 Hello. So, let us begin the fourth module today. In the first three modules, I have given a bird's eye view of what this course is about and a little bit about algebraic topology in general. I have told you about certain big problems that cannot be solved and about certain million price problems like Poincare conjecture, which we cannot discuss in this course in any depth. So, today, let me begin by telling you that some other big things that we can achieve in this course on the positive side. So, this is called Brauer's celebrated theorems. I have two of them here. One is the Jordan-Brauer separation theorem, which is the Jordan comes here for n equal to 2. For higher things, it is Brauer. That is why Jordan-Brauer separation theorem. Take a copy of Sn minus 1 in Rn for n greater than or equal to 2. Then the complement of this Sn minus, Sn minus 1, the copy of Sn minus 1, I am calling X. The complement of this has precisely two connected components and X happens to be the common boundary. So, in the case of n equal to 2, copy of Sn minus 1, one calls it as a Jordan curve, Jordan loop. So, Jordan loop separates the plane exactly in two components. One is inside, another is outside. Inside region is called, inside is what? The boundary region. That is the meaning of inside region. There is only one boundary region and there is only one unbounded region. And the loop happens to be the common boundary of both. This has been completely generalized by Brauer for all n. In this theorem, we will be able to prove in this course. Maybe it will take some time, but it will be proved. That is the whole idea. The next thing is, Brauer's invariance of domain. Invariance of domain means, you know, what is a mean of domain in calculus and so on. It is an open and connected subset. Open and connected subsets of Rn are called domains. So, if see something is a domain in some Rn, that n is invariant. That is the whole thing. That is the whole idea of Brauer's invariance of domain. Suppose you have u and v, some subspaces of Rn and they are homeomorphic. If one of them is a domain, that is one of them is open, then the other one is also open. So, that is like saying that invariance of domain. If something is a domain, then homeomorphic copies of that inside the same Rn, they are all domains. As an easy consequence of this, if you change the dimension, then they are not domains can also be observed. Namely, for n not equal to m, Rn will never be homeomorphic to Rn. So, this corollary is an easy corollary to theorem 1.3. I will let you think about it. Finally, we will solve this one. This corollary is not difficult from the theorem 1.3. Android method of proof of these two theorems is to obtain them as not too difficult consequences of singular homology theory. The singular homology theory will be taken up in the sequel to this course. On the other hand, in this course, what we shall do? We shall obtain a proof of the Brauer's invariance of domain as a consequence of simplicial approximation and some combinatorial result called Sperner lemma. There are of course, purely poinsettia colloidal proofs of this invariance of domain which are much too long and difficult. So, so-called dimension theory, books have been written on that. Notice that mere homotopy equivalence is not able to detect the fact that Rn and Rm are not homeomorphic for n not equal to m because both of them are contractable and therefore they are homotopy equivalent to each other. So, how homotopy helps? That is a strange thing, no? It does. It should be noted that any known proof of purely poinsettia colloidal invariance of domain is not too easy at all. All proofs are the more quite ignored and lengthy. But you can look into Engelken's book and Hurevis-Walmin book and so on. The general purpose of this course is to take a few steps which lead the student to the doorstep talks such great results in topology. We may not be able to see much of them. But once you have a couple of courses like this, you will be able to access all these results. Algebraic topological tools have been invented and sharpened by masters while attempting to solve topological problems. This requires the reader to master a formidable amount of technical tools even before understanding what the master is trying to do, master is trying to work out. We have tried to minimize this with shortcuts without missing out on important points which can have a certain permanent value. So, this is what we have tried to do in this course. So, in what follows, we shall keep acquiring new tools and sharpening the old tools so as to solve problems mentioned in question number 1 and question number 2 above and many other related problems. So, this is the summary of whatever we want to do. So, we will now start doing one by one those things. So, since we have already some technical definitions and so on here is a set of exercises which you should try to solve them on your own and submit and the tutors will check them and you know later on we can even discuss it in one of the open sessions, live sessions. But before that you have to submit and you have to participate. So, let me go through these exercises. First one is where is all they are simple