 Hello, welcome to module 61 of NPTEL NOC and introductory course on point theta poly part 2. So, in this chapter we will now study two dimension manifolds for short we call them surfaces. Later on you may restrict the word surface for a suitable class of submanifolds of dimension to also. So, this chapter is going to be not a very rigorous one, but intended to be a exposition for things to come like motivating you people to study algebraic topology and differential topology and so on. So, manifolds in general So, let us begin the study of surfaces with a good old technique of paper models which everybody uses. So, I begin with an example here which you are familiar with namely in general in mathematics we call a cylinder to be any space which looks like X cross an interval ok, modeled on the most familiar object namely when X equal to a circle. When X is a circle, circle cross an interval is a typical model for the cylinder that you have studied even in your 10 standard and so on all familiar with finishes and so on ok. So, the surface that we have here is you know not general X cross J, but some curve cross interval. So, that is the most general cylinder as such, but if you want real word cylinder let us stick to the classical notion when X itself is a circle. When you say circle it is only up to homeomorphism therefore, you can take ellipse also no problem ok. So, when you take circle cross J, J is an interval its boundary will be circle cross boundary of J ok at two different ends. For example, if J is just 0 cross 0 to 1 the closed interval then C cross 0 and C cross 1 are the boundaries ok. So, all these things can be verified. The point is I want to use a paper model for this one namely I start with a rectangular piece which is representative of you know the other way around now. We want the topology namely 0 1 cross 0 1 which represents the paper, but we were looking at modeling. So, you have to take a piece of paper which represents 0 1 cross 0 1 the plane topological space ok. And then what we are going to do to produce one factor as circle on the one factor namely 0 comma S will be identified with 1 comma S ok, where S C is varying over the second factor 0 comma 1. So, on this first factor the end points are identified 0 and 1. So, that I have to do for every S inside the second factor namely 0 comma 1 here ok. So, that is the identification. So, very straight forward identification which gives you a shield. These much you already know. On the other hand while identifying if I just turn around the two sides namely here I have this 0 cross the entire 0 cross 0 1 here. The second one is 1 cross 0 1 instead of S going to S I twist around namely S going to 1 minus S ok. So, 0 comma S is identifying with 1 comma 1 minus S then what I get is the so called Mobius band ok. We have you must have studied or if you have not you should not do that namely you know actually work it out and see that they are quite different objects whatever you want to say is quite different object they are you know even in the layman's language. So, what are the different topological aspects of them that itself is an interesting study. So, this you must have done if you have not done it we can do it at some time there is no problem. So, here is the way the you know this is the cylinder and the Mobius band are represented diagrammatically. So, take this model here namely this is 0 1 cross 0 1 there is no price for keeping the length the same there is no need and then you identify this edge with this edge as the arrow is indicated the arrow is indicating how you are going from one to the other here the arrows are reversed you see this is going to give you the Mobius band. Of course, when you actually identify the this one will give you the annulus which is equivalent to a cylinder and that will give you some surface like this where it will be twist here which is not a part of the plane here I have deliberately shown it as subspace of the plane itself which is possible for a cylinder which is same thing as an annular but this cannot be there will be some crossing over. So, this will actually hang around in the third dimension. So, that is a Mobius band alright. So, these are the paper schemes. So, you just indicate this way you have to understand that these two edges are identified. So, that is the meaning of paper scheme right. So, with a piece of paper you can actually perform this also. So, that is the one idea. But there are other things also which you have studied in your part one namely the torus and the Klein bottle. The torus is got by in the first model here namely again the same piece of paper rectangle piece of paper the opposite sides here are identified this and this the same orientation like that here also same orientation. Whereas, in the Klein bottle one pair is identified the same orientation whereas, the other one there is a twist the orientation is reversed here. So, this is this will give you Klein bottle this will give you the torus. Now, if you use a paper you cannot actually perform these torus forget about Klein bottle Klein bottle even with a rubber sheet you cannot perform inside R3. But this one if you have a rubber sheet you can perform okay like a cycle tube. But if you have a paper of course one side you can identify these two or you can identify these two to get a cylinder. Once you have got a cylinder you cannot bend it to identify the two circles there because there is some rigidity with the paper also you see it will crumple right. It will not be represent a embedded object as you represent. So, for that you will have to do different tricks. But so finally what I want to say you know the actual the so called experiments that I am going to