 Last time we introduced the notion of relative CW complexes, CW complexes and checked a lot of examples, standard examples such as spheres, disks and then later on projective spaces both real and complex. You would like to study many more examples. So, in two more modules, which I have denoted by module 4A and 4P, we will only study more and more examples. For example, DN itself you know can be given several different CW structures, each of them can be of use in different contexts. One such thing is thinking DN as the underlying space for mod delta n, where delta n is a n simplex in Rn plus 1. Remember a n simplex in Rn plus 1 is the convex hull of standard basic elements even E2 etcetera En plus 1. The endpoints of these vertices and points of these vectors will be taken as the vertices. Inductively having defined xk, we take the set of all k plus 1 phases of delta n as k plus 1 cells in xk plus 1 with the inclusion map as the attaching maps. The point is we are not building up the space here, but that the space is already there, we are decomposing this space into a CW complex, decomposing it into open cells, 0 cells they are open automatically. Then one cells their boundaries must be inside the 0 skeleton, then 2 cells their boundaries must be inside the 1 skeleton and so on. So, this phenomena we will keep observing. So, this happens to be a very special case of namely the simplexial complex. For any simplexial complex that is an associated CW complex, the simplexial complex itself can be thought of as CW complex, namely the vertices are the vertices of the 0 skeleton is the set of all vertices of the simplexial complex. Then the 1 simplex is there become 1 cells. So, only you are changing the terminology here that is all. Attaching maps for what? Attaching maps as well as the characteristic maps are the inclusion maps. So, this is what we did take any triangulated space. Say x is a triangulated space automatically it gives a CW structure. So, in all these the characteristic maps are all just the inclusion map because we have already built up the space, we do not have to build the space here. And inclusion maps happen to be injective monomorphisms whatever you want to say embeddings of homomorphisms that is why these are very special kind of CW complexes. Before considering the next set of examples, let us make a few more definitions. So, that we will become familiar with the definitions also. A CW complex x is said to be locally finite if for each closed cell sigma, closed cell is just the closure of a cell sigma. The number of closed cells intersecting sigma in x that must be finite. For example, take a vertex look at all one cells which may intersect that one which will have that vertex that 0 cell as one of their vertices. So, that must be finite. Not only that a two simple two cell which you attach the attaching map may find that this point this point whatever I have taken is in the image then I will count so those two cells also. So, all such cells two cells three cells n cells whatever all of them together must be a finite set. So, this should happen for every simple x every cell. The closure of a cell should have only finitely many other cells intersecting it. It x be a topological space and f be a collection of subsets of x. This is a more general definition from point set topology. We say f the collection of subsets is locally finite on x. If for each x in x there is an open set u x in x which will contain the point x and which would intersect only finitely many members of x. So, for each x there is a neighborhood u x of x which will intersect only finitely many members of x. As x be any topological space with two CW structures on x which I will denote them by x a and x b just temporary notation just to distinguish between the two of them. We say x a is finer than x b that is what it is. If each closed cell in x b is a sub complex of x a a sub complex may have different structure than the just a cell. If you take a closed cell along with all the phases and so on then only it will become a sub complex in x a. So, x a is finer than x b. So, if you have each closed cell in x b it is a complex of x a. For example, this will imply that if you take all the 0 cells of x b they must be inside x a. There is no other way that those 0 cells will be sub complexes of x a. So, all the 0 cells must be saturated contained inside the 0 cell of x a. But for k positive there is no such simple relation between the k cells in x a and k cells in x b. Except if you take the totality of x b that means the kth skeleton of x b must be contained inside kth skeleton of x as just topological spaces. So, this is the consequence of this definition one is finer than the other. So, you can think about these definitions and think about giving some typical easy examples and so on. So, let us take a more general example than just ad hoc typical example. Take a simple complex and a subdivision k prime of k. Consider as CW complexes on the same underlying space mod k because mod k prime is also mod k. Check that k prime is finer than k. So, this will require that you know CW complex, simple complex is already. So, but that is what I am assuming in this course anyway. Indeed about definition of something finer is modeled on this example. Instead of calling x a is a subdivision of x b which will be too much I am calling it as finer. So, just extracting certain properties of subdivision to call this as a something is finer than the other so that we can compare the two of them. So, here is a lemma which will ensure how to patch up two different CW structures on the union. So, suppose Z is the union of two topological subspaces x and y. Both of them are closed subsets of Z and have CW structures on them. Out of that we want to have a CW structure on Z. So, for this we have to assume one more condition. Suppose for the that on the intersection the two CW structures obtained as sub complexes from x and y respectively are such that one of them is finer than the other. There are there is a sub intersection y can be sub complex of x as well as sub complex of y. So, you get two of them you can call this one is a, r then it is b. So, one of them should be finer than the other. So, this is the condition I am assuming on the intersection. Then there is a CW structure on Z whose