 Today's topic is one special concept in covering space theory, which is again very classical. We are only touching just the definition and a few simple examples, even the definition may vary from author to author. So, this part we are doing just by examples, not much deeper study of this. The simplest model is when you are studying subgroups of a group, the cassette representation helps you a lot. This is the kind of thing that we want to do in topology also, when a group is acting on a topological space. However, the analogy stops there. We have to bring in more topology than choosing an arbitrarily picking up some cassette representatives. Okay? So, let me stop making comments. Let us first go through the definition and then see whether a few things make sense. Start with a group G acting on a connected topological space. Let me just take for the definiteness that the action is on the right. Then a connected subset D of X is called a fundamental domain. If the following happens, namely X is union of all the translates of D. D is the subset of X. You take all right translates of D, G and G, that must be covering the whole of X. This is similar to the choice of right cassettes. But there is no disjointness here. It is only union. The second part brings a little bit of disjointness. Namely, for any X interior of D, XG is in D implies G must be identity. In other words, the translates of interior of D, they are disjoint. Okay? The third point is that if you restrict the entire map, the quotient map from X to X by G, restrict it to the domain D, that itself must be a quotient map. So now you can see what is the idea. The idea is to cut down the top space X to something manageable, something smaller. All right? If we insist on percept representations like they are all disjoint, that is not possible because X itself is connected and that is not desirable also. So we allow minimal inter overlapping. Namely, in the interior, there should not be any overlapping. So the boundary there can be overlapping and that happens. The important thing is that since X is connected, we insist that D is connected. So now I want to tell you that the definitions may slightly vary. For example, if X is X by G is compact, then you may want to choose D to be compact. Secondly, there is no uniqueness in the choice of D. Each person depending upon the problem at hand, whatever you want to study, you may choose the fundamental domain differently. Okay? So let us just study a few examples how it helps to understand the quotient space and the action of action of G on X. So here is the picture which shows that if X is in the boundary, it translates may be also in the boundary, may be. If Y is in the interior, then its translate will not be in the interior. It will not even intersect. This is D. It will be outside of it. This should happen for all G not equal to identity. So this is the picture. That is all. Okay? So I have already told you condition one above tells us that the quotient map is surjective because translates of this D cover the whole of X. So there is a representative. It is like a positive representative. Condition two tells you that D to X by G is injective in the interior. Okay? And the condition three says that it is actually quotient map. It should take the whole thing. Alright? So let us take an example, the simplest example. All these examples are more or less familiar to you. The first example is the exponential function from R to S1 given by the action of the integers on R by translation R is a subgroup of R. So it is like a coset representative. Okay? So what you want to do is you can take any closed interval which we have been doing that any closed interval of length one then in the interior there will not be any identification. When you translate any interior point it will go outside the interval of length one but the boundary for a zero will go to when you add one it will go to one. So that is the only point of intersection between the translates of the interval close interval and itself. Okay? Either way, either way add of a track only one of them that intersect. After that there are no intersections at all. This interval close interval this happens to be connected that justifies the domain, the word domain here. So it is called a fundamental domain. But in this particular case it is also compact. Okay? We did not bargain for that compactness but because the quotient space which is the circle is compact this was possible. Obviously it is the fundamental domain itself is compact. The quotient which is a image of that will have to be compact. Okay? This is actually was this remark was used in proving that the projective space is compact. Let us come to that example then I will explain it again. Okay? So here is a comment says that most of the very interesting examples come similar to this Z contained inside R namely what are called as Lie groups and then inside that Lie group you are taking a discrete subgroup. Okay? When you take discrete subgroup inside a Lie group the quotient becomes a covering space projection and then you can talk about choosing a fundamental okay? To study what is happening to the action as well as the quotient space and so on. Alright? So this is the prototype of that example since we have not studied or we are not assuming any knowledge of the Lie groups and so on we cannot pursue that angle more than that. But we can take another simple example simpler than the exponential function namely from S1 to S1 or C star to C to C star or C star to C star itself. Okay? C star to C star namely that is going to zeta zeta where zeta is a primitive nth root of unity. If this nth root of if n is 2 then this is just an anti-polar action. If n is 3 you are multiplying by omega, omega square and