The Poincare conjectures are a family of questions about what space means, and about the landscape of the language we use to talk about space.

Not just outer space. Space. As in, things—that exist, and so on. The surface of the Earth is some type of space. The region near a black hole is some type of space. The vast interstellar vacuum is some type of space. A tiny Calabi-Yau manifold, which could be curled up inside your brain right now, is also some type of space.

How do we talk about such disparate things? More particularly, how do we talk about them mathematically?

We use the language of manifolds. A manifold is something that locally looks like familiar Euclidean space. Euclidean space is our intuitive picture of “flat” space: a straight line is one-dimensional and flat, the surface of a table two-dimensional and flat, the room I am sitting in three-dimensional and flat. Most space we encounter on a daily basis, as human beings, is flat space. Our brains are very good at understanding it.

But some types of spaces, such as the surface of the Earth, are not flat. Sorry, flat earthers. How do we do math on such spaces? Much of the geometry and calculus that is taught in high school and early college is restricted to flat space: your \(x-y\) plane, your functions of one real variable, and so on. But the very space we live on is not Euclidean! It is only “locally Euclidean”, that is, it looks flat near each point, but the “global” picture, literally, is not flat.

This is calculus on manifolds. A great deal of modern math is about this. For instance, calculus on manifolds is the absolute backbone of Einstein’s theory of general relativity—spacetime is a four-dimensional manifold.

Now you may ask the question: does calculus on manifolds “look like” the familiar Euclidean calculus? And the answer is the usual confusing one: it depends. If we are talking about smooth manifolds, we insist that calculus done on the manifold is reasonably well-behaved. But if we are talking about more general topological manifolds, we relax this requirement.

Now that I have two broad categories of manifolds, what do I do next? We develop some tools to study manifolds. One particularly important tool is the notion of homotopy.

Homotopy is the formalised mathematical way to talk about continuous deformations. Two things are homotopic if, vaguely speaking, one can be continuously deformed to the other one. For example, a line is homotopic to a point, I can squish it together from its ends until it becomes a point. A 2-dimensional sphere is homotopic to a cube. And so on. This is one way to understand when spaces are similar.

Of course, this is a rather weak notion, because a line is the same as a point in this world. So clearly, this doesn’t preserve stronger properties, like, say, finiteness (the technical term is compactness). A line is infinite, a point is clearly not.

A stronger notion of equivalence is called homeomorphism. Two spaces are homeomorphic if there is an invertible continuous-bijection between them. This is less visually intuitive, but what homeomorphism does is preserve many nice properties. If a space is connected, so is everything it is homeomorphic to. Same for “infiniteness” and “finiteness”, and a million other things. There is an even stronger notion of equivalence, called diffeomorphism, which tells you that the spaces, in addition to being homeomorphic, have the same kind of calculus on them.

There is a hierarchy to these notions of sameness. The weakest is homotopy equivalence, then homeomorphism, then diffeomorphism. This hierarchy tells us, for example, that diffeomorphic spaces are always homeomorphic, and homeomorphic spaces are always homotopy equivalent.

But can you go the other way? Let’s fix some type of space. Say abstract, \(n\)-dimensional spheres. Is it true that a space that is homotopy equivalent to an \(n\)-sphere must be homeomorphic to an \(n\)-sphere? Or even diffeomorphic to an \(n\)-sphere?

These are the Poincare conjectures.

You can ask this question for each category of manifolds, smooth and topological. In dimensions \(1\) and \(2\) the answer is yes, and is classical, known since the 19th century. But in higher dimensions, the answer may depend on the category. By 1980, it was known that in dimensions \(n \geq 5\), the answer is always yes in the topological category. But the answer is no in the smooth category in, for example, dimension \(7\). The counterexamples are called Milnor’s Exotic Spheres.

What about dimension \(3\) and dimension \(4\), the dimensions we seem to live in?

In 1981, Michael Freedman proved the topological Poincare conjecture in dimension 4. After 40 years of waiting, this proof has recently received a full exposition with all the gory details. The book is 496 pages long and costs over a hundred dollars! A small price to pay for some very deep knowledge. Freedman’s proof was exceptional, of course, but the family of Poincare conjectures was yet to witness its most notorious incident.

In November 2002, Grigori Perelman posted to the arXiv the first of three papers that would resolve the long-standing original Poincare conjecture: the question in dimension \(3\), where both categories coincide. This is, without doubt, the most famous mathematical achievement of the 21st century. Of course, Perelman received a lot of coverage in the wider media because he refused the $1 million prize awarded for the proof. It’s quite amusing that turning down money is far more newsworthy than solving a hundred-year-old problem.

The question for the smooth category in dimension \(4\) remains open. It is the last standing Poincare conjecture. I wonder if I will be alive to witness it fall. Probably no one expected Perelman to resolve dimension \(3\) so soon after Freedman did the topological case for dimension \(4\). But it’s hard to tell. Progress is often abrupt and instantaneous, and a paradigm shift is always lurking in the shadows.