exercises to show that a contractible space is always path connected. Second one is I have told you that there are lots of topological properties which are not homotopy invariants, which are not preserved in a homotopy. So, give a list list of a dozen topological properties I have done no half a dozen show that composite of two homotopy equivalents homotopy equivalents show that homotopy equivalents among the spaces is an equivalence relation. I have already told you how these things are, but now you have to write down full details of these exercises. Okay, exercises is only about that, there is nothing very hidden in this one. Now comes to a few more exercises. Suppose you have f and g x to y and y to x such that f composite g and g composite f are homotopy equivalences. I am not saying that f and g are homotopy inverses of each other. The composite I am not saying that the composite f composite is homotopic to identity of y, not g composite homotopic to identity of x, but they are themselves homotopy equivalences, then show that f and g are homotopy equivalences. So, I caution you, I do not mean that g is the homotopy inverse of f. It may not be, it may be, it does not matter. So, that is not the question here. Here you have to think a little bit. Okay, keep thinking. When I want to use one of these results in the exercise, given the exercises, by that time I will give you the solutions, but then otherwise you keep thinking about it. So, whenever you get a solution, you can submit it, the tutors and I will check them. So, similarly, next problem here, f from x to y, g from y to z be such that f and g composite f are both homotopy equivalences. Show that g is homotopy equivalence. It is like cancelling out one. If f is invertible, g composite f is invertible, then g is invertible. So, this is algebra of homotopy equivalences. Let us begin with now, prasthags. We want to do whatever we want to do. We will just start doing them. So, this section will contain the definition of the fundamental group and its fundamental, functorial properties. We shall also introduce two best methods of computing fundamental groups and use them to compute the fundamental group of spheres, the need objects. Once you have Rn, they are the simplest one. They are contractible. They do not have much homotopy properties. But the next objects are the spheres in them, unit spheres in the Euclidean spaces. Extensive study of these methods will be taken up later on. So, this is just now a trailer again to give you a flavor of what kind of things are coming up. So, that is what this section is about. But it already introduced you slightly deeper into the subject. We call that a path in a space can be thought of as the track of a moving point. The fact that we may move from one point to another point, that means what moving means what in a continuous way within a space that is described by saying that the space is path connected. We can go from one point to another point as path connected. Path connectivity is very, very old concept and which is very fundamental in all topological aspects. We know that the set of path component of a space is an important topological variant. We have introduced it as homotopy classes of maps from single point into X. The set of homotopy classes of maps from a single point into X. Now, even a path connected space, suppose X is path connected, we are now interested in looking at various different ways in which two given points may be joined. For example, suppose X is a two-dimensional disc. Let us take a unique disc. Then given any two points, the natural way to join them is to take the line segment. If we are not so economical, there will be a lot of nearby pathis, but they would all be in some sense the same. Even if you go a little bit away from the straight line, the straight lines are not always possible. Pathis are not always made straight lines except perhaps in deserts. But we keep the direction the same. So, it is more or less the same in some sense. So, that is the meaning of being same slightly. They are away, but like diversions in a given road, when there is some road constructings going on, they are all homotopy paths. But let us look at the picture S1. In S1, let us say take any two points. Then I want to say that there is no direct, there is no straight line now, but there are two different arcs from one point to another point. So, these arcs, you know, you cannot change from one arc to another arc continuously. So, how to make this one rigorous? That is the task now we have. A path in X is described by a continuous function from a closed interval which we have standardized as 01, the closed interval 01. The closed interval 01 itself is contractible. Therefore, we know that any path namely a function from I to X must be null homotopy. We have seen that once you have contractible space, any function from a contractible space into any other space is null homotopy. So, the homotopy that we have introduced is not very effective in determining the two arcs that are there in S1 which we want to distinguish. So, we need to sharpen the tool here. Then we want to study the the path is as such. That is the meaning of sharpening the tool. We