 Welcome to module 28. We now introduce quotient spaces. Last time we tried to understand what is the meaning of a quotient set in the three different ways. Just giving a surjective function on a given set to another set is one way. Defining an equivalence relation is another way and writing X as a disjoint union of non-empty open sets is the third way. So all these things are equivalent that is what we have seen. One of the important sources of these quotients sets, even at set theoretic level is what is called as what is known as group actions. Group action on a set gives you certain kind of decomposition of a set and that happens to be a big source, an important source, an interesting source of quotient sets. So let us recall this notion, maybe come under group actions while studying groups itself. So let us assume that you know what is the meaning of a group and so on. Let G be any group and X be any set. We have a map from G cross X to X. It is not a binary operation in the sense that it is not X cross X to X now. Neither G cross G to G, it is G cross X to X is mixture of this one. So G is acting on X. It is going to be defined via this map and this map is just, it is not F or G or something, it is standard notation. It is the CERC. Usually dot or CERC is used for writing compositions of you know, multiplications, multiplication inside a group and so on and that is why this is coming here. So what we will do is we will have a manageable and easy to understand notation. The CERC operating upon G, X we will write it as G CERC X. When even this CERC is too much, we will not just write anything, we may write it as G H also, G X also. Just like when we write multiplication of two real numbers or two complex numbers. So that will be the next stage of simplification. Right now we will keep the CERC. So G of, CERC of G comma X is written as G CERC X. So this operation, whatever you want to say action and so on has following two fundamental properties. The first one is the identity. Identity element E is an identity of G. Identity operates identically. E of X is X itself for all X where G is the identity element. The second one is the associativity. G H operating upon X is same thing as first G actually layer last G, G operating upon H of X, H operating upon X. So G CERC H CERC X is the G H of the bracket can be put in the first place or the other way around. So that is the associativity. So this must be true for all G and H inside G and all X inside X. It is all. As soon as you have such a map defined, you can see that fix a G, look at X going to G X, that becomes a bijection of the set X. Why? Because the inverse of this bijection, inverse of this function is nothing but X going to G inverse X. G is the element of the group. So it has G inverse. So G inverse operating upon G is the same thing as G G inverse which is identity is this one. So only these two operations are used. So you see that each G action of G, that is what it is called action of G sends elements of X here and there but in a bijective fashion that it means a permutation. So each element of the group G can be thought of as a permutation. Then associativity again says that this association from X G to the permutation groups that itself is a homomorphism. First G, first G H is same thing as first H and then G. So that is the meaning of this one. So what you get is, so first of all what you get is this one, rho G if I write just for fun, rho G of X is G X, that means it is action of G. So rho G is a now function from X to X and it is an invertebrate. So it is an element of the permutation groups sigma X. That rho is the association here G to permutation group that itself becomes a homomorphism. Identity elements, rho of identity, what is it? It is E, E operative upon X. So it is X itself, so it is identity map, identity permutation. And rho G inverse is why it is, these are all permutations. So here the decomposition is what I was talking. If precisely for each X, let us have this notation G X, what is called as orbit of, the word orbit is used as if these are satellites moving around, you see that is the whole idea. So G of X where G ranges, image of the point X other time X, G X, G 1 X, G 2 X, G 3 X, look at all those. So that is called the orbit of X called G X. The point is if you take another Y, either G Y will be the whole of G X itself or it will never intersect. So that is obvious because of the group G is there and identity of X is X itself. So that is why it forms a partition of X. It is easy to verify G X intersection G Y is non-empty means that element can be taken to span G X as well as G Y. So they are equal. If it is non-empty, that is the whole thing. That means it is partition. This is exactly the way you would have taken the subgroups and when you have a subgroup, it is cosets. Right cosets are all disjoint. Similarly left cosets are distant. So these are like cosets here. This notation can be used as the G cosets of X. So nobody says that there is a different word here. They are orbits of X. So clearly this forms a partition