Today I talk about something fairly complicated; but of great interest to me and all of those on a quest to better understand this universe. That is the manifold; a wonderful topological construction. It almost seems sinful that we have come this far without speaking about manifolds, when the logo in the top of the site is just that – an illustration of a certain type of manifold. Manifolds are a wonderful embodiment of how a mathematical construct can reveal so much about the world around us. I actually first got interested properly in manifolds following a talk I attended with an leading Professor currently at City University, who focuses on string theory and manifolds; and having had the opportunity to discuss further over a few drinks afterwards realized it was an area that was important.

The mathematics behind manifolds is actually (in my personal opinion) very complicated  due to the abstract nature, but understanding what they are and why they are useful need not be so challenging. What you need to get your head round with topology is that it is primarily an axiomatic discipline. Remember how I always talk about sets? I love them; in part because you can do what ever you want providing you define it. Let X be defined as the set {-1, 9, 412, 2}; unrelated numbers, in a horrible order but that’s fine the set belongs to you. How empowering. Now X is unlikely to be used by anyone ever again, because it has no use describing anything – but that does not make it any less real. We all know of set X and nobody can take that away. If I defined Y as the set of all rational numbers I would speculate Y would be a very busy set indeed; much more so than X because Y has applications all across the mathematical spectrum. So you see you can define things how you want to suit your needs; but the usefulness of the construction depends on the underlying rules and the correlation between them and the phenomena you wish to describe. As we start to look at manifolds you should try and understand the rules of the construction. If you don’t? Don’t worry just do as your told, like an ill-behaved child. Rules are rules.

Getting started 

There are a few important mathematical notions you should get comfortable with before continuing.

Euclidean Space – You may see Euclidean space denoted as either  Eⁿ or  Rⁿ depending on what you are reading – where n represents the number of dimensions. I recommend not getting seduced by extra dimensions and staying within the realms of three; lay concrete foundations before you try to build a skyscraper. Now in one dimensional Euclidean space, everywhere is defined by a single coordinate in the Cartesian plane; two dimensions you have two, and three you have three. I use Cartesian with no definition. If you have only ever studied one type of coordinates these are Cartesian; if you have studied more you know of what I speak. Here is three dimensional Euclidean space.

Euclidean space

In order for me and you to meet up in three dimensional Euclidean space (don’t get clever and think about time) all we need is an x and y and a z coordinate. Some or all of these may be 0, but I need to know that information; I need all three.

Topological space – This one is important; just remember these are just rules. A topological space is just like a set; but rather than a set of numbers it is more a set of points. The construction of this means that for any point in the set, along with a set of neighborhood points (local points in the set) the same rules all hold. Take a look at this definition (Wiki):

“Given a set , a topology on is a collection of subsets of (called open sets) with the following properties:

• The empty set and are both in .
• The union of any collection of open sets is an open set. That is, .
• The intersection of any finite collection of open sets is an open set. That is, .

The pair is called a topological space. If the topology is clear or does not need an explicit name (since we can just refer to sets in the topology as open sets), then we just say that is a topological space.”

If you understand that definition then skip this bit. If not let us pick at it a little. So X is our set and our topology T on X is a collection of subsets of X; that is quite easy we are saying given any set we define, a topology on that set is another subset contained within the original. Bullet point 1 – the empty set are in the set and topology; fine. The next one is a little harder; a union is quite simple though. If I have two sets, A and B, the union of the two sets is defines the set of points in A OR B. So what we are saying is the union of any two open sets is also an open set – where an open set is one which is not over a closed interval. Here is a nice example of this I found on Wolfram.

An open set of radius and center is the set of all points such that

That would be like an open ball. Finally we have an intersection rule. An intersection is like the above example with A and B, but now we have the set of point in A AND B. Here we say the intersection of a finite collection of open sets is itself an open set.

Okay so that was dense. Let’s step back – we have nice Euclidean space; “normal” geometry. We also have defined together a set and a topology within that set – obeying the above axioms. Don’t get too bogged down if you didn’t follow every bit of set theory, the rest is still very accessible.

A basic example

Take your brain to a planet. I will let you choose which one; it does not actually matter to me. You have woken up on this planet; it’s a huge one and you know nothing about it. You have no equipment so you can’t take any sophisticated measurements or do any detailed examination. You look to the left; you look to the right and you see flat planes of land as far as the eye can see. You may be forgiven for arriving at the conclusion that you are on a flat planet; exactly the same as our ancestors. A sheet of land, potentially infinite in dimension. After all it looks as though it is and you don’t really have much more to go on.