perform here they are partly experiments we can perform rest of them we have to remain only thought experiment okay. So, that is what you know thanks to the word you know this thought experiment is ever introduced by Einstein I think. So, this is what you can do finally alright. Now, let me recall that you have also defined of course the standard sphere S2 there is no need to define we have already verified that it is a surface right or Sn contained inside Rn plus 1 they are all n dimension manifold that we have seen. But now I am going to give you I am going to recall that you have also defined the projective spaces. In particular the projective space of dimension 2 P2 which can be obtained as a quotient of S2 by taking the action namely x going to minus x that is the Z2 action and go modular to action means identify x with minus x for each x. So, that is that will give you the the projective space let we can use this notation Q from S2 to P2 to denote the quotient map also we can use this square bracket to denote the equivalence class of x where x is inside S2 the square x the bracket x will be inside P2 right. Note that if you take all points x1 x2 x3 inside x2 namely summation xi square equal to 1 suppose you put x1 is positive or x2 is positive or x3 is positive and so on. So, those are three coordinate neighborhoods okay each of them will map homomorphically on to P2 because the moment one of the coordinate is positive when you take the action it will become negative that coordinate. So, those two will be disjoint so only one of them will be there so there is no identification inside that. So, that will project identically you know homomorphically on to an open subset of P2 and everything x1 x2 x3 okay once you start with that one you can check the one of the signs and make it positive so it will be covered. So, these three things these three open subsets cover P2 entirely and they are each of them homomorphic to an open subset of R2 okay because they are now part of the open subsets in S2 they are actually disks inside S2. So, they are homomorphic to some disks inside R2 okay therefore P2 is a Euclidean space is a locally Euclidean space also since S2 is compact P2 will be also compact because the quotient right image of a compact set under a continuous map is a compact that is also compact. So, compact and well you can immediately see that it is actually second countable also no problem but what is important here is to see that P2 is a hostile space then it will be a manifold right. So, for hostile there is a general thing you have studied under group action namely if you have finite group action on a on any hostile space the action is fixed at point free and so on then the quotient is automatically a hostile space in particular this action is just just to action x going to minus x you can immediately see that given any point x okay and minus x alright. So, you can separate them by this method now if you have 2 x comma y you can take the distance between x and y take half the distance and take balls of radius smaller than that right and with this property if all the all the coordinates are either positive or negative depending upon the starting point then you will get 4 different balls like this x minus x y minus y they are all disjoint. So, when you go down 2 of them will coincide 2 of them will get identified 2 of them will get identified. So, they will give you neighborhood disjoint for bracket x and bracket y. So, that will show that this P2 is actually half dog okay so you can directly verify it I have quoted a theorem also whole theorem that you have done. So, this is a very nice example of a somewhat non-trivial 2 dimensional manifold okay. So, remember this client bottle is even actually more complicated than the projective space the torus is simpler okay you can see the image of them inside R3 and so on. All of them are manifolds without boundary whereas this these 2 cylinder and this one which are much simpler objects in some sense they have boundary this has 2 boundary components this has only one boundary component which is obviously a circle okay. So, far all these examples are all familiar objects to us the whole idea is to get to the theory slowly through the example. So, do not hesitate if you have any questions you can raise right now. So, I do not want to state much theorems here but try to give you glimpses what are the things happening here okay. So, I have given you explicitly how to get the host torus and so on. So, schematically even S2 and P2 can be represented by some instead of rectangles you can just take some disk and identify some portions of the boundary. So, that is what I want to tell you here okay. Check that the coefficient map Q from S2 to P2 if you restrict it to just the upper half sphere all x1, x2, x3 so that x3 is greater than or equal to 0 okay. So, let us call this U3 bar instead of positive greater or equal to 0 then the entire thing below is covered because everything else is minus of that so you do not need that right minus of some point here we do not need that. So, if you just take U3 bar and then to restrict the quotient map that is such a quotient map therefore you can think of a P2 as a quotient of the upper hemisphere. Now the identification is occurring only on the boundary which is a circle contained inside R2 cross 0 right. So, what is the identification? Identification again x going to minus x right but it is performed only on the boundary okay. So, that is what you have got okay that means what you have disk the upper hemisphere is just like a disk and on the boundary you have an identification which I have shown like this the point x as it moves minus x will move on this side like this