and we have a quotient function put Y equal to the set of all partitions G X along to X. X goes to G X is the subjective function. So it is a quotient function. The partition is simply as giving quotient function or you can also say that X1 is related to X2 if and only if there is a G which will push X1 to X2, G of X1 equal to X2. This becomes a equivalence relation and these three concepts are the same. So you can use whichever one you like as pointed out last time for more general things. Now here is a special case arising out of the action of G on X. The orbit space is another name for this decomposition space, set of all orbits of the action. So that is I have denoted by Y temporarily because this is X. It may be something else. It may be R. It may be R2. It may be R2 minus 0. Various examples are there. So at least some examples we will have to discuss. So for each fixed G we have seen that the rho G namely action of G on X is a permutation. Conversely, suppose you start with a homomorphism from the group G to the set of permutations, the group of permutations. Suppose this is a homomorphism. Then I will define a SIRC operation corresponding to tau. That is why I have written as SIRC tau from G cross X to X by just like whatever I have got here just like rho G. So this is going to be tau G. So tau G of X will give you the action namely G SIRC tau X. So this is the formula for the action. Now if you take the corresponding rho here, it will be nothing but tau. For each starting with the action, I have defined a group homomorph in a rho. Similarly, starting with a group homomorph, I can define the action and this will be a bijective correspondence, one to one correspondence. Thus when you want to mention an action, you can just mention a group homomorphism from G to sigma X. For instance, suppose this homomorphism is trivial homomorphism. Trivial homomorphism is genuine homomorphism. It is allowed. What is the meaning of this? G of X is equal to X for all X, for all G. So this is called trivial action, no action. So it is inert action, inert here. So that is also allowed. Depending upon various properties of the homomorphism, there are various classes of actions, various types of actions. All that we will discuss whenever the situation arises. We just do not want to get stuck with group actions. This is one source, it is just to mention and we want to go ahead with our concept of quotient spaces. So there is a correspondence between group actions and group homomorphism from G to permutation. The special case, suppose G is a subgroup of G, H is a subgroup of G, then you have the group multiplication G cross G to G restricted to H cross G to G. As if you treat now G as a set and H as a group. But the multiplication is coming from G. So H G going to H into G. I am now writing no circuit right here. This is the action. This is called left action because H is getting multiplied on the left. I could have written it on the other side also. H comma G going to G H. Then that will be a right action. So left action, right action are all possible. The corresponding orbits of the left action are nothing but the right cosets. So you have to write H and little G where G belongs to G. Those are the orbits of this action. So they are right cosets. Similarly, if you take right action you will get left cosets. So be sure of that. So the things are left and right are getting interchanged here. An important special case is the action of additive subgroup Z contained inside the additive group R, the real numbers. See first I considered arbitrary group acting on an arbitrary set. Then I said you can take the set to be the group G and the group to be the subgroup, any subgroup. Now I am taking a further special case namely R is the additive group and Z is the subgroup. So this is integers. Look at the left cosets here. Namely, I have denoted again it by Y, denote the set of left cosets or right cosets. They will be same here because R is commutative. The additive group is abelian group. So left and right does not make a difference here that is all. Look at this map in a different way, entirely different way namely G going to G, the map G R going to e power sorry G, T going to e power 2 pi i t. 2 pi, e raise to 2 pi i t is an element of what complex number of modulus 1 which is S1. So you get a map from R to S1, G, T I have written as e power 2 pi i t. We know that this is surjective. We also know that G, T is equal to G S if and only if the difference is an integer. I have put 2 pi i t here. If and only if T is an integer. So this follows. Now it follows that the orbits of this action Y is in one-one correspondence with elements of S1. This G hat is a map from Y to S1 induced by map R to S1. The orbits of this G action are nothing but the fibers of G. That is the meaning of this one. They are fibers of this element. G, T equal to G S will go to the same point. The T minus S is an integer means they are you know S belongs to Z plus T or T belongs to Z plus S. T belongs to Z plus S. Z is also correct. So they are nothing