 Last time we introduced path is path homotopies and then examined or studied number of homotopical properties of this path composition. The path composition has two sided identities which are different as such because the path is have different end points. It is associative and it has an inverse also. Each path if you take the reverse way of tracing it, it will be the inverse. All these only up to homotopies which is what we have seen. Today we are specialized to the case when the end points are the same. X naught is equal to X1. So start with a space X and fix one base point X naught. Then we are going to define what is the meaning of this pi 1 of X X naught which is going to be a group. What is this group? It consists of homotopy classes, the path homotopy classes of loops. What is the meaning of loop? A path which has both its end points at X naught. Loops based at X naught. This is the set. The composition law which you have defined now becomes a binary operation on this set because if two loops are homotopy to each other and another two loops are homotopy to each other, their compositions will be homotopy to each other. This homotopy means path homotopy. So the composition law goes down to the set of homotopy classes of loops and that binary law we have seen must be become associative. Now the two sided identities which were namely the constant loop at X naught because both the sides are the same now, the end points are the same. That is why. And the inverse will be as it is, they trace the same loop in the opposite direction that will be the inverse. So automatically we have got a group associated to a space X together with a point, specific point X naught. A loop and its homotopy always contain the same path component of X. They will never get out of the path component of X which contains the point X naught because the loops have to start there. Homotopy will be also have to reverse respected point X naught and so on. Therefore, all the time will be inside the component C of X which contains the point X naught. For each point there is a component. So the group pi 1 of X X naught is identical to the group pi 1 of C X naught under the ordinary inclusion of C inside X. Because of this reason I could have assumed that X itself is path connected. It will never go out of one single path component. Wherever you are started it will remain in that component. So because of this reason and many other things which we do, all of them continuous maps from connected spaces, path connected spaces and so on. In algebraic topology it is customary to assume that a space is path connected. For the reason that each path component can be studied first and then you put them together you get the study of the whole space because a space is always divided into its path components. Now within a path component you may have take different points for defining pi 1 of X X naught. So let us assume X is path connected and suppose I take another point pi 1 of X X 1. What is the relation between pi 1 of X X 1 and pi 1 of X X naught? So this is what we want to study. This is called under change of base point what happens to the group. So start with a path from I instead of X naught and X 1 I have taken the notation a and b. The changing notation like this is somewhat dangerous first however but it is also good practice. So ultimately you will start thinking without reference to the actual notations. Suppose you have two points inside X you can take a path from one to the other. When you take a path from one to the other suppose you have a loop at one point then you can view it as a loop with a tail at the other point. A loop with a tail will be also a loop but its base point will be the end of the tail. So this is the picture you should want to have. Now when I speak like this I have not used any notation you see there is a whole point of changing notation. So suppose you have two points a, b its initial and terminal points for this path tau a is fixed, b is fixed, tau is fixed. Then what happens take a path omega in at a okay pre and post composite with tau and tau inverse. So start with tau inverse, tau inverse will start from b and come to a then you trace omega again you are at a now you go back by tau to b. So you are starting with b and you are ending with b so you get a loop at b. But omega is a loop at a its class will go to the class of tau inverse star omega star tau. Why? Because if omega is homotopic to some omega 1 then pre and post composing by a path tau inverse omega 1 tau will be homotopic to tau inverse omega tau. So this is what we have seen yesterday's class and previous class. So I can call this map as h tau. This is a set theoretic function now on the homotopic classes but I claim that this is a homomorphism. Very easy to prove this one using our earlier information what happens under compositions. Associativity law can be used here insert tau and tau inverse in between because it is homotopic to identity the entire class does not change. So if you use that trait then you can show that h tau is a homomorphism. What is the meaning of h tau is homomorphism? h tau of a class omega 1 into omega 2 star omega 2 must be h tau of omega 1 star h tau of omega 2. How to get this? You have omega 1 star omega 2 you can write omega 1 star tau inverse tau star tau inverse and then omega 2 tau composite tau inverse in between can be introduced because it is no homotopic. Now if you apply h tau tau inverse and tau will come again on both sides. The entire thing you can break it into two groups with brackets that will become h tau of omega 1 into h tau of omega 1. Exactly same way you can see that to take