 So, welcome to module 16. The topic today is the quotient constructions which are typical in algebraic topology. So, let us begin with the so called own construction. The subspace of R3 which is given by the equation x square plus y square equal to R square, no condition on z is a vertical cylinder of radius r, center axis will be the z axis. So, this is a model for what we call infinite cylinder ok. You can restrict the coordinate z say minus 5 to plus 5 then it will be cylinder of height 10 alright. So, you can you know z axis can be restricted to some extent that is also cylinder ok. So, what topologically what this one is nothing but S1 cross an interval. The interval could be infinite length, finite length closed interval or open interval all these things are called cylinders S1 cross j. So, these are called actually circular cylinders in physics ok. So, we will generalize this concept you can call any space cross an interval as a cylinder and the cylinder as x as a base instead of the circle. The x topological space cross 0 1 you can think of this as cylinder ok. Now, this kind of generalization we are going to do for what is known in in layman's parlor or in physics the usual definition of a cone ok. So, in algebraic topology we define the cone based on any topological space x what we do take x cross i. In fact, you can take any interval, but let us standardize to unit interval x cross i and then identify x comma t with x comma 0 or x comma 1 what ever y comma t this should be for all t not equal to 0. So, what is this happening here? x comma 0 is x comma t, x comma t is x comma t you know no identifications where t is not equal to 0, but when t equal to 0 all x comma 0 is identified with y comma 0 for every x y and x ok. So, this is the space for t not equal to 0 the only identification is with itself there is no identification x comma t is x comma t, but on the base namely x comma 0 in x cross i every point is identified with every other point. So, the entire x cross 0 is identified to a single point. So, this quotient space is called the cone over x ok. The point namely the entire x cross 0 which identify to a single point that is called vertex of the cone or apex of the cone ok. So, you can just denote it by simply by style. The space x itself can then be identified with the image of x cross 1 on the top ok that is a quotient space and this then refers to the base of the cone. You can think of to get a idea of cone you can always think of x as s 1 the circle it could be instead of circle it could be n lips also to I guessed it could be a curve and then you can talk about the cone over the curve. So, this is what the generalization is about ok. So, the base of the cone is the starting with the topological space that is a base, but then that will be subspace of the cone via several ways, but I have taken in one particular way namely x goes to x comma 1. So, here is a picture you start with x cross 0 and then the bottom x cross 0 x cross i here bottom x cross 0 is completely identified to a single point here. Whereas x cross 1 is can be thought of as the original x here. In fact, you could have put x cross half here there will be a copy of x to x here also. In the picture it will be somewhat shun but it is homomorphic to that copy or x cross t t naught equal to 0 there will be homomorphic to x itself namely x going to x comma d right. This cone construction is very fundamental in homotopy theory you will see why that is so important. So, for an arbitrary point of this cone over x in this notation can be represented in the form x comma t bracket curly bracket to usual and parenthesis would have been origin thing. Now, we have put square brackets for some x belong to x and t belong to i. Normally square brackets we did not the equivalence classes ok. The representation being unique if t is not equal to 0, but for t equal to 0 any x comma 0 will represent the same point ok. So, x comma 0 will represent the vertex 0 for all x. Though C x may not have any linear structure it is not a vector space, but something nice happens namely it makes sense to talk about line segments through any point x comma t of C x and star. Passing through star there are lines what are they namely the image of little x cross i so that whole thing is there. It is actually homomorphic to i itself for each x x comma t t going to x comma t will give you a embedding of the interval i inside x cross in x cross i to again C x ok. So, from star you can go to any other point of x t by a line segment ok. The points how what is the point somehow x t is there you take x comma s t when s is 0 it will be the star when x is 1 it will be your point x t. So, thus entire comb is star shaped in this sense the star acts as f x ok. So, this f x we have defined for a any star shaped set. So, here also we call it as f x we call it as a vertex also vertex of the point also that is also used ok. Observe that the map x t s going to x of s t ok just we have taken that map here defines the homotopy of the constant map star ok when s is 0 it is a constant map and s is 1 it is the identity