 Hello everybody, last time in the module 1 we told you about the course and about algebraic topology in general, we also showed how a fundamental problem in algebraic topology is shown to be not solvable by solving a corresponding problem in group theory, this was due to the only course. So, today we will introduce the concept of homotopy, the two problems that we have stated namely lifting problem and extension problem. So, let us see what homotopy is and how homotopy is going to affect this thing. So, let me tell you something about homotopy itself. In analysis often we need to approximate a given function by nicer functions, maybe some kind of linear maps or smooth maps and so on. This is one side, this approximation is something that we have to keep in mind. On the other hand in physics there are certain, there is one certain property namely those properties which have this this particular property, they are the ones which are more interesting one for physics, namely called stable property. In terms of for me who does not know much physics, what it means is the following. Suppose you take a system and disturb it a little bit, the system should return to the original position by a one and so on. That little bit is to be made precise and that precise way is nothing but homotopy, we will come to that one. The classical idea which was all the way back to Lagrange which describes this, the perturbation is what is called mathematically is the notion of homotopy. But the modern definition of homotopy that we are going to study is due to Brauer, okay? So, let us make a definition, take two typical spaces X and Y, let I denote the closed interval 0, 1. When any continuous function from X cross I to Y will be called a homotopy. Whenever you have a function like that, the first variable is the space variable, the second variable you can think of as time variable that is why we denote it by X comma t. So, when you fix a time variable t, you get a function from X to Y at the time t. So, these functions are called, are denoted by little ht. So, often people in the in classically they used to think of this as a family of functions from X to Y, the family being indexed by the set I. It is very important, it is not just a set, it is the space time space, namely the interval in some interval in R, that is that continuity of the time is very important. So, the continuity with respect to t also, it is not a separate continuity, continuity in X and continuity in t, that is not the correct thing. What we need is a joint continuity from the product space X cross I to Y, such a thing is called homotopy. So, this is the modern definition. Symbolically, whenever such a homotopy exists, the homotopy from H naught which we will, we may call it as F and H 1 at the time 1, it is called as G. So, whenever such a thing exists, we say F is homotopic to G and write it in this way. So, that is the definition. Now, if you have a homotopy H, you can define another homotopy H trigger by just reversing the time, namely t being replaced by 1 minus t. Then at time 0, it will be G first, then when time 1, it will be F, that will be homotopy from G to F. So, what this shows is, whenever there is a homotopy from F to G, there is a homotopy from G to F. So, the relation, if F is related to G by this equal, whatever this symbol, then that relation is symmetric, that is the meaning of this one. So, on the other hand, it is very easy to see that F is related to F, all that I have to do is H of X t, ignority is H of X t is F x. So, that will be the identity homotopy itself from F to F. So, the relation is reflexive. Finally, to see that it is transitive, suppose you have a homotopy G from G to H in addition to homotopy from F to G, then you can put these two things together, it is called juxtaposition one homotopy with another homotopy juxtaposing. So, what is the meaning of that? This is explained here, technically, completely, rigorously, namely, in the first half of the interval, you define it to be H. Only thing is, you fill up the interval by speeding up, you take the double speed, X going to 2t, F of X t going to H of X comma 2t. Exactly, similarly, you have to do it in the other half of the interval. Only thing is now the 0 is taken by half and 1 is taken by 1 only. So, you have to do a shifting also. So, this will become X comma 2t minus 1 applied to G applied to that one. In the interval half to 11 equal to 1. So, such a function is automatically continuous because the first half of X cross 0 to half, it is H that is continuous. In the second term, X cross half to 1, it is G it is continuous. And on the intersection, which is X cross half, it is both all these things are closed subsets. These two functions agree. So, there is a one single function though there are two formulas. So, this F becomes a homotopy from now H X 0, which is F to G X 1, which is H. This shows that this relation whatever you are wrote is an equivalence relation. Reflects you symmetric and transitive. So, here is a picture which tells you this X cross 0 to 1 and the half part X cross 0 to half, I have taken H then the second half I am putting it G. So, they patch up because on X cross half what is the function? It is the function G there little G. Here it is F, it is G here, it is here H. So, F homotopy to G, G homotopy to H means F is homotopy to H. Set of homotopy classes of functions maps from X to Y, it will be temporarily denoted by double bracket X Y. This is a set now. See already we have started doing algebraic topology here, started with spaces from X to Y and maps from X to Y. We have constructed purely a set of equivalence classes here of functions here. These are basic objects of study in algebraic topology. He says an element of this double bracket X Y is always represented by some map F from X to Y and that map will be denoted that class will be denoted by double bracket F. Now, we have to study what happens under compositions and so on and that is the next lemma. All right, next lemma says that suppose F and G are homotopy from X to Y. Now, suppose there is a map from W to X, then you can first take alpha and then follow by F, first take alpha, follow by G. So, if you have two different maps, they were themselves will be homotopy. Similarly, suppose there is a map beta from Y to Z. Now, you can take first F from X to Y, then beta or first take G and then beta. So, those will be also homotopy. All these follows if F is homotopy to G. Pre-composition and post-compositions of homotopies is a homotopy. So, this is the