but the right cosets or left cosets. So once you have an action, decomposition, surjective map. Equivalence relation. These are all equivalent. Now I am thinking of this as a surjective function. What is that function? It is precisely this. Integers are sent to one here. All the integers go to the complex number one here. This is the map. So this is a wonderful map. It has a lot of topological analytic properties that will be exploited to study this. This is very fundamental result here. Fundamental example. The right cosets of this action are in one-month correspondence with the set of complex numbers of modulus one. This example we will meet again and again. So here there is a topology also. Though we have never bothered about topology so far. This is subgroup of a group. That is all I have used. All right? So let us bring in the topology now. That is quotient space. Start with any topological space and a surjective function, surjective function to y. Put tau prime equal to all those subsets of y, v contents of y such that their inverse image inside x is an element of tau. That is inverse image of q or inverse image of b under q is open in x. You see why we want such a thing we want this function, the important function, the surjective function which is going to define partition or whatever. That function I want it to be continuous. There is a topology here. So I am using that topology. In that topology it must be continuous means I have to put a topology here. So that is the trick I am doing here. Take tau prime equal to b contained in third y such that q inverse of b is inside tau. Verify that tau prime is at a power of v on y. These kind of things we have done several of them. Several maps together we have also done at one time, q inverse of pi i inverse of x whatever. So this is very straightforward. If you take empty set, inverse image is empty. If you take the whole space y, inverse image is the whole space x. Take b1 intersection b2, q inverse of b1 intersection b2 is q inverse of b1 intersection q inverse of b2. Inverse of sets is a very well behaved operation. So you have a topology on y now. This topology is called the quotient topology. The name is induced by x which is actually wrong. But this is the term used. So I am using that. Later on we will see that. You do not have that one. It is actually co-induced by x is the correct term. So it is co-induced by later on we will keep saying co-induced. But this terminology is used. The quotient space induced by x and so on. So this is the map itself is now continuous surjection. This is called quotient map. Quotient map as soon as you say it is not just arbitrary continuous surjection. It is giving you the topology of y, topology on y. It prescribes what is it prescription? Something is open in y if and only q inverse b is inside tau. Something is open if and only q inverse is open. That is nothing else here. That is the only condition. That is the meaning of that quotient map. Given a topological space x and a surjective function q from x to y, the quotient topology on y which we have defined just now is the largest topology with respect to which q is continuous. You see I could have taken the indiscrete topology on y. Automatically q will be continuous. Maybe that is too small a topology but it is continuous. So there are topologies. You take the biggest one. If you put more open sets inside y then q will not be open set. q will not be continuous function. So that is the biggest one. You take the biggest one which exists which we know. So how to do that? Why this is biggest one? We will see. So this is a claim. This is not the definition. Definition we have already given. So let tau 1 be a topology on y such that q from x tau to y tau 1 is continuous. Then I am sure that tau 1 is contained inside this tau prime. tau prime is defined here. That is the meaning of this is the largest one. Given u in tau 1 because it is continuous q inverse of u belongs to tau. But the moment q inverse of something belongs to tau this that something will be inside tau prime. So if b equal to u here q inverse u is tau so that u will be inside tau and over. So that means that this tau prime is the largest topology. Next thing is suppose you have this quotient map. Now topology on x is fixed. Topology on y is fixed. So this is a quotient map. Suppose you have a function f from x to another space inside that. So it is a continuous map such that f x1 equal to f x2 whenever q x1 is equal to q x2. So q x1 equal to q x2 implies f x2. Then there exists a unique map f hat f quiddle from y to z such that this f quiddle composite q here first come q and then f quiddle. That is you are given f. f quiddle composite q is f. The existence of this map as a function we have already seen yesterday. Whenever q n equal to q x1 equal to q x2, this is a condition that gives you this map f quiddle. Why this f quiddle is continuous? That is what you have to see. It is existence and uniqueness we have already seen in the Lamine yesterday's lecture. So we have to prove that f quiddle is