map. So, in particular this gives you it is contractible of course you do not need this proof because once something star shaped we know that it is contractible, but that was in vector space. So, you better see what is the meaning of this one but similar to that one. Combs are always contractible irrespective of what the original space is. Suppose you take just two point space discrete space what is the cone over rate it will be just union of two lines ok at one single point because two points cross i is union of disjoint union of two lines two copies of i, but one of the point namely at 0 both the points will be identified. So, it will be again a line right. So, it is contractible x here it is two points it was not even connected. So, given any map f from x to y you can talk about the cone of f namely the map corresponding map induced on c x to c y it is a natural way to get a map on x cross i to x cross i x cross i to y cross i namely f of you know c f whatever x comma t goes to f x comma t namely f cross identity, but then you can pass down to the quotients. So, that is c f c of f of x t is f x comma t the second coordinate t part is not affected ok it has the property that if you restrict it to x namely x cross 1 then it is f it can be identified with f when you identify x with x comma 1 you can throw away that one then it is just f x so it is f. So, x to y you have a map c x and c y are larger spaces containing x and y then you can think of c f as an extension of f ok extension of f for the cones over x and cone over x and cone over y ok. So, this is what we we will have this picture namely x is sitting inside c x y is sitting inside c y namely x comma 1 and y comma 1 here ok. So, one of the nice property of c f is that c f will be a homeomorphism if filled only if f is a homeomorphism it is very easy to check the the more deeper thing here is that the cone construction is actually a factorial ok. So, if f from x to x is identity map c f will be identity map if x to y y to z you have maps ok f and g for example g composite f will be from x to z the c of g composite f is c f composite c g composite c f. So, that is the meaning of that the cone construction is actually a factorial map it is like our pi 1 pi 1 was was what what we it was a function of ROM in a topological space to the groups here it is space to space itself topological space to topological spaces ok. Now, the usefulness of the construction cone construction will come into picture in studying the homeomorphism theory. Any topological space x is contractible if filled only if the space is a retract of the cone you know cone itself is contractible always, but x itself is contractible if x is a retract of this one and conversely. So, this is the statement let us go through the proof of this take q from x cross i to c x which is the quotient map namely x cross 0 being identified as a single point ok this is just an notation for the quotient map. Suppose now you have a homeowner pH from x cross i to x such that it is identity it is a constant map at 0th level and identity at the t equal to 1. So, between 0 between the constant map and the identity map h is a homotopy ok corresponding to this there is a continuous map r from c x to c x ok defined by r of xt is equal to h of xt r of the class xt equal to h of xt you see this should be this should be independent of the class. So, let us verify if t is not 0 then xt is a single point so it is h of xt no problem if t is 0 all these is independent of x one single point ok, but h of xt depends upon x but if t is 0 this is the constant map. Therefore, this is well defined ok at all is well defined automatic collision semi continuous because it is continuous on x cross i and it boils down to c x remember c x is a quotient option it is a unique q composite this q composite r composite q is h of course that is how I have defined this ok it is boils down to the quotient space c x also this r on x comma 1 namely the subspace x it is the identity because h of x 1 is identity ok. So, we have identify x is x cross 1 this is what we are using here that is how we can talk about x as a retract of c x ok this is the meaning of retract if you want to recall it namely a function from the whole space to the subspace which is identity on elements of x and r must be continuous ok. We have used the retracts earlier we have shown that the disk d 2 in the disk d 2 the boundary namely s one is not retract of d 2 we have shown that one in proving Brouwer sweet point theorem all right. So, we have proved that if if the if the constant map is homotopic to identity map of x namely x is contractible then x is a retract of c x. Now, let us look with Converse suppose you have retraction c x to x ok then you just define h of x t by r of x t by same formula. Now, here I have used this one to define r now is r is given I am defining h ok automatically h will have the property when t is 0 it is a single point and t is 1 it is r of x t by definition it is r of x which is x ok so identity map. So, h gives you a homotopy of the constant map with the identity map of