h tau inverse namely the map with respect to tau inverse. Tau inverse will be starting from b to a therefore you will get a map from pi 1 of x b to pi 1 of x a. If you take first h tau and then take h tau inverse that is the same thing as composing with tau star tau inverse on the left as well as on the right. But tau star tau inverse is homotopic to identity constant one therefore it is nothing but identity map. It just means that h tau inverse is same thing as h tau inverse h of tau inverse is the same thing as tau inverse. That means h tau is an isomorphism. So, we have proved that changing base point changes the group by isomorphism. If the group what we are interested in isomorphism class of group then there is no problem. It will be displayed in slightly different way. Isomorphism of a group is a different to copy or isomorphism copy. So, you must understand the groups need not be the same but they may be isomorphism. Like two equilateral triangles of the same sides you draw two of them they are different triangles after all. For example, one may be containing the origin another may be containing some other point 100 100 comma 100 they will be different but as triangles they are isometric. Similarly, group theory it is important to understand that groups may be isomorphic yet they may could be different groups. Right? So, sometimes these difference does cause problems we have to be careful with them and then you would like to see what is that isomorphism which takes that one. So, then the isomorphism becomes important. So, this is not very you know just for imagination it actually happens in algebraic topology itself a little understanding of this is needed much later right now it will not come in our way in this course. Okay? So, this is what the sum of all this remark 2.6. While dealing with a path connected space we often need not mention the base point at which the fundamental group is being taken done. Why? Because they are all isomorphic. If your interest is only knowing the group up to isomorphism it should be noted that the isomorphism itself will depend upon what path you have taken within x joining a to b there may be several paths. For instance, if the two paths are homotopic path homotopic then the isomorphisms will be the same yet they may not be identity isomorphism there is no identity isomorphism between two different groups. Identity isomorphism makes only when the groups are the same but they may save isomorphisms if the two maps are two paths are path homotopic. If they are not then the isomorphisms may be different. Okay? So, this you should keep in mind in this course we will never meet this aspect at all. Let x be a path connected space we will make a definition now the modern definition of simply connectedness. So, the space x is said simply connected if the fundamental group pi 1 of x x naught is the trivial group consisting of one single element. This may happen at one point but then it will happen at all the points because at all other points is isomorphic to the trivial group. So, it is a trivial group. Okay? So, definition is independent of what point you take for a path connected space. If the fundamental group is non-zero at some point it will not be it will not be non-zero at all other points because all of them are isomorphic that is all I want. This simply connected definition is the most useful and the strongest definition you might have come across with many other hand-waving simply connectivity in as when you are doing complex analysis. In complex analysis you can have something like 10 definitions of simply connectivity if you want or even more. But when you come out of that arbitrary spaces most of those definitions will not work at all even if they work some of them they will be quite different than this simply connectivity. This definition is the strongest of them all. Okay? So, going back to examples we again come to the star-shaped regions inside Rn. Take a star-shaped region star-rated point with apex point x0. What does that mean? That means that if you take any say any point any other point in x then the line joining that point and x0 is completely contained in zx. Okay? Using this line segment you can easily show that every loop based at x0 is null-homotopic namely the constant map homotopic constant map which is at x0. Take any alpha okay which is a loop at x0 alpha t can be directly joined to constant loop constant loop x0 namely x0. What is it? t times alpha t sorry s times alpha t plus 1 minus s times x0. So, this line segment makes sense because there is entire line segment is inside x. Okay? So, this we have seen before right? So, I am just repeating this. So, all star-shaped subsets have trivial fundamental growth. So, they are simply connected. In particular every convex subset is simply connected. In particular the whole Rn is simply connected. R, R square they are all simply connected. Any open disk any closed disk they are all simply connected. Even regions inside a ellipse or ellipsoid and so on they are all simply connected because they are all convex subsets. Okay? Now, we will give you some different way of looking at pi1 of x, x0. Look at the map theta e power 2 pi i theta okay defined on the closed interval 0, 1. It is injective except at point 0 and 1 they go to the same point namely the unit vector 1, 1 comma 0 in the complex plane right in R2. That is the property of map e power 2 pi theta. We need this map very much. Okay? So, this means that the end points are identified and all other things are kept as they are I mean 1, 1 mapping. So, when you take the quotient space namely interval wherein end points