Terence Tao

And the sphere has this property, but a torus doesn't.

If you're on a torus and you take a rope that goes around, say, the outer diameter of a torus, there's no way, it can't get through the hole.

There's no way to contract it to a point.

So it turns out that the sphere is the only surface with this property of contractibility, up to like continuous deformations of the sphere.

So things that I want to go topologically equivalent to the sphere.

So Poincare asked the same question in higher dimensions.

So it becomes hard to visualize because,

A surface you can think of as embedded in three dimensions, but a curved free space, we don't have good intuition of 4D space to live in.

And then there are also 3D spaces that can't even fit into four dimensions.

You need five or six or higher.

But anyway, mathematically, you can still pose this question that if you have a bounded three-dimensional space now, which also has this simply connected property that every loop can be contracted, can you turn it into a three-dimensional version of a sphere?

And so this is the Poincaré conjecture.

Weirdly, in higher dimensions, 4 and 5, it was actually easier.

So it was solved first in higher dimensions.

There's somehow more room to do the deformation.

It's easier to move things around to a sphere.

But 3 was really hard.

So people tried many approaches.

There's sort of commentary approaches where you chop up the surface into little

triangles or tetrahedra, and you just try to argue based on how the faces interact with each other.