x this is tautology here right. Just I have used the fact that c x is a portion of x cross i where all the x cross 0 is identified single point and homotopy of the identity map to a single point also has this property therefore everything works fine. Now, here is an example let us take the circle more generally take any sphere you can even take s naught also ok. So, take the sphere s n minus 1 take the cone over that that will be homomorphic to the disc of one dimension higher if you start with s naught the cone over that will be homomorphic to the interval minus 1 to plus 1 if you take s 1 this will be the disc d 2 ok and so on for n equal to 2 this is the familiar polar coordinates polar coordinates of complex numbers ok r times cos theta sine theta the cos theta sine theta is a point of s 1 ok r corresponds to 0 1 so s 1 cross i you can say but when r is 0 it gives you one single point. So, the entire s 1 cross 0 is going to a single point that is a quotient value ok. So, that is why the cone over s n s 1 is d 2 for n greater or equal to 2 it is generalized polar coordinates you can when you have vector unit vector in s n minus 1 namely in r n a unit vector ok the rest of the r n can be thought of as r times unit vector when r is 0 it will give me a single point for r not equal to 0 it is 1 1 s ok. So, what we do s n minus 1 cross i d n we just write x comma t going to t times x this map is clearly continuous rejection and 1 to 1 except for points of the form s n minus 1 cross 0 the entire thing is mapped to a single point. Therefore, this map p factors down to the cone over s n minus 1 to d n this is a quotient map. So, it gives you a map which is a quotient space c s n minus 2 p bar is a unique map defined by p ok. So, p bar q composite p bar is p since the domain is compact d n is compact the ray sorry c c of s n minus 1 is compact plus s n minus is compact the product with i is compact therefore the quotient is compact. So, the domain is compact and this is Hausdorff ok continuous bijection is a homeomorphism. So, this we keep using all the time. So, we identify x cross 0 to a point ok. So, we got an ice cream cone like that you could have an x cross 1 also to a point then you would have what a tent both of them are cones homeomorphically all that you can x comma t you can change x comma 1 minus t then you would get the other one. So, there is no problem in fact what we can do is any interval also I have told you but we have just standardized these things so that again and again you do not have to keep on changing coordinates that is all. Now, let me tell you about another important construction called adjunction space. So, pay attention to the definitions you should not have any confusion in the definition you start with a space and a closed subspace z z is a closed subspace of x ok sorry the other way around z is x is a closed subspace of z let us do that one let me let me stick to the notation that I have already introduced do not change it here. Let x be a closed subspace of z ok then the function is there is a continuous map from x to y y is another space on the subspace you have a continuous function on this closed subset you have a continuous function ok now the adjunction space is defined as disjoint union of z and y ok modulo some relation so it is a quotient of z disjoint union y modulo some relation and I am going to denote it by af or z union over f y so these are the notations for the final quotient space it is going to be disjoint union of y and z then you have to make identification what is the what are the relation every point x in x will be identified with fx in y x will be identified as image effects for every x in x so this is the identification or no other points there are you know if z is a point not in x then it is not identified with any other point for all z is equivalent to z itself that is all all y is equivalent to y itself there is no identification inside y ok it is a point of y and a point of x they are identified when namely y must be equal to fx then only those points are identified ok this space is also called the space obtained by attaching the whole space z to the space y y are the map f ok so this this description of attaching this one this will come very very much into into operation maybe much later in algebraic biology when you study cell complexes and so on ok y can be identified with the close of space of af namely f is a junction space namely each y being its class there is no identification each y itself is a class there ok so that inclusion map from the from the disjoint union to the quotient space ok that will be a closed subspace why because inverse image is just y is disjoint you know why you insert that is closed here this disjoint union of two spaces both y and z are closed subspaces ok only thing is we won't get a copy of z in z z you and this one because there is some identifications inside x suppose fx is equal to fx1 equal to fx2 then fx1 fx2 x1 and x2 will be identified with fx1 fx2 right so it depends upon whether f is injective or not so z may not be a subspace of the entire