Now you could keep walking in a straight line until you eventually get back to the same point; but firstly depending on the planet size this may take an unfeasible length of time and secondly you would need a very solid frame of reference in order to confirm you had certainly arrived back at your starting point. Even then, you have no tools, you could have walked a large circle on a flat plane rather than a straight line on a sphere. It is after all quite difficult to know, unless you have a tool to confirm you have gone exactly straight. What is this witchcraft? It’s the problem with being stuck on a 2-dimensional manifold. Have a look at this picture.

A 2-D surface

You can think of each one of these little square as being much further than the eye can see. This is the trickery; each square looks like 2D Euclidean space – so we arrive at the conclusion we are in 2-D Euclidian space. It is like lots of 2D panels have been amalgamated into a sphere to trick us. This is the very essence of topology; we have lots of different point where the localized points obey certain characteristics which can reveal the true nature when you are either much smaller or much larger than the manifold.

A more rigorous definition

A topological manifold is a topological space (as above) which represents n-dimensional Euclidean space in the vicinity of any point on the manifold – as discussed above. We say that the manifold is locally Euclidean (adore that phrase) if for every non-negative integer within the topological space there is an equivalent point in Euclidean space of the same dimension. I know that is horribly clunky wording; but appreciate that crudely speaking what we end up with is  a 1-manifold as a point, a 2-manifold as a surface, a 3-manifold as a volume. When we start to define manifolds of higher dimensions not only does the Mathematics spiral out of control, but the visualization is hard if not impossible.

Why do so many Mathematicians  struggle with their weight? They don’t know the difference between a cup of coffee and a donut. Homeomorpism is a final piece of excitement to flesh out the definition. At the most basic level when we say things are homeomorphic we are saying they are the same, in a sense. Two topological spaces are homeomorphic if there exists a continuous function that maps between the topological spaces preserving all the features; with a continuous inverse function function. There is a function from A to B and B to A preserving all properties. This is quite easy – it’s just matching things up. Here are two homeomorphic tori  (a surface formed by rotating a curve around a line).

Get it – coffee and a donut. In fun words, if it were made of Plasticine; if I can make one from the other without making new holes or tearing it, it is probably homeomorphic.

So what we have now is a fairly good appreciation of the mathematical construction of a manifold and the slightly abstract way of looking at spaces where two spaces are the same if we can preserve the properties of them; we don’t get over excited about the visual appearance. After all, you could be living on a Klein bottle and life wouldn’t seem that different. That is the important thing to appreciate – day to day life would look very much the same (ignoring the obvious fact that gravity would be odd – stay in the imaginary kingdom of fanciful topology).

Klein Bottle

Applications

Manifolds may seem like a playful mathematical construction but in fact they are becoming of increasing importance in theoretical physics; so much so that it is almost impossible to study some of the most cutting edge areas without getting a grasp of them. The manifolds we have discussed so far have been simple; but none the less have allowed you to gain an appreciation of them.

The Mathematical Physics group at the University of Oxford are largely considered to be the world leaders in manifold computations and their applications to superstring theory. They spend a lot of time with a certain type of manifold; the Calabi-Yau manifold which you may fondly recognize from the top of our site. Fair warning: the mathematics is above undergraduate level; but in nice basic words you just add lots of complicated rules and properties the manifold must obey on top of what we already discussed.

Calabi-Yau manifold

Often when you hear people talk about string theory having 10 dimensions, but the other ones are really small or “curled up” this is where it comes from. The general framework of the manifold says that the four dimensions should be prominent; ordinary space time and the next 6 would be small. This isn’t made to fit the problem it is a natural part of the manifolds construction; and by small we mean Planck length small. This is very useful. String theory requires the “curling” of these extra dimensions to fit in a specific way; which meet the conditions of the Calabi-Yau manifold and hence why this toplogical construction found itself at the cutting edge of theoretical physics. These manifolds have been proven to exist (see Shing-Tung Yau) which has lead to a rich and fruitful cooperation between topology and particle physics. Different models of the Calabi-Yau manifold, with different topology lead to mini models of the universe; within them we can naturally fit the laws of quantum mechanics and gravity. Currently researchers are working tirelessly to understand the connection between different manifolds, and a construction that describes our universe.

You can see a research paper published by the very person who got me interested in this here. It’s  stupidly hard.