so on so this is a group of order 3. Right? We have studied on S1 itself you can take the primitive nth root of unity what you get as a quotient space is again S1. Same thing happens in C star also the polar coordinate representative the norm of the vector does not get affected because zeta is of unit vector, is of unit length. Okay? So what will be the fundamental domain for C star? You will have to choose a sector, sector take any line say for example the positive real axis and take another line passing through origin which makes an angle 2 pi by n and everything lying in between that is called a sector all theta between 0 to you know e power 2 pi i theta where theta is between 0 to 2 pi by n. So that forms a fundamental domain. In the interior there will not be any identification but on the boundary the whole line the x axis part of the x axis positive x axis is turned into the next line. Okay? You keep turning it n times you will come back to the real axis. Next example which is little more interesting or little more complicated is start with a two dimensional vector space r2 pick up any two vectors as basis for r2. Okay? Any two vectors as basis for r2. Now you take the subgroup generated by these two vectors. Abelian subgroup not the vector space they generate the vector space r2 but I think subgroup generated by two that will be a free Abelian group of rank two. So I am writing it as a square namely all elements which look like m times u plus n times v are m and n are integers. Okay? So that happens to be a discrete subgroup. Okay? You can mark these things on starting with u and v make a parallelogram. So you get four vertices of the parallelogram keep translating this parallelogram both up and down right and left and so on. So you get all those lattice points. Okay? So that is the subgroup. Caution is now is just the quotient of one of these parallelograms. Any one of the parallel, close the parallelogram is good enough to cover the entire thing. Okay? And the quotient is again a covering projection. Indeed the quotient space is nothing but omomorphic to S1 cross S1 again just like if you have taken the u and v to be a special case as they you know e1 and e2 two perpendicular vectors. Okay? The importance of this one is that no matter what you what your choice of u and v that they must be independent and that is all. The quotient space always S1 cross S1. S1 cross S1 it is homomorphic to the torus. But remember that R2 can be thought of as a complex plane with complex structure. Then there is a way to give complex structure to the torus which will depend on what vectors for generating vectors you have taken. Indeed it will depend upon just the angle depending on the angle choice of the angle you will get different complex structures on the torus. They are all called elliptic curves. In fact they are all smooth elliptic curves. They are all of them, smooth elliptic curves. This is a very classical subject and extremely important in other areas of mathematics also like this was used in solving a formal asterium also. And this is classical in sense it goes back to Weistra's and even Abel and so on. The study of the elliptic curves. And it goes back to Riemann also and you know this is these are examples of Riemann's services. Not the first one. The first one is a sphere. The next set of examples, the simplest examples are these torii with a complex structure. So beyond that I cannot touch this one. This may be just a motivation to study these things because the study of elliptic curves itself is a very, very, very deep subject. One can study the whole thing for the entire of now one's life. But we can now from S1 to S1 cross S1 we generalize. We can immediately generalize it to all dimension S1 cross S1 cross S1. All that you have to do is take n vectors in Rn as a basis vectors. Any basis that will give you as a group of Rn which is of you know Abel is a group is of rank n. That will be a discrete subgroup. The quotient will be again a compact space and that is nothing but S1 cross S1 cross S1. Now what will be the fundamental domain here? Once again you look at the box. The box given by all these 0, v1, v2, v3, v4, etc. your n vectors and like a parallelogram you generate the box parallel of it. So that will become a fundamental domain. In dimension 3 it will be which is like a cube. If you take vectors to be perpendicular then they will be cube. The more familiar example which you have studied already is projective space. Projective space we have defined first as quotient of Rn minus 0. But then you can choose S2 such that the projection from S2 to that one is also surjection. Now even S2 is not a fundamental domain because there are points interior which may be related. So all that you have to do is cut it down in half, take only the upper hemisphere then in interior there is no identification. On the boundary x going to minus x is still identified. That is all. So this was used in understanding the projective space inductively. I have discussed this one earlier. For example in the case of n equal to 1 this will immediately tell you that the projective space p1 is again homomorphic to S1 because then the fundamental domain is just an arc the upper hemisphere circle and 1 and minus 1 are identified. When you identify a string the end points are identified by a single point what is that circle. And that will give you picture of S1. It will also give you picture for p2. p2 is now nothing but a disc being attached to the circle. Why is it going to z square? Namely 1 and minus 1 on the boundary are identified. So that is the picture of pn. Unfortunately p2 unfortunately it is easy to describe that but you cannot construct a model with thin remaining inside r3 because p2 is not immediately in