets glued onto a doughnut, you still have a doughnut — with a blister. But if two doughnuts merge together, they form a two-holed shape. To mathematicians, that’s something else completely.

That quality makes spheres a crucial test case for geometers. Mathematicians can often transfer lessons learned on spheres to more complex shapes by looking at what happens when you sew the two together. In fact, they can apply this technique to any manifold — a class of mathematical objects that includes simple shapes like spheres and doughnuts, as well as infinite structures like a two-dimensional plane or three-dimensional space.

Spheres are especially important in a subdiscipline of geometry known as contact geometry. In contact geometry, every point on a three-dimensional manifold — such as the 3D space we live in — corresponds to a plane. The planes can tilt and twist from point to point. If they do so in a way that satisfies certain mathematical criteria, the entire set of planes is called a contact structure. A manifold (like 3D space) together with a contact structure (all the planes) is called a contact manifold.

Though contact structures might seem to amount to little more than decoration, they bring fundamental insights into the manifolds they live on, as well as links to physics. Modern mathematicians can use contact manifolds to reformulate theories about how light behaves and about the way water flows through space.

Results about three-dimensional contact manifolds frequently come back to spheres. If you glue a contact sphere onto another contact manifold, such as a 3D doughnut, the 3D version of the sphere can donate parts of its contact structure to the union. If you want to prove that a doughnut can have a contact structure whose planes twist a thousand times as they circle the doughnut hole, you can first build that structure on the sphere and then add it to the doughnut by cutting a small hole in both shapes and patching them together along the edges. Mathematicians exploring which contact structures can exist on a given manifold frequently rely on this framework, said John Etnyre, a mathematician at the Georgia Institute of Technology. “They do a lot of work to reduce the problem to understanding what happens on the sphere,” he said.

As Jonathan Bowden, a mathematician at the University of Regensburg, puts it: “If you can’t understand a sphere, how can I possibly understand anything else?”

We tend to think of spheres as simple shapes: They’re merely all the points that are a fixed distance from a center point. Examples include a circle, which is one-dimensional, as well as the two-dimensional surface of an ordinary ball like a basketball. But when you add in contact structures, spheres can get more complicated than you might expect. And as mathematicians attempt to sort through a disorganized ocean of contact manifolds, new types of spheres can give them clues about what they might fish out of the depths.

In a recent paper that was substantively updated last week, four mathematicians — Bowden, Fabio Gironella, Agustin Moreno and Zhengyi Zhou — have uncovered a new type of contact sphere and, with it, an infinite number of new contact manifolds.

**Full Contact Sport**

As a field, contact geometry emerged gradually over the course of centuries. Though modern mathematicians looking back see hints of contact geometry in the study of optics in the 17th century and thermodynamics in the 19th, only in the 1950s was the phrase “contact manifold” first used in a paper, according to the mathematician Hansjörg Geiges’ history of the subject.

By that time, mathematicians were already aware of some examples of contact manifolds. For technical reasons, contact manifolds only come in odd dimensions. Standard three-dimensional space has a contact structure consisting of rows of planes that gradually tilt forward. This structure naturally extends to what mathematicians call the three-dimensional sphere. (This is the surface of a four-dimensional ball, much as the two-dimensional mathematical sphere is the surface of an ordinary three-dimensional ball.)

Starting in the late 1960s, mathematicians began to present new examples of contact manifolds. In 1968 Mikhael Gromov made progress on finding new contact structures on certain manifolds, such as three-dimensional space, and Jean Martinet followed in 1971 with examples on so-called compact shapes (which are finite with a clear boundary) like the 3D sphere. In 1977, Robert Lutz figured out how to create a new contact structure on any three-dimensional manifold. Lutz’s construction involved slicing open the contact manifold, twisting it up, and sewing it back together in a way that kept the underlying shape the same, but forced the contact structure into a new configuration. It resulted in a new contact structure for infinite 3D space, the 3D sphere, and any number of even stranger objects, such as a cube where, if you stick your hand through the bottom, you’ll see it dangle down from the top.

Still, those results left late-20th-century mathematicians with many unanswered questions about contact manifolds. What kinds of contact structures were out there? How should they be categorized? “When mathematicians come to some subject, they always want to classify or understand objects,” said Yakov Eliashberg, a mathematician at Stanford University who was instrumental in the early development of contact geometry.

In dimensions five and higher — remember, contact manifolds can only have an odd number of dimensions — these questions still aren’t answered. In the three-dimensional case, much of the progress was made almost single-handedly by Eliashberg, who arrived in Berkeley, California, in the 1980s as an immigrant from the Soviet Union.