okay if x is here minus x will be here x is minus x will be here. So, this is the circle and on the boundary you have two arrows the boundary is divided into two parts this arrow this entire R is identified with this R by a homeomorphism namely x goes to minus x here. So, that will produce the projective space not very easy you cannot perform that. On the other hand look at this picture here again it is a circle and you have a disk here identification is taking place like this as point moves like this its image will be moving like that that is the homeomorphism what is it it is just z going to that bar you can say or just x y going to x minus y not minus x minus y but x minus y sorry here in this picture it will be x y going to minus x y so x coordinate is getting interchange this matter. So, this is another paper model we can say for the sphere and this is for the projective space right. So, everything is now so I would like to again this if you take a paper it is not even if it is difficult to imagine what you can do. So, this is just like a lady's purse which you know snap purse just close it up like that right. So, if something like this you have seen the purse like that then you close it up like that. So, it that will give you a sphere representative of the sphere if you identify like this we have seen you know this we have seen logically you cannot see it actually perform this is the projective space. There are deeper reasons the projective space cannot be embedded inside R3. So, you cannot have a model representing a projective space inside R3. So, motivated by this experiment in a simple situation in this simple situation we now consider the following general process for obtaining compact surfaces with or without boundary. Anything which you want to perform it has to be the finite process. So, compactness creeps in. So, you start with a piece of paper. So, it is compact. So, everything is compact now. Compact with or without boundary both of them we will consider. The mobius band and cylinder etcetera are easy examples right. Let us not exclude them let them be there okay. So, begin with a two disc that itself is a two dimensional surface okay it is a disc. So, it has a boundary. So, boundary is non-empty. So, that is the starting point very nice. Now, what we want it any homeomer will copy we will also call it a disc okay. It need not be inside R2 of course. When you bend it and so on it will be at least inside R3 okay you do not have to go beyond that. So, let us fix an orientation on the boundary once for all. The boundary is a one single circle. So, for definiteness let us say counter clockwise sense that the standard orientation okay. Next you select the finite number of points at least two of them. So, that the entire circle is cut off you know cut out you do not actually cut it but you just imagine you can just take. So, like if you have an interval cut it into any parts means what you are taking a partition right T0 less than T1 less than T2 and so on. So, that less than less than less than is not possible in the circle but orientation will take place of this less than less than less than okay. You can take points say Z1, Z2, Z3 and so on then you can call label them in a counter clockwise sense. So, these clockwise sense this labeling is unique up to a cyclic permutation. Why? Because you can start from any point and then you have to end before that guess when you had again that point do not repeat that is all. So, that is the way you label the points right points which you have chosen on the circle okay. Let us call them as vertices and then the resulting arcs there we will call them edges just for convenience okay. Now label these arcs not the letter with letter not vertices with letter and write them sequentially in a counter clockwise sense that is what I am saying okay. So, that is because you may start at any one of these while labeling then you will get a different labeling but that labeling will be just differ by a cyclic permutation that is all. So, you are allowed now in the labeling you are same letter you can use at least at most twice okay at most twice you can repeat it once more that is all. So, a, b, c, d and so on but the next one may be again a next one may be b now you should never use a and b okay you have to use different letters and so on that is the way. So, at most twice so next look at a letter x which occurs twice in your labeling you have done it okay nobody else has told you. So, look at it and you have a freedom to put the script plus 1 or minus 1 okay plus 1 is as if you do not write like 5 when you write it is understood that is plus 5 right but whereas minus 5 you write it similarly do not write the script minus 1 plus 1 okay if it is minus then you write it as minus 1 such as instead of just writing x the second letter you may or may not okay you may write it as x inverse or you may write it as x also I will tell you why. So, these are the things that you are permitted to do okay. So, when it occurs second time you can do it either you either you change the letter to x inverse or you leave it as x that is all okay. So, here is an example one sequence consisting of two letters a, a inverse another one just letter 2, a, a the third one is a, c, a inverse d the fourth one here a, c, a, d. So, here I have not I have repeated but I have not changed the sign here right. So, the exponent is not changed similarly a, b, a inverse b inverse a, b, a, b inverse. Now, I have 1, 2, 3, 4, 5, 6 of them right here right none of them is just imaginary things they all have occurred right now let me show you. So, here see this is a and that is coming this way. So, it is a you know here it is a and it is repeated again a