Tell me about math's. Anything. I'm so curious
Okay! I will tell you about one way we can test for holes in a topological space!
I will first clarify what I mean by a loop because it's important to be precise and they are the star of this show! A loop in a topological space X is continuous map from the interval [0,1] to X which starts and ends at the same point.
To motivate our test we shall look at two examples!
First imagine a plane (ℝ²). Intuitively this has no holes. If we consider a loop in the plane we can imagine shrinking this loops down to a point. This is sort of like placing a rubber band on a table and squishing it down as close as you can (except rubber bands have a physical limitation. Even if you can keep squishing it, you'd eventually make a black hole). It's important to point out that the loop lives in the space rather than on top of it like a rubber band on a table.
Now imagine an annulus (see picture)
This obviously has a hole in the middle. Now we can consider loops in an annulus. We can think about two kinds of loops!
The first kind doesn't go around the hole and we can still shrink these to a point. The other kind goes around the hole and when we try to shrink it, it snags on the hole.
So the idea is that we can find the presence of a hole in a space by shrinking loops to see whether they can be made into a point or not. We can adapt the rubber band analogy by adding the extra rule that the rubber band must always touch the table at all its points. So if we were to cut a hole into the table, we would no longer be able to shrink the rubber band.
Let's try another example! We shall look at a torus. This is the surface of a ring doughnut (that is, it doesn't have anything on this inside).
First consider the red and green loops. All we can do with these is move them around the tube. We can't shrink them. Similar, we can't shrink the orange and blue loops. We also can't deform a blue/orange loop into a green/red loop and vice versa! So a torus must have some holes!
One thing that might seem weird is that holes can have different dimensions! The part of our space which bounds the hole can have a different dimension which means the hole is fundamentally different. In this context the dimension of a space is to do with what the space locally looks like. That is, if you were to zoom in closely the space would sorta look like a flat Euclidean space, i.e. a line or a plane or 3D space etc. For example, the torus is 2 dimensional since locally it looks like a plane. A circle is 1 dimensional since locally it looks like a line. Another way to think of this is how many different independent directions could you walk if you lived in that space. Another example is the sphere, think the surface of the earth. This is two dimensional because we only require two numbers to describe positions on it!
The reason I bring this up is our test can't always detect the presence of holes! This is because our test is great at picking up on 1 dimensional holes, but it doesn't always detect higher dimensional holes. A good example here is the sphere. We can always shrink a loop on a sphere but it's fairly easy to see that the sphere bounds a region that isn't a part of the sphere itself. There is a 2 dimensional hole in the sphere.
One more neat thing we can do with loops in spaces is we can use them to define a nice algebraic structure! By algebraic structure, I mean anything that involves a set and an operation between elements of that set which produces another element of that set. An example of this is the integers with addition. We can add two integers to get another integer. The integers have some nice properties. There is an element 0 such that 0+n=n+0=n. We can also take inverses, i.e. we for any integer n there exists another integer m such that n+m=0. We also have a property called associativity. This is the rule that says (n+m)+k=n+(m+k). This makes the integers what's called a group!
We can make a group using loops in a topological space! We first pick a basepoint which every loop will start at. Then we define out operations on the loops to be concatenation. That is, given two loops f and g, we define f*g to be the loop we get by first going around f then going around g. We also have the added rule that we consider loops that can be deformed into each other to be the same. The identity element is the constant loop, i.e. the loop e such that e(t)=x for all values of t, where x is our basepoint. The group that we get is called the fundamental group (kinda pompous but it really is important!).
We can see an example of this using the annulus from earlier! We can consider an anticlockwise loop around the hole to correspond to the number 1. We can get successive positive numbers by going around this loop the right number of times! We get negative numbers using clockwise loops! (Equally we can make clockwise loops correspond to positive numbers and anticlockwise to negative numbers).
The fundamental group turns out to be a very powerful tool! It turns out that if topological spaces have different fundamental groups they can't be the same space (strictly speaking, they can't be homeomorphic or even homotopy equivalent). And we can prove some useful results using fundamental groups too!
This is all formalised in the area of maths called Algebraic Topology (the area I hope to do research in!). Making this all rigorous is no easy task (I have written a few formal posts about it on my maths blog!)
This was a lot longer than I had planned originally haha. I've been writing for about an hour and a half. I hope you find it interesting!