have the homotopy concept, but we want to modify it as per our requirement. So, we fix two points X0 and X1 belonging to any space X. You can assume X is path connected, there is no other way. And look at the space of all paths from X0 to X1 inside X. So, that I have written it by omega X X0 comma X1. All paths. So, this is the collection of all paths. They are starting at X0 and ending at X1. All of them are in X. So, such a space can be given a neat topology what we call as compact upon topology. What is the meaning of compact upon topology? I will tell you later on. There is some topology. We may then look at path is in this space. Path component of this space. This turns out to be nothing but classes of homotopies in this space. Namely, I have to change from, I have to change the given path to another path. But all the time we are in this space means the end points X0 and X1 remain the same. So, homotopy keeps the end points same. So, this is the extra homotopy that we are going to introduce. A modified homotopy that we are going to introduce. So, once we see what we will try to do, then we can do that. We have to understand what we want to do first of all. So, this leads to the concept of fundamental of the space X. So, let us introduce this concept of path homotopy. Following the simple commonsense rule of tracing one curve until its end point and then tracing the another curve which begins at that point. We get a binary operation on the set of all loops at a given point in a space. To take a point and then look at a loop at that point means the end point and the starting point are the same. X1 is equal to X0. Take that special case. Then take a loop. Take another loop. You can compose it by this method. It is called concatenation of the loops which is just the extension of homotopy that we have done. Constant loop, you know, you see any constant loop is a funny thing. Geometrically, you would like to have a loop as a continuous function from an interval into the space X with end points are same. But if the end point, all the points are the same that is also looped by our definition. Why we allow this one? This is very nice thing to allow this one. A constant loop will now, perhaps I would like to say that it will act as the identity for this operation. Because after you trace a curve and come back and then you do not do anything. You keep there all the time. It is less like you have traced that curve that is all. So that is the meaning of operation being this constant loop being an identity element for this operation. It is a right-sided identity, left-sided identity also. But there are problems. We are just now making a demand, making, you know, anticipating something, how to do this one, trying to do something. So we have to keep sharpening our definition, how to make these things and finally it should work. So another thing is, if you trace a path in the opposite direction, it should be treated as the inverse of the path. You have gone through this one, but finally you have come back the same way. So it is as if you have done nothing. But this kind of thing one has to do. However, our expectations are met only when we pass on to the homotopy classes of loops. Otherwise as functions they are never the same. So this is what we want to emphasize. We obtain a powerful notion namely fundamental group only when we go to homotopy classes of loops. So which is going to play a very important role in the topological behavior of a space. So let us make a formal definition of path homotopy. Before that let me make a formal definition of a path also now so that we have no confusion now. A path is just a continuous function from a closed interval from the closed interval 01 to X. All the time we have fixed the domain to be the interval 01, closed interval 01. If omega is a path, omega 0 is called the initial point. Omega 1 will be called the terminal point. Both of them together can be called as endpoints. Okay, when the endpoints coincide, such a path is called a loop. And what is the base point, the base rate at what point namely omega 0 which is the same thing as omega 1. Okay, so these are some basic terms. So I have defined what will be a path, initial point, terminal point, end point and a loop. Okay, so let us now make a a path homotopy definition of path homotopy. Take two paths, the same endpoints, omega 0 is equal to tau 0 equal to X, not let us call omega 1 equal to tau 1 equal to X, 1 let us call. Then a path homotopy from omega to tau is first of all a homotopy. Homotopy of these maps, remember if the map is from X to Y, then homotopy was taken X cross I to Y. Now the maps are from I to X. So homotopy will be from I cross I to X. So H is a continuous function from I cross I to X such that when you take H 0 for all points S, the starting thing is X naught. H 1 for all points S is X 1. Okay, so these two points do not move at all. The second coordinate showing is, is moving. Okay, so they do not move at all for every point 0 less than X cross S 1. H of t 0, it is the first path that is omega. H of t 1, it is the last path, it is omega tau t, sorry. Okay, so if this happens, then we call omega path homotopy to tau. All right, and we use