why I am repeating because I am going to identify this that is the whole purpose of repeating a letter okay. Let us look at the previous one look at this one this may be we can call it as a I am not going to identify this one with this one. So, that will be but this one I am going to identify. So, I will put this one as a but it is in the opposite direction now you see counter clockwise you have to follow but this arrow is in the opposite. So, I will put a a, b, a inverse and then what is this one this is b inverse. So, this is the torus here a, b, a, b inverse. So, both of them have occurred there right. So, even these things have occurred a, b, a inverse what is this one this is some other letter this is not identified. So, a, b, a inverse c this you can say a is the other way around or maybe you can start this way whatever a inverse okay then b then what is this one this is a inverse this is all coming. So, this a inverse if this is a so better call this as a b again a this one d no identification. So, all these things I have listed here the 6 of them okay. So, they are my starting examples easy examples a, b this one this one I have redrawn them I do not need now any of these rectangle and shape and so on one single shape will do namely the disk itself. So, all of them are drawn on the disk now. So, this is first one a, a, inverse this will give you a sphere this one a, a, inverse will you project your space sorry a, a this one this is a, a, inverse this will give you a, c, a, inverse d this will give you what a, a, inverse b identified this will give you cylinder this one will give you Mobius band a, c, a, d. So, c and d are left out but together what happens after identification they will give you one single circle here these two identified will give two different circles on the boundary. So, this is now the torus and that is the client model schematic representation of 6 of the surfaces that we can access in some sense easily okay. Now, let me try to generalize this kind of thing this is the preparation needed to perform the next step edge identification models, paper or scheme models I have just chosen some scheme what I am going to do I am going to perform some identification how namely follow these instruction whatever you have whenever edge is repeated identify those two edges how do I identify depending upon the exponent a, a in the same way orientation you have to fix right in the beginning okay a, if a identify in that way a, inverse identify other way so this is what you have to do okay. So, identify means what you have to use homeomorphism now this is where you have to tell that only orientation preserving or orientation reversing matter okay there are many homeomorphism okay among us the orientation preserving some of them orientation reversing some of them homeomorphism from one interval to another close interval to another close interval. So, the only thing that matters is whether torus is preserving orientation reversing okay some edges are never identified in the scheme because they are not repeated so those things are called free edges so what will happen to them when you perform all the identification those free edges will remain free so they will be the boundary they will be the boundary part this is what we want to see now okay let us see so it does not take much effort to see that the homeomorphism type of the quotient space only depends upon the isotope class of the homeomorphism used in the identification process so this is what I meant by those exercises 12.42, 43, 44, 45 whatever I hope you had some time to spend it even if you have not solved them what is the meaning of isotope etc you must have understood by now okay since there are exactly two isotope classes of homeomorphism of edges you can call them orientation preserving and orientation reversing that there are only two classes like that they have been encoded in the rubber sheet scheme by pivot by putting an exponent okay so it follows that each rubber sheet scheme defined defines a unique quotient space so there is no ambiguity in the definition of a quotient space obtained by using a scheme see in other words for all matters the scheme represents the entire topology of the quotient space so this is the underlying principle here I am not proving any of these statements here by the way huh okay it does not take much effort to see that the homeomorphism type of quotient space only depends upon the isotope class so I repeat just that but I do not prove it now why is the quotient obtained a two manifold that is more serious that we should try to understand okay manifold with or without boundary so I already told you the role of free edges free edges at the most the two end points may get identified because of the corresponding edges there are many edges right at each point there are two edges even if this is not identified the other one may be identified with something so in that case the end points may get identified okay then it will become a circle okay you can see that the points the point one side is there the surface on the other side there is nothing so you can see that it is actually a boundary point okay we will we will make it more clear so clearly since there is no identification at an interior point of the entire disk take the interior point there is no identification so in a small neighborhood that neighborhood will go injectively homeomorphic to a neighborhood of the corresponding point in the quotient space therefore at all those points our quotient space is locally Euclidean of dimension 2 okay so now move to a point on the interior of an edge like this this is an edge but do not look at the vertices look at an interior point of the edge okay at that