**Twist and Shout**

Prompted by a question from a new Berkeley acquaintance named Jesús Gonzalo Pérez, who had been studying Lutz’s technique for creating new contact manifolds, Eliashberg noticed that all the three-dimensional contact manifolds you could get using Lutz’s strategy had certain commonalities. In 1989, he published a seminal paper describing these manifolds in detail. He called the new class of contact manifolds “overtwisted” because of the way the planes of the contact structure rotated multiple times, beyond the twisting required to qualify as a contact structure. Eliashberg’s 1989 paper answered practically any questions mathematicians might have about overtwisted manifolds in three dimensions, but any other contact manifold — which Eliashberg called “tight” because of how little its contact structure twisted — was much harder to get at.

“Whereas overtwisted structures exist in abundance, tight contact structures are more rare or, at least, way more poorly understood,” said Moreno, a mathematician at Heidelberg University.

One distinction between overtwisted and tight contact manifolds becomes clear if we view a manifold as the boundary of a larger space. Since contact manifolds are odd-dimensional, they always form the edge of an even-dimensional manifold. (Think of how the one-dimensional curve of a circle surrounds a two-dimensional disk, or how an infinite line cuts the two-dimensional plane into two separate halves.) Contact geometry has an even-dimensional counterpart called symplectic geometry. Mathematicians wanted to know if the interior of a contact manifold — which is always even-dimensional — forms a symplectic manifold or not.

If it does, the original contact manifold is called “fillable.” Fillability is a special property. Results of Eliashberg and Gromov from the 1980s and early 1990s implied that fillable contact manifolds cannot be overtwisted — they must be tight. But the reverse scenario was murkier — could a manifold be tight but not fillable?

“For a long time, it was possible that maybe being tight was really just a reflection of being fillable,” Etnyre said. Eliashberg had proved that a three-dimensional sphere only has one tight contact structure, which is also fillable. But in 2002, together with Ko Honda of the University of California, Los Angeles, Etnyre found an example of a three-dimensional contact manifold that was tight but nonfillable.

In higher-dimensional cases, things were uncertain. “We have a lot of tools to study contact structures in dimension three, and we have virtually none in high dimensions. And that’s a real problem,” Etnyre said.

“In contact topology, higher dimensions is really the wild west. People really don’t know almost anything about what goes on,” Honda said. The question became: Are there tight but nonfillable contact manifolds in high dimensions? And if so, what do they look like?

**Keeping It Tight**

In 2013, three mathematicians found a way to create such manifolds, but “the manifolds that they constructed were actually very, very complicated,” Etnyre said. It was unknown, he added, whether that level of complexity was necessary. If so, there might still be a close link between tightness and fillability for simple manifolds like spheres.

In 2015, Bowden, then at the Ludwig Maximilian University of Munich, and two collaborators showed that certain contact manifolds could be carefully carved up and patched together to form a sphere without sacrificing their contact structures. Their work suggested that mathematicians could not only transfer a contact structure from a sphere into a more complicated contact manifold — the usual direction of things — but also create a brand-new contact structure on a sphere by starting with a more complicated example.

By 2019 he had begun to work with Gironella and Moreno. That year, they published a paper building on techniques by several previous mathematicians. The three found examples of contact manifolds which had symplectic fillings, but fickle ones: The fillings, called “weak fillings,” vanished if the contact manifold was tweaked in just the right way.

After the start of the pandemic, they began to suspect that they would be able to construct spheres with the desired properties. They took some of the contact manifolds and carefully reworked them into spheres: cutting a hole here, patching it up there. When they were finished, they had an infinite collection of tight but nonfillable spheres. And because spheres can transfer parts of their contact structures into other manifolds, this created tight but nonfillable contact manifolds of all shapes and varieties.

The three showed Zhou an early draft of their paper in mid-2022, hoping he would proofread some of their calculations. Zhou had previously collaborated with both Moreno and Gironella, and was familiar with some of the techniques their draft used. “I read through the paper, and I realized that this had huge potential to get even stronger results,” said Zhou, a mathematician at the Chinese Academy of Sciences. He returned to them full of new ideas.

The group incorporated Zhou’s insights into their paper, and the four of them posted it online in November 2022. Their work shows that tight but nonfillable spheres in dimensions five and above are possible. Then last week they updated the paper with an important generalization. They are now able to find tight but nonfillable contact structures for any manifold with dimension seven or higher.

Even though their proof uncovers an infinite number of new examples, the study of higher-dimensional contact manifolds — and even of higher-dimensional spheres — is only just starting.

“This is giving us a glimpse into what seems to be a very wild and sort of complicated world,” Moreno said, later adding: “Higher dimensions is going to eat the attention of several generations to come, I would say.”

“Right now, you’re just trying to find any examples; you’re trying to distinguish things; you’re just trying to get a sense of what’s there. And understanding things on the sphere is kind of the germ, or the seed that might help you understand other situations,” Etnyre said. “We don’t really have the tools to take that next step yet.”

**Correction: **November 8, 2023

*This story has been updated to clarify the types of contact structures shown to exist in a new paper.*