 I am going to present this course on algebraic topology part one under this NPTEL portal. My team members are Dr. B. Subash Aysar Thirupati, Dr. K. Ramesh IITM, Vidhan Paul, Shiva Shankar, Vinay Sipani all from IITM. This course is presented in 62 modules of approximately 30 minutes each in a span of 12 weeks. Along with these slides and lectures, there will be a number of live sessions for doubt clearing sessions. The content of the lectures can be divided into these seven chapters you may call Introduction, Fundamental Group, Functions-Paced Topology, Relative Homotopy, Financial Complex is part one, Sympirical Complex is part two and Covering Space and Fundamental Group and Covering Space and G-Coverings and Fundamental Group. I will be assuming that the student of this course is well acquainted with the rudiments of point topology. Also, this course is at the level MSc. Therefore, a certain amount of mathematical maturity will be resumed. We have strived hard to bring this advanced subject to your door by giving some completely elementary treatment to this one and also explaining beyond all written textbooks. The one good reference for this material is my own book on basic algebraic topology. However, I will give you along with all the material in this cover in the PDF format in which there will be further references for further study. As far as the present course is concerned, the duration of the present course, you can stick to those notes which will be available on the NPTEL website. So, let us come to what is being a basic problem in topology. Central problem in topology is to determine if two given topological spaces are homeomorphic or not. For instance, hope you all know that any two open intervals in R are homeomorphic to each other. Since we can actually write down the homeomorphism in each case, take, give me any two intervals a, b and c, d. We can actually write down a linear homeomorphism which sends a to c and v to d. And then you can also write a homeomorphism from open interval a, b to the entire of the real line. So, this will take care of most of the cases, but then you can study as the case is solved. By actually writing down homeomorphism, no problem. So, on the other hand, we also know that closed interval and open interval are not homeomorphic to each other because one is the closed interval is compact whereas the other one is not compact. So, compactness which is a topological invariant is good enough to distinguish between an open interval and a closed interval. There may be other methods, do not worry. So, it is one method. But we have, we have been able to tell that a closed interval and an open interval are not homeomorphic. In general, displaying a homeomorphism between topological spaces becomes very difficult. It is not like just intervals case. On the other hand, it is fruitful and easier to find out that there is no homeomorphism between two given spaces x and y, if that is the case. A standard method is to look for a suitable topological invariant such as compactness, connectedness, etc. So, these things you must have seen in topology course. Just I am recalling to motivate what I am going to do in this course. So, let us take another example of how one is using connectivity. An example, very simple example, namely let us look at r and r cross r which is r2 real line and the plane. These are not homeomorphic. So, how do you proceed? Suppose there is a homeomorphism, say let us call it as f from r2, r2. Throw away the 0 from the origin, throw away the image of 0 from the domain r2. Whatever is left out, f takes r minus 0 to r2 minus f0 and that itself will be a 1-1 map, on-to-map, continuous, inverse is also continuous. So, it is a homeomorphism. Any restriction to the corresponding subsets is also homeomorphism. Now what is happening? The domain of f is not connected whereas the range is connected. You must be knowing that if you throw away a single point from r2, it is still connected. Whereas any single from r, if you throw away one single point, it is disconnected. Therefore, right in the beginning there could not have been any function f which is homeomorphic from r2. Then this method may not work very far. I have used connectivity, I have used compactness and so on. But you can go on using a number of them. There are situations wherein none of these things, the so-called topological invariants may fail to distinguish between two spaces. Such a thing can happen. Let us just think about, say suppose Rn and Rm, do not have to go too far where n and m are just arbitrary. Say m is bigger than n. Can you say that Rn and Rm are not homeomorphic? No simple points at topological fact will be able to tell you this. You know, just connectivity etc. will not be enough. You may succeed to do something with r2 or something but again r3, r4, r5 etc. it will not work. However, very deep topological results are there. I mentioned theory will tell you that Rn and Rm are not homeomorphic. So, there are. But they are quite demanding even inside points at topology. They are not easy. So, the role of algebraic topology is now, it is good to have at least some idea what the fundamental problems in discipline are. Whenever you want to study some discipline, you must know what is it about. In most cases, these problems are, problems remain unsolved or you will know that someday that the problem is too difficult or problem is just unsolvable. Even then the concept of the central problem or fundamental problem has so much of role to play namely that there will be always some related problems or modified problems which will demand our attention. And this is the case with topology also in general and algebraic topology also in particular. Central objects of study in topology in the mainstream mathematics are not just topological spaces, it is too large. But let us take what are called as manifolds. These manifolds are something like modeled on Rn, Rn and Rm I am discussing. So, they are locally in a small neighborhood, they look like Rn and Rn. Their rigorous definition will be introduced much later. The problem is then to determine whether two manifolds are homeomorphic or not. Don't worry about arbitrary spaces. Even this problem is known to be unsolvable. So, I want to tell you the algebraic topology is associated with certain algebraic objects such as groups and rings rather than compactness, separability, etc., which are more qualitative. So, the algebraic invariants that we are going to introduce, they are much more flexible than homeomorphic invariants. Obviously, these invariants will become additional tools for us while distinguishing two topological spaces. So, I told you the classification problem, classification problem means what? Determining whether two of two given spaces are homeomorphic or not. Up to classic, up to homeomorphism we are putting them in different classes. This has known to be, it is known to be unsolvable. The negative result itself has some implications for us. It is useful for us. So, let us take a few minutes to look at how it was established. So, this is where already algebraic topology comes into picture, not points at topology. Each path connected SpaceX, one associates a group called the fundamental group. We are going to study this fundamental group very rigorously in this course. So, right now you take it for a granted. What we have done? Take a topological space which is path connected, you get a group out of it. This association has the property that if you have a function from x to y, then there will be homomorphism from the group pi 1 of x to pi 1 of y. So, association is so natural that if another map from y to z is there, okay, then you have two maps f check and g check which are group