continuous. So let a be an open set inside. This set is an arbitrary space. I want to show that its inverse image is open here. When a subset is open here, its inverse image here must be open. So what is f quiddle inverse followed by q inverse? That is nothing but inverse of f quiddle composite q operating upon that set. But f quiddle composite q is f. So it is f inverse. But f is continuous. So start with an open set. Its inverse image here is continuous. But that is the hypothesis. Now why f quiddle is continuous? Take an open set here. Come here. This is open. You fill down the inverse image of this one under q is open. That is where the role of f inverse of that one is open comes. f inverse of some a is q inverse of f quiddle of a. q inverse of f quiddle of a will be open because it is q inverse. It will be open because it is f inverse of a. Therefore f quiddle inverse of a is open. Take a coefficient map given any function from y to z, where z is a topological space. This function is continuous if and only composite f is continuous. Against this picture, you have some function here. Here when I have this function f, it introduce a function here. This is continuous if this is continuous. Now what I am doing? I am taking a function here. When this one is continuous is what I am asking. If this is continuous, composite q, that will be continuous because complete of continuous function is continuous. But here is a case even if you do not assume this is continuous. Suppose the composite is continuous, then f quiddle itself will continue. That is the hypothesis here. That is the statement here. Even now if f is from y to z, this f was from x to z here. So this f is playing the role of f quiddle here. So notations are different here. If since q is continuous, f is continuous implies the composite is continuous. That is okay. Conversely suppose f is continuous. Now I want to show that f is continuous. Given any open set u in z, we have q inverse of f inverse of u is f composite q inverse of u. The rule is the same, you see. Now this is open is given. Therefore the inside thing is open here by definition of quotient powers. The proof similar looks similar but it is given different results here. Every surjective continuous open map is a quotient map. Instead of open, you can put a closed rose. A closed map is also satisfied this property. When you say closed map, I always mean continuous and closed. When I say open map, I mean continuous and open. The extra thing is that they must be surjective. Unless you have surjectivity, you cannot get a quotient. So, surjectivity is a theory. Continuity is needed. Open map is extra. That will ensure that it is a quotient map. Just continuous surjection does not imply that the map is a quotient map. So, let us go through the proof here. Start with any open set here. Start with any set here. Show that it is open if and only q inverse is open. Both ways you have to start with any set u. If it is open, then q inverse is open. If q inverse is open, then u is open. That is what you have to show. So, first assume u is open. Then continuity of q will give you q inverse is open. So, here it is easy part. Now, suppose q inverse of u is open. Why u must be open? If it is a quotient topology, then it will be. But we do not know that. We are proving that it is a quotient topology. That is why we have to prove this. Suppose q inverse u is open. We have to show that u is open. Now, first suppose you take the case when it is open mapping. I have to show it for closed mapping also. First take the case where q is open mapping. Open mapping means what image of an open set is open. q of q inverse of u is open. Now, it is a lucky break that q inverse of q operating upon u is exactly u. Why? Because q is surjective. In general, it is not true. It may be subset of u. So, surjectivity again plays a role here. So, this is equal to u. The q inverse u is open. q of that is open because q is open mapping. Therefore, what we have proved is that u is open inside y if and only if its inverse image is open in x. Which means that it is a quotient map. Now, the same thing can be done for closed set also. By just little demorgan law I have to use. I will skip it. You can read it. The quotient map construction opens the floodgates of geometric topology to us. We can now study a large class of very interesting geometric objects via topology. As anticipated, group action is one of the important sources of quotient spaces. There are many other sources. So, we have to study one by one some of these examples. Okay? So, group action, what is the topology coming, where is it coming? That is what I have to explain. Otherwise, you have only quotient set, right? So, this is where we make a definition. Take a topological space and an action of a group on it, namely mentioned by a group homomorphism. This group homomorphism, I am assuming extra condition. I told you extra conditions on the group homomorphism will be different types of actions, right? So, I am putting an extra condition here. What is that? This homomorphism