What is a manifold?

In this section we are going to describe the general class of things we want to study. We are going to start with a large class of things and then add more and more structure to it so that it is more and more limited. In mathematics we often are interested in classifying objects up to some form of equivalence. For instance, we may want to be able to distinguish TYPES of US coins. In other words, we want to know how many different types of coins there are, and if we have two, we want to know, for instance, if they are both pennies or if one is a nickel. A mathematician might call this classifying up to the type of coin (this is the equivalence). To contrast, we may want to classify up to type of coin and year it was minted. In the first case, if we have a 1986 penny and a 1997 penny, they are the same up to equivalence. But in the second case they would be different up to equivalence. Thus it is important to know what we consider to be the same and what we consider to be different. In the first case, our equivalence was very weak, since we only consider two things different if they were different types of coins. Thus the only objects we have are pennies, nickels, dimes, quarters, half dollars, and dollars (6 objects total). In the second case our equivalence was stronger because we consider two things different if their years of minting are different. In this case we have a lot of different objects, including 1999 pennies, 1990 pennies, 1987 dimes, 2000 nickels, 1967 quarters, 1990 dimes, etc. Now we are going to consider objects the same based on a number of different equivalences. With manifolds, we are mostly going to be concerned with two being "topologically" the same and being "geometrically" the same. These two terms are actually not as specific as we might hope, but I don't want to get into the specifics of different kinds of topological equivalence or geometric equivalence. I will describe the relevant ones. Now the idea of two things being topologically the same is perhaps one which you have not encountered before. Two things are topologically the same if they have the same shape in the sense that one can be deformed into the other one without "breaking" it. I'm being a little vague because the actual definition is not very instructive to the beginner. But you already understand the basics of this concept. Now consider if you had a perfect circle that you drew with a compass and a circle that you drew freehand. Now only one is REALLY a circle, right? The other one isn't a perfect circle in that there is no one point in the center which is EXACTLY the same distance from every point on the circle. But, nonetheless, if you were to show it to someone, they would recognize it as a circle, even though it's not a circle. This is in some sense the difference between being topologically the same and geometrically the same. Geometrically, the second is not a circle, since there is no center. But topologically they are equivalent. Topologically both are the same, because we could change one of the "circles" slightly to make it into the other. Now when I say change, I am only allowing certain kinds of changes. For instance, you are not allowed to cut the circle unless you reattach the two ends to each other. So, for instance, if your circle were a rubber band you could stretch it but not let it break. You could even cut it, tie a knot in it, and reattach the two ends. These would all be topologically equivalent. This idea that you can stretch things however much you want has caused topology to be dubbed clay geometry. The typical example given is that topologically a coffee mug is the same as a doughnut. (Think about why!) But really we just removed some of our restrictions on two things being the same. We are just about ready to learn what a manifold is. We first need to understand what Euclidean space is. You already know what Euclidean space is. It's just what you consider to be space. For instance, the infinite line (number line) is a Euclidean space. So is the infinite plane. So is the space we live in. So what is it exactly? Well, the number line is just the set of all real numbers (so we allow all decimal expansions, even infinite ones that give irrational numbers like $\sqrt{2}$ and $\pi$). This is one dimensional Euclidean space since there is only one degree of freedom. You can only go forward or backward if you live on the line. Now consider the plane. Here we have two degrees of freedom. We can go forward/backward or left/right. This is two dimensional Euclidean space. It is important that forward/backward is not the same direction as left/right, for then we would have a line, right? Now, if we want to describe a location on the plane, we can fix a point and then say how far forward/backward to go and how far left/right to go. Thus there are two real numbers associated to a position in the plane. This is what we really mean by two degrees of freedom or two dimensions. Now in space we have three directions we can go: forward/backward, left/right, and up/down. Thus we get three dimensional Euclidean space. You can similarly imagine Euclidean space of any dimension. All you need is some number of different degrees of freedom and a location (called a point) is determined by that many different numbers, which tell you how far you are in each direction from some fixed point which we will call the origin (note that the origin is 0 units away from itself in all directions). Note that topologically the size of the units is unimportant. Now a manifold is just something which locally looks like Euclidean space. Consider, for instance, the surface of the Earth. For ages it was believed to be a plane. Why? Because from what we can see it is a plane. (Topologically, hills, mountains, and valleys don't change this, right?) But we know that it actually curves back on itself and is topologically a sphere instead of a plane. So it is a manifold. Locally it looks like a plane (Euclidean two dimensional space). We would call it a two dimensional manifold because it is locally like two dimensional Euclidean space. Now here is a question: is the universe we live in Euclidean three space or is it some other three dimensional manifold? As far as I can tell, this is still unknown. One goal of mathematics, however, is to see what it could possibly be and then maybe we could eliminate things on the list with experiments. (This is, of course, under the assumption that the everywhere in the universe looks essentially like the way we see it, which seems a reasonable assumption.)