I’m in Paris, stumbling on streets named after mathématiciennes and growing closer to Marie Curie, because of French mathematician Henri Poincaré. My spouse is working at the Institut Henri Poincaré right now, and I came along for the ride. I’ve been wanting to write about the Poincaré homology sphere in his honor since getting here, but every time I’ve tried it’s gotten overwhelming, and I’ve written about something else instead.

There are at least eight ways to define the Poincaré homology sphere (pdf), and at least seven of them require a lot of fancy mathematical machinery to understand. They all lead to the same space, but the connections between the different definitions of the space are hard to see. I don’t want to put off the Poincaré homology sphere forever, so I’m going to jump in. But instead of describing the space right away, I’m going to tell you why I care about it.

A central theme in topology—and mathematics in general—is determining whether two spaces are the same or not. Topological sameness is fairly lenient. Two spaces are topologically equivalent if you can make one into the other by stretching or squishing, as long as you don’t tear or glue anything. As the joke goes, a topologist can’t tell a coffee cup from a donut because they both have one hole, and if you squish the liquid-containing part of the cup down, you can make a coffee cup into a less-delicious version of a donut.

It can be hard to assess topological equivalence because it is so flexible. Just because you don’t immediately see a way to squish one shape into another doesn’t mean you won’t eventually find one. There are infinitely many ways to stretch and squish any shape, and you can’t try all of them to figure out whether two spaces are equivalent. And if you think the problem might be difficult in three dimensions, imagine that you’re trying to assess topological equivalence of two impossible-to-visualize high-dimensional objects. Intuition might not get you very far.

Early topologists wanted to try to find ways of distinguishing spaces by finding invariants: numbers or other mathematical objects that could be assigned to each space. Ideally, two spaces with the same invariant would be the same space, and two spaces with different invariants would be different spaces. Poincaré came up with Betti numbers, which are informally a way to catalogue the holes of different dimensions in a space, and torsion coefficients, which sort of keep track of twistedness. In a 1900 paper, Poincaré conjectured that these Betti numbers and torsion coefficients (also known today as homology) could tell you whether or not a space was a sphere. 

A few years later, Poincaré showed that he was wrong. He came up with the first of what are called homology spheres: spaces that have the same homology as spheres but are not topologically equivalent to spheres.

You might be picturing a beach ball, when I say sphere, but there are spheres in every dimension, and Poincaré was specifically looking at the 3-dimensional sphere, which sits naturally in 4-dimensional space. (Mathematicians describe the beach ball as being a 2-dimensional sphere because a tiny creature living on its surface would feel like it lived on a two-dimensional plane.)

Poincaré’s discovery of a homology sphere led him to refine his conjecture to what is now known as the Poincaré conjecture. He added another invariant, known as the fundamental group, and believed that if a manifold had the same homotopy and fundamental group as a sphere, it had to be a sphere. Poincaré used the fundamental group of the homology sphere to show that it was topologically different from a sphere.

The Poincaré conjecture was one of the most important unsolved conjectures In 2006, this conjecture was finally proved, with Russian mathematician Grigori Perelman putting the finishing touches on the proof. (He famously refused the million-dollar bounty the problem had as one of the Millennium Prize problems. In fact, the prize money he refused is currently funding the program that brought my spouse to the Insitut Henri Poincaré.) The solution to the Poincaré conjecture was the most important mathematical breakthrough of the 21st century so far.

So the Poincaré homology sphere was an important figure in the story of one of the most fascinating areas of mathematical research in the last 100 years. But what is it? Poincaré’s original approach to defining a homology sphere was via a technique called Heegaard splitting. It basically involves looking at two solid two-holed donuts and gluing them together in a carefully prescribed way. The diagram at the top of this post is from his 1904 paper first describing it. A better-known way to define it now starts with a dodecahedron, a shape made of twelve pentagons. (And in my experience, that’s what mathematicians will think of if you mention the Poincaré homology sphere.) Rather than reinvent the wheel, I’ll refer you to Yen Duong’s blog posts about it at Baking and Math (Part 1, Part 2).

For those with an extensive math background who want to know more about the Poincaré homology sphere, there’s a good article by Klaus Volkert at the Manifold Atlas and a page about it at the (French language) website Analysis Situs, which focuses on Poincaré’s foundational papers.

Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Torus
The Three-Torus
The Möbius Strip
The Long Line
Space-Filling Curves
The Wallis Sieve
Two Tori Glued along a Slit
The Empty Set
The Menger Sponge
The Connected Sum of Four Hopf Links
Borromean Rings
The Sierpinski Triangle
Lexicographic Ordering on the Unit Square
The SNCF Metric
The Mandelbrot Set
Fatou's Pancake
The Pseudosphere
The Douady Rabbit