thing all right but why is a subspace if the image of f is closed in a y it's a extra hypothesis then z will be closed in f that means image of z will be closed in af ok also if f is injective then z can be identified as image af if f is injective then there will not be any identity within points of z then the z goes into its class it's a single point that's each class a single point ok so then z can be identified z will be also a subspace ok the quotient this adjunction space is a so wide definition it it gives you lots of examples in some special case have their special names now all right now I am going to make several claims here which are all topological claims which are all of easy claims but unless you verify them each you know step by step you will not give the full picture of what are these new concepts so you have to spend that much of time ok so reader is requested to verify the validity of each and every claim made by the above paragraph you have to see everything close up space blah blah blah so you have to verify for instance y is j and embedding ok why j y is close subspace of af etc all this you be sure that you have verified them yourself so we will take some special cases now the mapping cylinder remember just now we have defined the cylinder now I am going to define a mapping cylinder f from x to y is a function say is a just a continuous map ok then the mapping cylinder is first of all on x I have a cylinder x cross y then I take this joint union with y just like I took z cross disjoint union z then sorry z and y disjoint union in the adjunction space is y disjoint union x cross y then I have identification namely x comma one the other one of the end points is identified the effects ok for every excellence so this is a a special case of the adjunction space namely z dis x cross y x cross i here and x is x cross one on x cross one think of f being identified f being defined there ok if you put z equal to x cross i and x equal to x cross one here in the earlier example adjunction space you get the mapping cylinder of f so this is a special case this is called a mapping cylinder of f and this denoted by m s the reader may think of this as a special case of adjunction space construction that is what I have told so here is the details of that before that let me show you the picture for mapping cylinder so you had x here and x cross i ok but before that I had a map x to y in this picture I am showing x as a disk and y as just a curve here so that is the image of f ok so image of f is a curve y is a is this square is the rectangle y is a rectangle x is a disk but the image is only just a curve here ok the mapping cylinder is I have taken x cross i but at the x cross one x cross one does not remain as it is x cross one because each point x cross one has to be get identified with its image under f so in this point goes to this point then x cross one will be identified at this point and so on so this is the picture for the mapping cylinder m f ok m for mapping cylinder of f ok so there are various inclusions and so on because of the importance of this let us pay some attention to this one just like in the case of adjunctions pay let us say map j from y to m f it is always an inclusion because the space y is not disturbed ok there is a map from inclusion map i from x to m f this is at x cross 0 or any other x cross if not x cross one x cross one there are identification ok so this inclusion is x cross 0 the i of x is x cross 0 and then there is f hat from m f to y f itself is extended ok f hat of x t is f x f hat of if you say x 0 it is just f f and x one thing turn one will get identified with all these effects f hat of y i have defined it is just y ok is y is you know if it is f hat of x then this is this f f x will be y of course so this map f hat is continuous you have to verify i have defined on x cross i i have defined f hat like this and on y i have defined this way whenever the there is identification they are ok because f x here is x comma one let us identify with f x that is why this works ok so f hat is an extension of of it is x contained inside m f x cross 0 is an extension of that so here are the claims i and j are embeddings ok i is embedding of of x and j is embedding of y we use them i have to identify x and y as subspaces of m f x and y are arbitrary spacing there is a map between them now m f will include these both domain and codomain of f as subspaces of this space ok so they it brings them together this is the whole idea now f hat is defines a continuous function on m f such that f hat composed i x is f hat of x is f hat for every x check that f composite j is identity of y that is how we have defined it here right f hat of y is y which will soon see that in a very strong sense the mapping cylinder can be used to device as a device to replace the continuous function f from x to y by inclusion map from x to m f what is the meaning of this replacement we will let you know later soon the mapping cylinder is a special case of the exemption space there are some more things we shall consider them next time thank you