closed cells are dose of x or y. So, you do not have to do any more work. Look at all those cells which are which are in x or in y then put them together. So, this I am putting it as an assignment just because I want you to participate in this so that if you start thinking and working on them. So, that will these things will become more familiar. So, in this lemma I have already used the definition that I have made here. So, while proving that lemma you will be automatically become more familiar with that definition. Now I come to a class of structures which are very nice structures CW structures which fall short of being CW complexes just a little bit. Fall short of being a simple complex is just a little bit. So, here are those examples. They are very close to being mod k of a simple complex except that instead of being built up of simple complexes or what? Tetradrons, triangles and so on. So, here there will be cubes, squares and cubes of cubes of higher and higher dimension and so on. Only in 0 dimension and 1 dimension the simplex as well as cells coincide. So, all these n cubes are built of inside RNA everything is happening inside SMRN built up of n cubes with their sides parallel to the coordinate planes. So, these things will be very useful in analysis all the time you cut RNA into smaller and smaller subdivisions and that is precisely what we are going to do here and then put them to produce for several complexes. So, we will start with this subdivisions namely what are called as lattice points. So, Lk, Lk fix I am capital N is arc power n that is the dimension I am fixing that n. So, I am not writing sometimes that n but whenever there is confusion I will write. So, Lk is same thing as Lk power n here. This is a notation all points inside Xn whose all of the coordinates look like some integer Ri divided by 2 power k. So, this k is fixed here. So, the same k should come here 2 power k. The numerator is an integer denominator is 2 power k. For example, it may be 2 power 0 which means denominator is 1. So, all integers are allowed. Then say integer 2, 3 divided by 4, 3 divided by 8, 6 divided by 8 is same thing as 3 divided by 4. So, all those things should be allowed here. So, those are the coordinates of this point X such coordinates. Let pk denote the set of all closed n cells. So, generally I am denoting them by sigma. The n cells have side length 1 by 2 power k. So, that k, k is fixed here 1 by 2 power k. And with corners of sigma inside the lattice cell k. So, that is pk. Take all the pk for p written equal to 0 that we will denote by p. This is just notation. Again and again, we will have to use this notation. I am setting it up. That is all. What you have to know is that lk, all the points, the lk consists of only points, with coordinates are like this. This is what we call it as lattice. It contains lk plus 1, contains lk plus 2 and so on. However, for i between 1 and n, if tau is an i dimension phase, phase means what? You take a square. The square has phases all the sides and vertices. A cube has phases all the, there are 6, 2 phases and 8, 4 plus 4, 12, 1 edges and 8 corners. So, they are all called phases. So, i dimension phase of a member of PR, start with a PR. Take all the phases between 1 and i. It will not be a phase of any member of PS for s not equal to r. Because s not equal to r just means the side length will change. So, 1 by 2 cannot be thought of as a cell of, like 1 by 4 or the other way around. Whereas, vertices, once you have a vertices in higher things higher, more and more divisions, more and more vertices will come. The old vertices will still remain there. So, for example, 0, 0, 0, 0, 0 is in all of them. Similarly, integers are in all of them. All, suppose some coordinate has all the integer, then it is in all lk's. So, that is what we have to remember. An important property of P, which is worth noticing and remembering is take two cells, n cubes, both of them. I am taking P, that means the sizes could be different, n cubes in P. Then interior sigma, intersection, interior sigma 2 is non-empty, would imply one of them is contained in the other or this one. This is what I have to say. You can also say that if they are equal, then these two are same. But I am taking two different ones. So, either sigma 1 is contained as sigma 2, sigma 2 is contained as sigma 1. So, first of all, if both of them are in P r for the same r, then of course, interior sigma and interior sigma 2 are definitely disjoint. So, this does not happen at all. This can happen only when one of them is smaller, say one is in P1, another is P2 or P5 and so on. Then this can happen, only this way it can happen. The interior non-empty would imply sigma 1 is contained as sigma 2 or sigma 2 is contained as sigma 1. So, following lemma is easy to see. Each P r, when r is fixed or k is fixed, v, p, k whatever you want to say, P r along with its n cells and all the phases of this n cells defines a CW structure on R n which is pure and locally finite. Okay. The local finiteness is obvious because around a point in P, inside R n, how many edges you will have, how many edges will come there, squares will come there, you just think about that. In r, every point has exactly two edges emanating from there. In R2, it will be 2 power 2, there is four of them. Okay, that is it. So, you can actually see how many are there. If you take one cell in R2, whatever P r is, there will be exactly two cells on either side. Think about that. So, local finiteness of these structures is obvious and these we have been using. So, all the n cells, their interiors do not overlap. So, they will themselves be n cells. Okay. Their boundaries will consist of n minus 1 cells, several n minus 1 cells. Their boundaries are several, several more n minus 2 cells and so on down. So, this is what the structure, CW structure on R n, if you keep changing R, you will get different CW structures on R capital N. Okay. So, these things we keep referring to them in short lattice structures. Okay. So, out of these, you can get several interesting examples. The first example is take a k cube sigma where k is between 0 and n. The 0 is the most interesting example but you can take that when k is 0, it is just a single term. Right. So, that is also a nice thing. All right. Which is a phase