message of this lemma 11, fundamental lemma. Proof is very easy. All that you have to do is corresponding compositions of homotopies. Starting with the homotopy H from F to G, take H, composite alpha cross identity. What does it mean by alpha cross identity on the W part? Remember alpha is the map from W to X, right, alpha. So, on the W part, it is alpha. On the I part, it is the identity. So, alpha of W comma T is nothing but alpha W comma T. Then apply H. That will give you homotopy in what between F composite alpha and G composite alpha. Similarly, if you take beta composite H, that will give you homotopy of beta composite F and beta composite G. So, those are the proofs which is very straightforward. Now, we can sum up what the effect of this lemma on these sets. Look at the starting with two a pair of topological spaces. We immediately wrote down this set. What is this set? Take a continuous function from here to here. Take the homotopy class. So, this is a set of all such homotopy classes. This assignment has some wonderful natural properties with respect to compositions. If you pre-compose alpha, you pre-compose this class with an alpha from W to X, a function from X to Y will become a function from W to Y. So, what happens to the class? The class will become a whole class. There is no change. That is the whole point here. So, alpha check from X, Y to W, Y, these are now sets will be well defined. A homotopy class here goes to homotopy class here because if two functions are homotopic here, alpha composite, F composite alpha, G composite alpha, they will be homotopic here. So, that is the whole idea. Similarly, you can compose with once more. So, alpha and beta from Z to W, then alpha from W to X, then what you get is alpha composite beta check is beta composite alpha check. So, this is all pre-compositions. Exactly same thing happens on the compositions on the right also. Then we get new composite gamma. Now, this time we have notation, the checks are suffixes. Here we are superscripts. Here it is pre-scripts equal to new check composite gamma check. Moreover, one more important property is that if you take identity map, then identity check is nothing but identity, whichever way you take either pre-composite or post-component. Okay. How to get this one? I have told you. It is just lemma 1.1 that we have just seen along with one simple property namely associativity of functional compositions. F composite G bracket, bracket report, composite H, you can put the brackets on the other side. F composite, bracket G composite H. So, that is associativity. That is all you have to keep using. In fact, let us just clarify this one. There is no hanky-panky is being done. Okay. Just to be very sure. So, let us verify one of the property in the first one namely start with alpha composite beta. The upper check I want to show is equal to beta check composite alpha check. It interchanges the slots. That is very important. Whereas if you take the lower check, then it will be alpha check lower, beta check lower in the same order. So, let us verify this. So, what I have to do? Operate on F. F is a function from x to y on both sides and verify that they are the same. So, I start with this one. What is the definition of alpha composite beta as a function? Check F. It is F composite this function alpha composite beta. Now use the associativity of this composition. It is the same thing as F composite alpha composite beta. But then this is namely beta check of this function. F of alpha, F composite alpha. But F composite alpha is alpha check of F. So, beta check of alpha check of F. It is just beta composite alpha check check. Exactly same way you can verify the other one also. As I have told you, I keep repeating this. The workspace of algebraic topologies in the whole later on we can make it smaller and smaller. In the whole is all topological spaces. And for each pair of topological spaces, the set is namely homotopy class of maps from x to y of homotopy classes of maps from x to y. This is the one which you are interested in. Not exactly the maps but the homotopy classes. Modern mathematics is full of such collections of assignment. So, we have assigned, starting with a pair of topological spaces we assigned. So, you know we study this set. This set when you take special cases will have lots of structures, different kinds of structures. And that is what we will have to study. So, the property of this assignment which we have seen just in the previous remark. So, those properties are themselves called, you know, they are categorized. They are, you can say, they have been put inside a kind of discipline which is called categories and functions, which is part and parcel of algebraic topology. These categories and functions gives you the basic language required to express so many complicated ideas into simple language. This has been, you know, this is a very modern discovery I would say. People like Euler, Gauss, they never had this kind of language. Even Riemann did not have, even Hilbert did not have this. So, if you try to read their papers, it will be very, very difficult for us. It is entirely different kind of language. Okay, let us take a closer look at these classes. Suppose I take a special case namely the domain is just a single point that I denote by your bracket star, very single point, singleton space. Then what is this space, set of homotopic lines of maps from x to y. To understand this first of all, you must see what are maps from a singleton space. Any function from a singleton space will be automatically continuous. There is only one topology there. So, what are all functions? A singleton space, what is the meaning of function? You have to just mention one point in the space y. As soon as you mention that the function is well defined, that the domain singleton goes to that point. Right? Therefore, this is nothing but set of points in y. Now, what is a homotopy? A star cross i is nothing but homomorphic to an i itself. So, they are functions from i, of course, continuous functions, maps. So, these maps you must have studied, they are called paths. The starting point is some point that is one function, end point is another function. So, these things are connected by a path. The equivalence classes of such things are nothing but now path components of y. So, this is one of the simplest algebraic topology invariant. This is a homotopy invariant which we have constructed now. So, this was there already in point set topology. The path connected components, a number of