is actually taking its values inside a smaller subgroup, namely group of homomorphism suffixes, self-homomorphisms. X to X, any homomorphism is a bijection. It is a permutation. But I do not want discontinuous ones. They must be continuous. Inverse must be also continuous. That is the definition of a homomorphism. So, take only those homomorphism, that is the forms a subgroup is very clear. Because composite of two homomorphisms is a homomorphism and inverse of a homomorphism is a homomorphism by a definition. When you have this, we call it a topological action. You may wonder why all this is necessary. You just take the quotient topological space is there. There is a quotient set is there. You can always give the quotient topology. So, that will have something to nothing to do with the group action. Group action is just set theory or group theory, whatever algebra and isotopology is two different things. You want to bring them together and this is one way off. There may be many other ways also, okay? Depending upon the context that you want. This is the way. Take that may, now what does it mean is that all these rogies are homomorphisms, let us solve. Remember, all these rogies are elements of permutation, but now they are homomorphism. Associated quotient map for x to y, where y denotes the orbit space of x, this will have some special properties which was not possible in the general situation of a group action, okay? So, that is what we want to study now. Let q from x to y be the associated quotient map of a topological action of a group on the topological spaces, okay? Then this q itself is an open mapping. In the previous theorem, we proved that open surjective, continuous surjective open mapping is a quotient map, okay? So, these quotient maps which arise here, they are open mappings. Remember, a quotient map, there is no condition that it should be open. If it is open, it will be quotient map is what we have verified, okay? All quotients arising out of topological group action, they are open mappings. So, they are stronger questions in that sense. They have better properties, okay? Let us verify why this is an open mapping. It is one-line proof. See, all of things are one-line proofs here. Take an open set u, we have to show that q u is open inside y. When this q u open, q inverse of that is open inside x. What is q inverse of q u? What is q inverse of q u? We have to understand. That is nothing but, you see, all elements which are related to some element in q u, they will be there. So, if you understand that, then under rho g and this rho action, whatever, this is just all the cosets of g, rho g of u, taken together union g over g, okay? So, this is just verified, this one. What is q u? u is some subset of x. Anything, this is the equivalence classes here. When you take equivalence classes, what is the inverse image of equivalence classes? It is the union of all those classes. That is the meaning of this, okay? These are classes where members range over u. So, it is rho g of u, union. Now, what is rho g of u? Rho g is a homeomorphism from x to x. Just concentrate on one element g, okay? Multiplication by that on the left or right, that is an action. That is a homeomorphism, self-homomorphism of x to x. So, rho g of u is also open for all g, right? So, for union is also open. So, that is all. So, we have proved that the action given by a topological action, sorry, the topological action and the quotient space given, quotient topology given by that, the quotient map is an open map. Again, I come back to some examples here. This example is again the same as e power 2 pi i t, okay? So, coming back to 2.56, the additive action, z cross r to r, namely integer comma a real number going to n plus t. It is additive action, right? Corresponds to group homomorphism, z going to h r. h r is what? All homeomorphisms of r. Here, there are special homeomorphism, namely the translation by n. So, t n. Each n defines a translation. What it does is here, t is pushed to n plus t, that is all. These are called translations, okay? So, t n of capital T n of t is n plus t. These are translations. What are the inverse image? Inverse is minus n. T of minus n is inverse of that. Inverse is here, what? Additive inverse, okay? The translation homeomorphism. So, negative of t n is minus n. So, this defines a topological action of z on r. Let q from r to y be the association quotient map, where y is the orbit space, okay? We have already seen that this y is nothing but, you know, in one-one correspondence with s1, the unit complex number. We have already seen that, right? So, we have also seen that exponential map g from r to s1 induces a bijection from y to s1, the right cosets of this action. So, by theorem 2.29, it follows that one, just now g is continuous. g hat, which was just a bijection, is now continuous, okay? g hat is a continuous map. Now, extra thing we have, g is an open mapping. Why? Because this is given by, this quotient map is given by group action, right? Therefore, g is an open mapping. It follows that