are identified interval module 0, 1 that is my notation here. So, this will be ohmium or fictor circle S1 by this map e power 2 pi i theta. So, it follows that when you have a loop namely omega from i to x wherein 0 and 1 go to the same point that map will factor from factor through this quotient space i, i by 0, 1 that is 0 and 1 are identified right which is the same thing as having a map from the circle into x. 0, 1 where a specific point here that has gone to the point 1 the unit vector unit complex number 1 in S1. Therefore, every loop can be thought of as a function from S1 to x and its base point being the image of 1. Therefore, instead of looking at the way we have done path is from a closed interval we can take this set namely set of all continuous functions from S1 to x which takes the base point 1 here to the base point x. Convertingly, if you have such a map you can compose it with e power 2 pi i theta and you get a map from interval into x which sends both 0 and 1 to the point a. Therefore, under this identification what you get is a new way of looking at pi 1 of x a which is nothing but homotopy classes of maps from S1 to x wherein the homotopy is taken with respect to the base point this base point whenever moves omega of 1 is a and the entire homotopy is also always fixing this point a under that entire homotopy 1 is fixed at 1 goes to always a. Okay, so this way loops are nothing but images of S1 that is what you have to think about. You fix a base point there so that you know on a circle you could have thought of any point as a base point right it is still a circle. So that freedom is there but you have to fix a point because now that will become base point for arbitrary loops inside x. Okay, now what we have is starting with a pointed space the pointed topological space xa we have given a group. This assignment has what I keep calling factorial properties. So let me repeat what is this functional properties. Suppose you have a map continuous function from x to y such that f of a is b then what happens if you take a loop at a composing with f you get a loop at b. If two loops are homotopic here f of that and f of another they will be homotopic to each other inside y. Therefore a map like this induces a homomorphism by check from pi 1 of xa to pi 1 of yb. The definition is f check of omega is the thing but the class of f composite omega. Because of the associativity of the compositions okay actually it is much more than that f of omega 1 star omega 2 is just the composition how star is defined by concatenation. So everywhere f of omega t are taking so it will be f of omega 1 star f of omega 2. Okay this is not associativity sorry. So this makes it a homomorphism okay f of omega star tau is f of omega star f of tau it is like a distributivity okay. So any continuous function induces a homomorphism of the corresponding groups okay with the additional property that if you have identity map what does identity induce? Induce homomorphism is also identity all right. Suppose you have another map from yb to zc f from x to y g from y to z then I can talk about g composite f from xa to zc right under this g composite f what happens g composite f check will be g composite g g check composite f check is that clear because all that I have to apply is take an omega on this side g composite f check of omega is nothing but g of f of omega right. So this is associativity so g of f of something is g check of f of that so it is g check f check of omega so if f and g are homotopic themselves as maps from xa to yb where the base point a does not move this is extra hypothesis we have to put not just homotopic maps okay they are homotopic maps relative to the base point which we have not defined relative but I am telling you what is the meaning of this this just means that a does not move during the during the homotopy f of a is always b all the homotopies must have f of a equal to b g of a is also b okay f or some other see g here was in in item 2 g was something different here I am taking f and g are homotopic to maps but they are from xa to yb both of them okay then the homomorphisms on the fundamental groups they are the same okay that means f check is equal to g check so these properties are going to play a very crucial role throughout the study of fundamental for instance what we get is suppose two spaces where homomorphic pick up a homomorphism let us call f is a homomorphism from x to y okay take a base point here a call b f of a b is f of a then I have already an isomorphism from pi 1 of xa to pi 1 of xb why because f is a homomorphism f inverse will give you the inverse of the f check f check inverse is the same thing as f inverse check right therefore homomorphic spaces must have isomorphic fundamental group right so if you know if you suppose you take two spaces and compute their fundamental groups and find it that they are not isomorphic then you have solved a big problem the spaces that you have been given are not homomorphic okay so this is the way it is used already in our introduction I have told you how this was used to solve a big problem in topology namely the classification problem cannot be solved okay however we do not know any space yet for which pi 1 is non trivial all our examples were convex sets and star-shaped sets and so on okay we don't know any example where pi 1 of x is non trivial so we will do that by computing pi 1 of the circle in some sense this is the simplest example and actual computation is an illustration of a powerful you know notion that I also we are going to study namely the covering space okay so that we will do in the next module thank you