a simple notation omega seem equal to tau. A general notation for homotopy was there was a twiddle and an air, or remember that here, this was a different notation that time. So this is a different equivalence. This is a different relation to a different symbol. So here is a picture starting with omega here, ending in tau here. Endpoints are fixed here. So this is some, you know, t equal to t 1, t 2, t 3, various stages dotted lines are a picture. So this is how a homotopy is supposed to look like. Okay, it seems two points X 1, X naught and X 1 on the set of all paths in X with initial point X naught and point X 1. It is easily seen that path homotopy is an equivalence relation. The proof is exactly same as how functions equivalence of homotopy equivalence is the same as this one. We are everywhere, we have now taken endpoints fixed at same thing. So every time it will be fixed same thing. Transitivity, reflexivity and symmetry, all they are, we can verify we can verify the same way. Okay, so path homotopy is a equivalence relation among the class of paths which have same endpoint. That is what is important. Okay, notice that path homotopy is more restrictive than the homotopy of maps, which you have defined in the previous section. So it is natural that both the properties of the domain and the core domain will influence the nature of maps between them. We have witnessed this in theorem 1.1, namely if the domain is contractible, then the function is null homotopy. Similarly, code domain is contractible, any function into that null homotopy. So even for path is and path homotopy, there must be something happening irrespective of where I am taking X, X is the space I am taking the path is in that and path homotopy. Let us first understand what are these essential homotopies between path is, of course, endpoints must be the same. Okay, so let us first take away this path. After that, we can talk about what happens inside X. Right now, irrespective of what happens to X, where X is, some path is, these path homotopy must have certain property. Let us understand that. So this leads us to what is called as reparameterization. Take a path omega from I to X. Reparameterization of omega, we mean a path which is omega composite alpha, where alpha itself is another map from I to I, such that 0 goes to 0 and 1 goes to 1 under alpha, any path. And then you change it, namely, omega instead of omega t, you take omega alpha t. That will be called reparameterization of the path omega. One of the simplest thing is the image of omega and image of omega alpha does not change, it is the same thing. Okay, so for Lehmann's point of view, both the paths are the same. But for mathematician's point of view, they may not be the same. But in fact, they are not the same if alpha is not identity map. Okay. But the Lehmann's point of view should be respected and what happens is, these two paths should be always path homotopic to each other. So weaker equivalence is there. Any reparameterization will not produce any new path is in that sense. They will all path homotopic to the original omega. Let us see how. All that I have to do is look at this homotopy A times A of t s. A of t s is 1 minus s times alpha t plus s times t. So I am joining t and alpha t where are they? They are inside closed interval 0, 1. Therefore, line segment makes sense. 1 minus t times this, 1 minus s times this and plus s times that one. It is again inside the closed interval. Therefore, this gives you a homotopy. When you put s equal to 0, it is alpha. When you put s equal to 1, it is the identity map t going to t. So alpha is homotopic to identity map relative to the end point 0 and 1. No matter what s is, when you put t equal to 0, what do we get? Alpha 0 is also 0 and t is 0. So A t s is 0 for all s. Similarly, t equal to 1, alpha 1 is 1 and t is 1. 1 minus s times 1 plus s is equal to 1. So this homotopy A is a homotopy of the identity map with alpha keeping the end point fixed. Therefore, when you apply omega to it, what do you get? You will get a homotopy of omega composed alpha with omega composed identity which is omega. No matter what omega is. Or no matter what alpha is, reparameterization of all path is is path homotopic to the original one. This is the concept. Now I want to warn you that you might have studied in differential geometry or even in complex analysis and so on. When there is a reparameterization, first of all those maps are not just maps. They are smooth maps or piecewise smooth maps. Similarly, the reparameterization must be smooth maps with extra condition namely the derivative at every point must be positive. So this is the standard condition in differential geometry, also in integration theory and so on in complex analysis. But in algebraic topology, we do not need those conditions. We are taking all continuous functions and we do not want only end points are the same, you see enough for us. If these conditions are also satisfied, there is no problem. Of course, we do not need to bother about them because our spaces are arbitrary spaces. The derivative does not make sense there. So I will stop here and we will presume from this point onwards in the next module. Thank you.