g hat is also an open mapping, right? See, g hat, different notations are there, otherwise you can use this. If this is g, this is g hat. This happens to be s1 now, okay? This is an open mapping. This should be an open mapping. Why? Take an open set here, inverse image is open here. Take the image of that, it will be the same thing as image of this one, okay? So, since f is open, it should be also open. So, g is an open mapping, f g hat will be open. So, open continuous bijection. So, that is a homeomorphism. So, the orbit space here, okay? It is actually a homeomorphism, bijection we had proved. Now, we have proved that it is homeomorphic to s1. So, under this homeomorphism, you can say that the quotient of x, namely r, modulo, the action of integers is precisely equal to s1, okay? Here is some algebra also coming here. R is some group. Z is a subgroup which is actually normal because every subgroup is, you know, the whole group is abelian. So, every subgroup is normal. Quotient has a group structure also. What is the group structure? Multiplication of complex numbers. You know that exponential function takes additivity, additive things to multiplication operation. t plus s, e power 2 pi i, t plus s is same thing as e power 2 pi i t into e power 2 pi i s because multiplication, right? So, this s1 is the quotient group of r just by group theoretically. Now, we have put an extra condition. It is the quotient space also, okay? So, this example will be, again and again will be with us all the time. I will do one more example. Only touch upon this one a little bit. This again group action. First consider x to be the n plus one-dimensional real space, okay? Euclidean space minus the origin. You can put the origin for a while but it does not help. So, minus the origin, okay? Now, you define the action here by scalar multiplication. Again, throw away the 0. So, r minus 0, okay? Scalar multiplication. That is what I am doing. x naught, x1, xn is an element of rn plus 1 not equal to 0. Similarly, why not why? These two are equal, equivalence relation I am deliberately doing this one. If there is a non-zero real number such that y naught, y1, yn is lambda times x naught. So, this is vector, this scalar multiplication. This lambda must be non-zero. Non-zero real numbers form a group under multiplication of that group I am using. Action of that on this one. This time, rn plus 1 minus 0 is not a group, okay? This is a genuine group action. There is no subgroup and so on. This is not the special case as r contained inside, z contained inside r, okay? Those set of equivalence classes together with a quotient topology that becomes a space vector that becomes a topological space. It is called n dimension real projective space. That is the definition and the notation is p power n. There is a deliberate notation. It is not a mistake. It is not a misprint. You start with n plus 1 and then you write the corresponding means of pn here. Observe that equivalence classes can also be thought of as representatives of one-dimensional subspace is in rn plus 1. Why? Because take an element, take its equivalence class. They are all the same line, okay? Minus the origin. You put back the origin just to make it a subspace. That is why I told you, you can keep the 0 for a while, but it does not help. Instead of that, whenever you want it, you put it back. So when you think of this as a line passing through the origin, you put back the origin only for that purpose. That is all. Okay? So they represent, you see, you want to represent the whole line. What do you want? You cannot represent it by 0. You take any other non-zero vector, the whole line is determined. That is the geometry here. So that is what we want to consolidate with this equivalence relation. Observe that the equivalence classes can be thought of as a representative of one-dimensional subspace is of rn plus 1. If q from x to pn is a corresponding quotient map, let us denote qx by the bracket text. The question is, is q an open mapping? You should have one line prove here. What is that? I already told you that this equivalence relation is given by group action. What is the group? r minus 0. Is that action a topological action? That is what you have to see. What is the mean of topological action? Multiplication by non-zero element should define a homeomorphism. Of course, its multiplication is continuous and its inverse is multiplication by lambda inverse and that is also continuous. Therefore, each multiplication defines a homeomorphism. Therefore, it is a topological action. Then you use the other theorem that we had. Whenever you have topological action, the quotient map is an open map. So that is the answer. There may be different ways of seeing it. You try to think of the different ways also. It is very good because you would like to have different views of this projective space which is difficult to imagine and very difficult to draw pictures also. Once again, I tell you that we will come back to this example again to study more properties of this. Thank you very much.