path connected components is something different from all other concepts like compactness, separateness, T1-ness, T2-ness, various things. So, this is a quantitative invariant as compared to qualitative invariants. So, all topological invariants in algebraic topology, they will be of this type. They will be quantitative, not qualitative. Quantity just doesn't mean that just the cardinality, there is much more than that. So, you can say it is algebra. There are structures there, of course, like groups, abelian groups, rings and so on. All right? So, we produce in algebraic topology and study various such invariants of topological spaces which are not only sets but more algebraic structures on them, rings, modules and so on. This basic idea of assigning algebraic invariant such as groups and rings instead of numbers goes back to a mean order. Before that, people were just counting the numbers like path connected components, number of, okay. If not, number of arcs needed to cut down a given something. For example, look at r, you remove one point, it has disconnected. Look at the circle. You remove one point, it is still connected. So, you need two points. So, this is the kind of things they are studying, okay. So, M. Minoiter says, don't worry about such things. Best thing is to do, assign group structures to them, okay. So, this was one of the landmark contribution of M. Minoiter. M. Minoiter has done a lot of physics also. So, now it is time for us to reformulate our fundamental topological questions in terms of homotopy, okay. So, first question we are going to change. That was lifting problem. Remember that. So, we are going to make it into homotopy lifting property. So, homotopy lifting problem, then things which have satisfied certain things will become homotopy lifting property. So, fix a map P from E to B. Given a function X, F from X to B, question is, does there exist a map G from X to E such that P composite G is homotopy to F instead of equality. Everything else is the same. Instead of equality here, replace it by homotopy equivalence, okay. So, here we are asking can F be lifted through P up to homotopy. This is the meaning of, this is another way of saying the same problem. Obviously, the original question when you have to equality here, okay, has an affirmative answer, then the change your question, I mean Q1A I am talking about, it is also a affirmative answer because equality implies homotopy equivalence also. But the converse is not correct. Even this is equivalent, if this is solved, there may not be any function G is a set, P composite G is equal to F, okay. But this simple observation that homotopy equivalence, okay, will be assured if there was actually a map, homotopy lifting, if this is not true, namely you cannot lift even the homotopy, then of course you cannot lift up to, you cannot lift the map at all, right. Therefore, a negative question, negative answer would have given you negative answer. So, even this much surveys given by this, this map is of importance to us, okay. But what we want to use is, we will ignore this portion, we will only solve or answer whether something is homotopy, homotopy equivalence can be lifted or not and we are satisfied by that. So, we will pretend as if the problem is over when you have lifted a function up to homotopy. What does that mean? We just make a hypothesis, namely assume if something can be lifted up to homotopy, then it can be lifted, okay. So, this is called homotopy lifting property, this is an axiom, you know, on the function, on the map from P, P from E to B, okay. So, let me state this one correctly. Regarously. So, start with a map P from A to B. So, we are given a homotopy from X cross I to B, okay. And a map G from X to E such that when you take P composite G, that is a map from X to B, right, it is the function f of X0, it means starting point of this homotopy. The starting point of this homotopy, namely suppose this is f of X0 small fx, then that map has been already lifted, that is G, okay. So, this is part of the hypothesis. So, this all this is called homotopy lifting data, this much is given, okay. We say P satisfies the homotopy lifting property if each such data, homotopy lifting data will give rise to an actual homotopy from G from now X cross I to E, see this homotopy was X cross I to B. Now, the lifting, this whole thing is gone up to E such that when you compose with G, compose with P it comes to f and the starting point of this homotopy G by X0 is the given function G from X to E, okay. If this happens for every X and every f and every G, there must be capital G like this, then we say P satisfies homotopy lifting property, understand. So, let me put this one in a figure in a picture which you will just easy to remember, this is a picture. So, here you have X cross 0, G to E and here you have X cross I, okay and this is your inclusion map data, X cross 0 goes to X cross 0 under this inclusion map. f is the homotopy at the starting point it is G, compose with P it is this f, this much is given as soon as you are given here you must have this dotted arrow capital G from X cross I to E when you compose it must be f. So, this triangle is commutative and this triangle is also commutative it is same thing as if you restrict it to X cross 0 it must be horizontal. So, this is the conclusion and this is the data for every such data if you can come here from this part this part that means P is a as homotopy lifting property, okay. So, this is a definition so this will take care of all our botherations about a pointed topology and we can do only on algebra topology this is the whole idea. Such special classes of maps are you may say very rare no that is not true there will be plenty of them and one such class we are going to study very rigorously in this course namely covering projections. The notion of homotopy lifting property is important enough to make it another definition namely such things are called Hurevid fibrations or just fibrations this is because Hurevid was the first one to notice it and study it quite deeply so people call it Hurevid fibrations, okay. What is the meaning of fibrations it is a map from one space to another space which has a homotopy lifting property, okay. So, this is what I am telling you Hurevid you are the first to recognize this one. So, one important case of fibrations will be studied in this course and that is covering projection, all right. So, we will take up the second question a little later namely the next module. So, today this is enough thank you.