A friend of mine got harem pants and was talking about how much more comfortable they were than a lungi in Chennai’s current weather. A lungi is a long cylinder made of cloth (open at both ends of course) commonly worn by men in South India.
Five minutes later, our conversation included this statement:
2-manifolds with the same genus are homeomorphic.
Here’s how we got there, and a little more.
My friend’s a theoretical physicist. He works on string theory, which is a set of mathematical tools physicists use to solve problems about space and time.
To a physicist, a manifold is any surface. There are some specially defined manifolds that physicists use to understand how forces work.
For example, we’ve heard so much talk about Albert Einstein’s general theory of relativity, which describes how gravity works. When working with this theory, physicists assume that gravity is acting on the surface of the spacetime continuum. This surface is in the form of a so called Lorentzian manifold.
A numerical prefix to the manifold indicates the number of dimensions the surface has.
Say there’s an ant moving around on a sheet of paper. You can describe the ant’s position on the paper using two numbers: its distance from the length of the paper and its distance from the breadth.
For the ant, the surface it’s on has two dimensions – so it’s called a 2-manifold.
For humans, the surface of Earth is a 2-manifold. Humans can describe any point on Earth’s surface using two numbers: the latitude and the longitude coordinates.
Let’s take a slightly different shape called the torus.
A torus is a tube connected on itself, with a hole in the middle. Its surface is also a 2-manifold. According to the picture below, you can tell where you are on the torus by specifying the position of the red circle and your position on the red circle.
Now, let’s stick three toruses together like a fidget-spinner:
Its surface is still a 2-manifold because you still need only two numbers to describe your position on it: the position of a circle moving across the entire triple torus and your position on the circle.
Both a normal torus and a triple torus are 2-manifolds. However, they have an important difference: one has one hole and the other, three. This difference is important to physicists who study manifolds.
The number of holes in an object, as far as the physicist is concerned, is called the genus. The normal torus has genus 1. The triple torus has genus 3. A sphere has genus 0.
Let’s revisit the statement from above:
2-manifolds with the same genus are homeomorphic.
If two solids are homeomorphic, then one solid can be deformed such that it forms the other solid. One example is a lungi and a torus.
So what my friend’s saying with his statement is that if two solids whose surfaces are 2-manifolds also have the same number of holes, then one solid can be deformed into the other.
A famous example of this is the torus and the coffee mug. Both their surfaces are 2-manifolds. Both of them have the same genus, 1. (The coffee mug’s opening at the top is not considered a hole because it is closed at the other end.)
This is where the conversation between my friend and myself took an interesting turn.
The reason 2-manifolds with the same genus are homeomorphic is because all of them can be constructed using a combination of objects shaped like a pair of pants.
Mathematicians don’t have a different name for these objects. They are, in fact, called a pair of pants.
If you closed up the waist-rim of the pants and joined the two cuffs together, you’d get a normal torus. If you joined two pairs of pants by their waist-rims and joined the cuffs together at their respective ends. You’d get a double torus. Do this with three pairs of pants and you’d get a triple torus.
Some combination of these ‘pair of pants’ objects can be used to yield all the different kinds of 2-manifolds you can think of. So each pair of pants is like a nuclear unit, just like different combinations of protons and neutrons make up the nucleus of every different kind of atom in the world.
At this point, I proceeded to ask my friend about what kind of nuclear units make up 3-manifolds, surfaces on which you’d need three numbers to pinpoint your location.
He told me that it was a big unsolved problem in mathematics and physics, that mathematicians and physicists actually didn’t know.
The issue is with knowing how many different kinds of 3-manifolds there are. According to my friend, there could be millions upon millions – and that if you up came with a number, someone else would find a different 3-manifold that isn’t included in your set.
But there must be some way, some lead or indication of how we could go about it, I asked.
He said that mathematicians had been able to come up with a partial solution.
In our example, we used the genus as a differentiator. That is, 2-manifolds with different genuses were considered to be different kinds of 2-manifolds.
Instead, he said, mathematicians have used different differentiators other than genuses to describe the types of 3-manifolds.
They’ve found that if two 3-manifolds can be described by a fixed group of differentiators, then they may or may not be homeomorphic.
However, if two 3-manifolds can’t be described by the same group of differentiators, then they’re definitely nothomeomorphic.
It’s a sort of definition by exclusion, and that’s the best we have.
The Lorentzian manifold I mentioned above – the surface of the spacetime continuum on which gravity is thought to act – has four dimensions. It’s a 4-manifold. We have absolutely no idea how many types of 4-manifolds there are.
As I wonder on that, I’m going to get out my pair of pants, into my lungi and crash for the night. It’s so hot out here…