r3. The next example is of more interest to topologies. It is called Klein bottle. It is again two dimensional thing. So I will explain this one because it seems that many people have wrong conception of what is Klein bottle. Consider the subgroup G of rigid motions in R2. What is a rigid motion? Which preserve the distance in R2? Translations, rotations and such things. The rigid motions in R2 generated by following two elements. One is just a translation along the X axis. X goes to X plus 1, Y remains the same. The second one first the first variable is reflected in the line in the point 1 by 2. So it is X goes to 1 minus X. The second coordinate shifts Y goes to Y plus half. So this is called a glide reflection gliding along the Y axis, reflecting along the X axis but not exactly X axis but here X equal to 1 by 2. So such things are called glide reflections in complex analysis. It is a rigid motion. It can be thought of as two composite of two things namely one reflection and then and translation. Reflection is also a rigid motion after all. Only thing is it is not orientation preserved. So such thing is a glide reflection. So look at just square of this, R square of this 1 minus X, 1 minus 1 minus X is X. Y plus half plus half will be Y plus 1. So which will become a translation along the Y axis. Hence if you take T, T is the translation along the X axis, R2 square along the Y axis that will be just like your standard Z2 inside R2. Generated by 1 comma 1 right as basis I mean E1 and E2 that is what it is. So generate a free abelian group of rank 2 in R2 which we have studied above. So quotient would have been S1 cross S1. But now I am going to take subgroup generated by T and R. T and R square will be a subgroup of that. So now I am going to quotient R2 by a larger group. A larger group, a group which is Z2 happens to be a subgroup of index 2 inside there. So it follows that H is a subgroup of index 2 in this G, H is Z direction. Check that the action of G on R2 is E1. So this is fairly easy. All that you have to do is take the neighborhood of any point say 00 for example and take small enough neighborhood, very small enough neighborhood such that either reflection or shifting does not intersect with theta that is all. So details I will leave to you. So check that the action is indeed I have written already. So you can 01 cross 0 half is a fundamental domain. So rectangle of length 1 and half along the y axis. So that will give a fundamental domain for this action. So here is a fundamental domain for this one. This is a fundamental domain for the torus action. This is for what we are going to get a Klein bottom. The identification you get here x axis is shifted along the y axis. What happens when you go shifted? Anyway this is torus action that is clear. Namely this way I shift, this way I shift. X axis is shifting like this, y axis is pushing this way. Halfway through you are rotating this way. So what you have to do is when you come here this will be 1 minus x. See this is half, half will go to half but 0 would have gone to 1. So say 1 by 1 third would have gone to 2 third and so on. It is coming this way. So that is why I am coming this way and I am going this way and coming this way. Of course remaining here it would have been different but I am shifting also. So a point here would have gone to point here and point here would have gone point here. On the line y could x equal to half this will be just shifting from here to here. So action is like this and this is shifted like this. So identification is precisely this line segment is identified this one after rotating not just like that. This one is identified x as it is. You can perform this one first when you get a cylinder. While you performing this one you cannot just bend it down and glue it like this. You have to bend it and go inside and glue it like this and that is why you cannot perform this one remaining inside R3. So this is a Klein model. One knows that Klein model cannot be made in R3. You will learn this in an advanced algebraic topology course that it cannot be made in R3. It is not a part of this course. You cannot handle that one. Putting theta equal to 2 pi t psi 2 pi s consider from R2 to R4. In R4 I am giving you a embedding of R2. I am defining R2 to R4 this map. It is of see that is 2 pi t there must be a comma here psi equal to 2 pi s. I am putting theta equal to 2 pi t psi equal to 2 pi s. Then f of t s is 2 plus cos theta which means 2 plus cos 2 pi t is writing 2 pi etcetera becomes 2 bits that is why I have written put that one. And then cos 2 psi that means this s is replaced by 2 pi of that which means 4 pi s. So write like this. You take sin theta cos theta and so on. What you get is in theta it is of period 2 pi and in psi it is a period of 4 pi. So you will get an action, you will get a map which respects the torus which respects the action of t and g here the group g whatever I have written down here. So it goes down quotient and defines a quotient from the Klein bottle inside R4. That map you have to show that it is injective okay. Once you show that it is injective that is enough because one can show that the quotient is already a covering projection and quotient is a compact space. This is a horse torque space so automatically it will inhibit. So this is elementary checking but many people make mistakes here okay elementary mistakes. So I have written down this one. So please check that I have not made a mistake that is all okay. Rest of the thing that I have put here we will discuss it some other time. We are all counter examples here. So they do not play much role in the theory that we are developing but they are good for understanding what is going on. So we will cover it in some other time. Thank you.