homomorphisms. The g composite f check, that means you first composite g and f and then take the associated map that is nothing but the composite of the corresponding associate maps to begin with g check, f check. Moreover, if you take the identity map from x to x, the associated map will be the identity homomorphism from pi 1 of x to pi 1 of x, okay. So, all this machinery is very, very fundamental in the construction of this, the fundamental group. And that is what part of algebraic call the essence of algebraic methodology, okay. So, the point is once you have such a association, if f is a homomorphism, then f check becomes an isomorphism. Why? Because look at a inverse to g, f composite g is identity. Therefore, g check, composite f check will be identity. So, this way g check becomes inverse of f check as a homomorphism. So, a homeomorphism in this is a isomorphism. It just means that if you started the two spaces which are homeomorphic, the corresponding groups must be isomorphic. Therefore, if for some reason I know that the groups are not isomorphic, then you know that the corresponding spaces are not homeomorphic, okay. Now, how does one use this to show that the classification problem itself is unsolvable by showing that the corresponding classification in group theory is not solvable, okay. So, that itself is a huge thing. Even a restricted problem in group theory, namely what are called as finitely generated groups with finitely many relations, we are called finitely presented groups. Suppose I have given you a set of relation, set of generators and set of relations that should describe a group. On the other hand, another set of generators, another set of relations that should specify another group. The world problem is that to determine whether these two groups are isomorphic or not. It is known that this world problem cannot be solved in general. What is the meaning of that? Suppose you write an algorithm to solve such a thing. As soon as you display your algorithm, we can give a set of examples namely two such groups which cannot be determined by this algorithm. The isomorphism between that whether the two are isomorphic or not cannot be solved. This is the meaning of the world problem in group theory cannot be solved. A consequence of this is the homeomorphism problem for manifolds cannot be solved. There is a small gap here namely how does one go back to homeomorphism by constructing manifolds of any particular take any particular group then construct a manifold with that as fundamental group. So, these problems are all problems in algebraic topology not general topology and unfortunately we will not go this much deep in this course. So, this is only a trailer for you to study more and more algebraic topology. So, this problem in group theory was solved by a topologist P.S. Noyako in 1955. This however does not close the subject altogether. Topology is still very lively subject extending a helping hand in solving problems in various areas of mathematics. Process such as associating the fundamental group to a space S as considered above is called constructing a funta. Algebraic topology may be described as at the outset study of such processes. In this course we shall construct and study one most important funta called the fundamental group and we will have number of applications. Coming back to the fundamental problem there are many interesting related problems other than the central one. For example, you want to find homeomorphism from one space to another space. Before that you would have to find some nice functions. A part of finding a homeomorphism a function from some nice function which is continuous or maybe it is more property open mapping and so on. So, instead of listing all these problems I will list here take up two central problems again it is in this one just two basic ones. So, they are like called one is called a lifting problem other one is called extension problem. The underlying themes in these two problems occurs repeatedly in the study of basic notions of algebraic topology. So, let us get familiar with these concepts slowly. So, start with a triangle of functions x, y, z are sets f, g and h are functions what is the meaning of this triangle is commutative look at starting from x you can go to y and then come to z the arrows indicate where to go from where okay the arrow is a function that right so x to y y to z but you can start directly come from x to z these two functions must be the same namely g composite f must be h that is the meaning of a diagram like this is commutative you will have lots of such diagrams not necessarily triangles you may have rectangles you may have beta ones and so on. Okay in particular if x, y, z are topological spaces and f and g are continuous functions then you would like to have maps but commutativity is just a satiric notion g composite f must be h okay the diagram is called commutative diagram even two of the three maps the question is can you consider the problem of finding one or all maps which fit the third arrow so this is a general problem okay if f, g are given then h can be taken to be g composite f there is no other choice but suppose h is given and f is given then g could be many other possibilities similarly if h is given and g is given there may be many possibilities for f there may not be any possibilities for that this is the problem that I want to attack now okay so let us make accordingly let us make two different cases the first case is now I am going to give specific names here so that we remember it again and again instead of x, y, z now I have take x, e and b e to b there is a function p there is a map okay and the concentration is on this map given any function x to b that is f can you find a function g such that p composite g is equal to f this is called the lifting problem f being lifted to g via or through p p composite g must f so the question is on this dotted dotted dotted line other things are data does g exist is a question that is why I have put it dotted understand that one okay second question is the other way around namely now I have a to x a map f a map eta so concentration is on a to x this eta is fixed now suppose a to y there is some functions given can you have a map down in a fact from x to y so this is called factoring so this f is factored through eta there earlier we are lifting problems factoring problem but it is actually can be thought of as an extension problem especially when a to x this map eta is an inclusion map of topology spaces then what is the meaning of having a function a on a subspace having a function f at is on the larger space so eta is the inclusion map so this is a very special case so but the special case name is given to the entire problem you can say a part is a factorization problem and we part we part is an inclusion problem the factorization comes when instead of an inclusion map you have a surjective map and this is a quotient map okay both these cases are you can call them as extension problem factorization problem is easier extension problems difficult so you concentrate on extension problem later on okay so factorization problem we will decide very easily this in this course that extension problem is the problem okay algebraic topology steps in here by bringing a big twist to these questions which we shall take in a next module these these questions again we will discuss but we take twist namely what is algebraic topology is going to do this is a topological problem point set topological problem given a subset and a function and a larger space can you extend the function to the whole space and so on okay this is just a point set problem so what is algebraic topology doing that we will discuss in the next session thank you