point what happens it is possible that this a is repeated elsewhere in the diagram that means this edge and this edge are getting identified what happens this half disk neighborhood here and half disk neighborhood here okay they will get identified only along the edge to give you a full disk in the quotient space the image of this point and this point they are identified to a single point that point becomes an interior point with an open neighborhood homeomorphic to an open disk inside R2 so therefore these points are also at these points also we have got the local Euclidean there what is left out image of all these vertices that is not very easy to see that when you take the image of all these vertices okay a neighborhood of the quotient in the quotient space the neighborhood is homeomorphic to again a disk or it may be half disk where this point is not identified with anything else so therefore this is not identified with anything else it can be just half disk here this entire thing like this okay so that is possible but these are the only cases is what you have to see I will skip the proof for that but I will assure you that you do not have to put any more conditions okay my paper scheme is well defined it will always produce a manifold with boundary provided there are free edges otherwise it will produce a manifold with without boundary sorry all right so I want to tell you that do not perform any extra identifications of vertices on their own you know you are not supposed to just identify these two edges after doing all or before the identification of edges will come because I am identifying for example this edge with this edge this one of the vertex will be identified with the one of the vertex here automatically that is all because identification is taken place on the closed edges not on open edges okay so that is the point here now I want to state something and this is a big big theorem actually now immediately a classical name for a rubber sheet scheme in which all edges are repeated is called a canonical polygon everything is repeated I mean repeated only once right so a a a a a inverse c c inverse or something like that no free edges are there that is the meaning of it such a thing you will be called canonical polygon I have not much use for this this terminology but I will use it because I am want to want to state something among this canonical polygon there are certain some of them which are called reduced canonical polygon I will list them and the beauty is that list is so complete that it will represent all the compact surfaces compact connected surfaces with or without boundary okay sorry without boundary because there are no free edges okay the floating free edges are all removed so only repeated things are there and I am going to list them the first one is a one single in this list there is only one one element it is a very typical element it is like it is like you have zero and so on right natural numbers and zero is something different so a a inverse you know what it is we have already done that a a inverse represents the sphere okay then there is a there is a infinite list here g inside n indexed by natural numbers a 1 b 1 a 1 inverse b 1 inverse a 2 b 2 a 2 inverse b 2 inverse dot dot dot a g b g a g inverse b g inverse g inside n there is a definite purpose denoting this number g by g okay this g is short for genus okay so that is a standard name for this one all these things are going to be tori of genus g suppose I stop at 1 then I know that this is nothing but our standard torus so that torus is called torus with genus 1 okay you can call this one the first one as torus with genus 0 but nobody used that kind of terminology so these are all what are called as closed surfaces that means they are compact and without boundary okay and they are oriented orientable okay so this is this list gives you all of them as g varies 1 2 3 4 indexed by the integer positive integer over natural number over the third one is again an infinite sequence again indexed by natural number but here I will use ordinary n here this is much simpler a 1 a 1 a 2 a 2 a n a n don't write a 1 square a 2 square that will be a funny straight funny representation so these are sequence these are not algebra here a 1 a 1 a 2 a 2 a n a n okay so as n varies you know the first one a 1 a 1 is nothing but the projective space okay so you may wonder where is the Klein bottle it is neither here nor here no the story is that the Klein bottle and all those non orientable surfaces are all hidden here so a 1 a 1 a 2 a 2 will represent the Klein bottle you understand so in general what is happening if you don't use reduced canonical over here several paper model several rubber sheet model may represent the same surface so here the list is shortened and now the the claim is that every member here is a distinct member up to actually homotopy type actually homomorphism type actually defumomorphism type so that is the strongest theorem here two of them will be different if they have been given different number here first of all list a b c they are they themselves will represent different homotopy types homotopy type is the strongest now if two things are not homotopy to each other they will not be homomorphic to each other if they are not homomorphic teacher they will not be defumomorphic to each other okay so this classification is very strong classification okay up to homotopy type they are different all right so that is the meaning of this one so let us stop here today I will tell you a little more about how this these things actually to arrive and what how do they they look like how the Klein bottle is hidden here so those things I will try to tell you next time thank you