In an old Indian parable, six blind men each touch a different part of an elephant. They disagree about what the elephant must look like: Is it smooth or rough? Is it like a snake (so thinks the man touching the trunk) or a fan (as the man touching the ear proposes)? If the blind men had combined their insights, they might have been able to give a correct account of the nature of the elephant. Instead, they end up fighting.
For decades, topologists have hoped to avoid falling into a similar trap. They thought they could characterize mathematical shapes by synthesizing numerous local measurements. But newly discovered, paradoxically curved spaces show that this isn’t always possible. “Things can be much more wild than what we thought,” said Elia Bruè of Bocconi University in Italy, who worked with two other mathematicians to demonstrate this.
Topologists stretch and compress the shapes they study. An infinitely thin rubber band, from a topological perspective, is equivalent to a circle, because you can easily deform it into a circular shape. Topologists tend to characterize shapes according to their global properties: Do they have holes, like a doughnut? Do they go on forever, like an infinite plane, or are they “compact” like the surface of a sphere? Do their “straight” lines go on indefinitely — making them what mathematicians call “complete” — or are there dead ends?
But as with the elephant in the parable, it can be hard to directly perceive the global nature of topological shapes. And so mathematicians want to understand their relationship to local geometric properties, like curvature. What can you say about a shape’s global topology, given information about how it curves at every point?
In 1968, John Milnor, a renowned mathematician then at Princeton University, conjectured that an average sense of a complete shape’s curvature was enough to tell us that it couldn’t have infinitely many holes. For the next 50 years, many results supported his claim. “You were tempted to believe it was true, because it was true in so many realistic cases,” said Jeff Cheeger of the Courant Institute of Mathematical Sciences at New York University. “And how in God’s name could you construct a counterexample to it?”
In this area of mathematics, said Vitali Kapovitch of the University of Toronto, “the Milnor conjecture was probably the biggest open problem.”
And so in 2020, Bruè and two colleagues set out to prove it. They ended up finding a counterexample instead — and built an entirely new kind of topological shape in the process. “It’s fantastic work,” Cheeger said. “A landmark.”
A Holey Grail of Topology
To understand Milnor’s conjecture, it helps to first consider how topologists and geometers think about curvature.
Both study manifolds — spaces that look flat when you zoom in on them. A tiny ant on the surface of a sphere, doughnut or other two-dimensional manifold will perceive its immediate neighborhood to be no different from a two-dimensional plane. But if the ant moves a little bit in any direction, it might notice that the space begins to shift, or curve. The idea of a locally flat manifold generalizes easily into higher dimensions. But curvature is tougher to define.
Take, for example, the simplest case: a one-dimensional object such as a circle. Surprisingly, these one-dimensional spaces cannot, in a mathematical sense, be intrinsically curved. A one-dimensional geometer walking along a circle, unable to perceive more than one dimension, would think she was traveling in a straight line — and would be surprised to find herself retracing her steps.
But if you embed a circle in a two-dimensional plane, it’s apparent that it has constant, positive extrinsic curvature. (The relevant distinction here is between intrinsic and extrinsic curvature: what you can see if you’re stuck inside the space, versus what you can see from outside it.)
Smaller circles bend more quickly as you move around them, and therefore have higher extrinsic curvature; bigger circles have lower curvature. (A straight line, in this sense, is like an infinitely big circle. Its curvature is zero, indicating that it’s completely flat.) We can also apply this definition to more complicated shapes that have changing curvature, by considering how big a circle you would need to match the shape at any given point. In this way, curvature is a local property: Every point on a manifold has an associated curvature.
For a surface — a two-dimensional manifold — there are many ways to place circles so that they match the surface’s curves. At a given point, you can measure curvature in any direction by placing an appropriately sized circle in that direction. But, surprisingly, it’s possible to define the surface’s curvature at that point with just one number. If you find the directions that give you the biggest and smallest curvature values, and multiply those values together, you get a number called the Gaussian curvature. This number summarizes information about how the surface bends in a useful way. Even more surprisingly, the Gaussian curvature turns out to be an intrinsic property: It doesn’t depend on any higher-dimensional background space the surface might be placed into. In this sense, paradoxically, cylinders are not intrinsically curved, though spheres are.
