1 Lecture 4: September 8 Homeomorphisms. As I mentioned in the first lecture, the purpose of topology is to look at qualitative properties of geometric objects that do not depend on the exact shape of an object, but more on how the object is put together. We formalized this idea by definiting topological spaces; but what does it mean to say that two di↵erent topological spaces (such as a circle and a square) are really “the same”? Definition 4.1. Let f : X Y be a bijective function between topological spaces. ! 1 If both f and the inverse function f − : Y X are continuous, then f is called a homeomorphism, and X and Y are said to! be homeomorphic. Intuitively, think of X as being made from some elastic material (like a balloon), and think of stretching, bending, or otherwise changing the shape of X without tearing the material. Any Y that you get in this way will be homeomorphic to the original X. Note that the actual definition is both more precise and more general, since we are allowing arbitrary functions. Suppose that f : X Y is a homeomorphism. For each open set U X,we ! 1 ✓ are assuming that its inverse image under f − : Y X is open in X; but because f is bijective, this is the same as the image of U! under f. In other words, a homeomorphism is a bijective function f : X Y such that f(U) is open if and only if U is open. We therefore get a bijective! correspondence not only between the points of X and Y , but also between the open sets in both topologies. So any question about the topology of X or Y will have the same answer on both sides; we may therefore think of X and Y as being essentially the same topological space. Example 4.2. The real numbers R are homemorphic to the open interval (0, 1). One possible choice of homeomorphism is the function ex f : R (0, 1),f(x)= ! ex +1 1 Both f and the inverse function f − (y) = log(y) log(1 y) are continuous. − − Example 4.3. Consider the function 1 f :[0, 1) S ,f(t) = (cos t, sin t) ! that takes the interval (with the subspace topology from R)totheunitcircle (with the subspace topology from R2). It is bijective and continuous, but not a homeomorphism: [0, 1/2) is open in [0, 1), but its image is not open in S1. Example 4.4. Let us classify the letters of the English alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ up to homeomorphism. Here we think of each letter as being made from line segments in R2; the topology is the subspace topology. By inspection, there are eight homeomorphism classes, depending on the number of loops and line segments in each letter: BARPQDOCGIJLMNSUVWZEFTYHKX For example, W can be bent to make I, and so the two are homeomorphic. On the other hand, there is no homeomorphism between T and I: if we remove the crossing point, we are left with three intervals in the case of T, but removing one point from I produces at most two intervals. (Think about how one can say this in terms of the topology on each letter.) 2 Topological manifolds. In the remainder of today’s class, I want to introduce three additional examples of topological spaces. The first one is topological mani- folds. A manifold is a space X that “locally” looks like Euclidean space: if you sit at any point of X, and only look at points nearby, you may think that you are in Rn. Here is the precise definition. Definition 4.5. An n-dimensional topological manifold is a Hausdor↵topological space X with the following property: every point x X has a neighborhood that 2 is homeomorphic to an open subset in Rn. In geometry, people look at other classes of manifolds that are obtained by working with a smaller class of functions. For example, if a function and its inverse function are both di↵erentiable, it is called a di↵eomorphism; di↵erentiable man- ifolds are defined by replacing “homeomorphic” by “di↵eomorphic” in the above definition. In algebraic geometry, there is a similar definition with polynomials. At this point, somebody asked why we need the Hausdor↵condition; the answer is that we do not want to allow something like taking two copies of R and gluing them together along R 0 . (More about this example later on, when we discuss \{ } quotient spaces.) Later in the semester, we will show that an open subset in Rn can never be homeomorphic to an open subset in Rm for m = n; this means that the dimension of a manifold really is a well-defined notion. 6 Example 4.6. The square and the circle are both one-dimensional manifolds; a homeomorphism between them is given by drawing the square inside the circle and projecting one to the other from their common center. Example 4.7. The n-sphere n n+1 2 2 2 S = (x0,x1,...,xn) R x + x + + x =1 , 2 0 1 ··· n with the subspace topology coming from Rn+1 , is an n-dimensional manifold. Intu- itively, this is clear; let me prove it for n = 2 by using stereographic projection. The plane z = 1 is tangent to the sphere at the south pole; given any point (x, y, z) not equal to− the north pole (0, 0, 1), we can see where the line connecting (0, 0, 1) and (x, y, z) intersects the plane z = 1. In this way, we get a bijection − 2 2 f : S (0, 0, 1) R . \{ }! It is easy to work out the formulas to see that f and its inverse are continuous. The points on the line are parametrized by (0, 0, 1) + t(x, y, z 1), with t R;the intersection point with the plane has − 2 2 1+t(z 1) = 1 or t = , − − 1 z − which means that 2x 2y f(x, y, z)= , . 1 z 1 z −1 − One can show in a similar manner that f − is continuous. Since we can also do stereographic projection from the south pole, every point of S2 has a neighborhood 2 that is homeomorphic to R . Å ã Example 4.8. The inverse function theorem from analysis gives us one way to define manifolds. Suppose that f : R2 R is a continuously di↵erentiable function. The ! inverse function theorem says that if f(x0,y0) = 0, and if the partial derivative 3 @f/@y does not vanish at the point (x0,y0), then all nearby solutions of the equation f(x, y) = 0 are of the form y = '(x) for a continously di↵erentiable function ':(x0 ", x0 + ") R with '(x0)=y0. This function ' gives us a homeomorphism between− a small! neighborhood of the 1 1 point (x0,y0)inthesetf − (0) and an open interval in R. This shows that f − (0) is a one-dimensional manifold, provided that at least one of the two partial derivatives 1 @f/@x or @f/@y is nonzero at every point of f − (0). Example 4.9. If M1 and M2 are manifolds of dimension n1 and n2,respectively, then their product M1 M2 (with the product topology) is a manifold of dimension ⇥ 1 1 n1 + n2. The proof is left as an exercise. For instance, the product S S is a two-dimensional manifold called the torus. ⇥ An important general problem is to classify manifolds (or more general topolog- ical spaces) up to homeomorphism. In general, this is only possible if we impose sufficiently many other conditions (such as connectedness or compactness) to limit the class of topological spaces we are looking at. We will come back to this problem later in the semester. Quotient spaces and the quotient topology. In geometry, it is common to describe spaces by “cut-and-paste” constructions like the following. Example 4.10. If we start from the unit square and paste opposite edges (with the same orientation), we get the torus. If we start from the closed unit disk in R2 and collapse the entire boundary into a point, we obtain S2. To make a M¨obius band, we take a strip of paper, twist one end by 180◦, and then glue the two ends together. We can make a torus with two holes by taking two copies of the torus, removing a small disk from each, and then pasting them together along the two boundary circles. In each of these cases, the result should again be a topological space. To formalize this type of construction, we start with a topological space X and an equivalence relation on it; intuitively, tells us which points of X should be glued together. (Recall that⇠ an equivalence⇠ relation is the same thing as a partition of X into disjoint subsets, namely the equivalence classes; two points are equivalent if and only if they are in the same equivalence class.) What we want to do is to build a new topological space in which each equivalence class becomes just one point. To do this, we let X/ be the set of equivalence classes; there is an obvious function ⇠ p: X X/ , ! ⇠ which takes a point x X to the equivalence class containing x. Now there is a natural way to make X/2 into a topological space. ⇠ Proposition 4.11. The collection of sets 1 T = U X/ p− (U) is open in X ✓ ⇠ defines a topology on X/ ,calledthequotient topology. ⇠ Proof. We have to check that the three conditions in the definition of topology are 1 1 satisfied.