of some n cube belonging to PR. Each n cube belongs to some PR. Okay. So, I am fixing this PR here. So, take a sigma, take a k cube sigma, which is a member of, which is a phase of some member of PR. Then each PS, by the way, this curly PS denotes the CW structure here, PR. Okay. The round, the ordinary P will denote only the n cubes. Okay. So, this is just to facilitate again and again. I am not talking about just the n cubes but the entire structure here. Okay. Then for each PS, as given to R, okay, each of them induces an obvious subcomplex structure on sigma. Just sigma I am looking at. For example, suppose sigma is some edge in PR, the P2. When you go to P3, this edge will get divided into two edges. If you go to P4, it will get divided into four edges and so on. Okay. You started P1 for example, an edge of length 1. In P2 itself, well edge, so P0, length 1 is P0. In P1 itself, it gets divided into two edges. Okay. So, this way, you will get CW structures coming from PS on each k cube. Okay. The same holds for, let us look at WF. Okay. What is WF? I am defining. All sigma, collection of all sigma is set. Sigma is in F, where F is some family, where F is a collection of some faces, I am taking the union of all of them. Like I may take some finite limit point and then some cubes and then some higher dimensional cubes or maybe in between some one simplex, two cells, two cells and so on, various things, but finite. Okay. Any collection of faces of members of PR, actually there is no finiteness in this one. R is fixed. Okay. For each S greater than or to R, PS will induce a CW structure on WS. Okay. Why? On each of this one, it induces and they will be compatible because they are all belonging to one single PS. Okay. To fix a PR and then for each PS, there will be a structure. Okay. So, this generalizes the earlier observation that in Rn, you can have different structure, P1, P2, P3 and so on. Okay. This curly P1, P2, P3, P4. A little more generally, let X be a union of finitely many faces of n cubes in P of various dimensions and various sizes. Okay. Faces of n cubes. P, members of PR n cubes. If you take faces, there may be 0 dimension, 2 dimension, 3 dimension and so on. Take all this, but finitely many. Okay. So, what I am doing here, here there was no finitely many, but R was fixed. Now, I am ranging R. I mean I am allowing R to be different, but I put a condition X must be finite, finite collection. Okay. Then it will again get a CW structure by some S. What is that S? Bigger than or to R. What is that R? I have to tell you. Namely, choose R to be the maximum of all T such that one of the n cubes containing a member of X belongs to PT. Okay. Look at all the integers T such that there is a member PT of which some face is inside X. Otherwise, you do not take it. Look at all those where X is finite. So, set of all such T will be finite. So, you take the maximum. Okay. Maximum of all such T. That you call it as R. Then, if S is bigger than or to R, PS will divide all of them very nicely. So, that is the whole idea. That will give you a structure on this X itself. Okay. So, here is one example here. Alright. So, here you see the smallest thing occurs here. So, I have divided all of them like that. The X is shown by heavy lines here. Okay. Union of this large square here, this small square here, this square, this big square. This is the smallest one. So, you cut them like this one. So, that will be the structure on this entire thing. It is very easy to see. Okay. A finite case. Very easy to see. Alright. So, here is slowly we are going to build up on this theme. Towers making it more and more meaningful and more and more useful. For any k phase sigma of a member PR, okay, take a member of PR, take a phase of that. Let f sigma denote the set of all i phases of sigma. So, i phases of sigma will be smaller than k, right? i less than k. I do not want to once, zero cells here. So, one less than one less. This is the collection of f, this collection f sigma. On f sigma, take a positive integer function such that f tau is bigger than equal to this R for all tau. Tau contains at tau 2, tau 1 contain at tau 2 implies f tau 1 is bigger than equal to f tau 2. So, f is order reversing. Okay. The smaller the phase, the larger the integer. That is what I want. Alright. Take such a function. For each tau in f sigma, give the CW structure induced by the corresponding number here P of f tau. Okay. We claim that all these structures patch up to define a CW structure, okay, on sigma. Okay. Sigma is what? Sigma is a one fixed member, one simple phase and all these f sigmas were phases of it. Alright. So, on all of them I have a CW structure. They will put together on sigma itself gets a CW structure which is finer than the CW structure coming from P of the sigma itself has some f, right? P of sigma f. Okay. We should denote this CW structure by sigma f, sigma upper square upper suffix f just to denote the function f there. Okay. See, P f sigma, this will divide the n cell sigma, the k cell sigma according to this f sigma number. But for the smaller things, the division may be even finer because the numbers are larger or at least that much. At least that much is important. Okay. It may be finer. It does not matter. If I have a pentagon or a hexagon or whatever or just a triangle, okay, the empty triangle, I can all of them I can attach a two cell to get typologically the same picture. Okay. CW structures will be different. So, this is what you have to keep in mind. Okay. So, we have, we get a CW structure on sigma itself depending on this function. The function should have these two properties. That is all. Okay. So, this is germ of these k cells, how to cut them into finer pieces and so on. This will be of some use soon. So, we know, we do this by induction on k. I am explaining this, how this is working. Because the way it is done here, this is what you have to understand to patch up further more complicated things also. Okay. So, let us do it by induction on k. k is what? k is the dimension of sigma itself. Suppose it is 0. That means it is just a singleton. There is nothing to do, a singleton will never get divided at all, no matter what sigma you choose for it. That is what one observation I have made, namely once a point is in Lk, it is there in k plus 1 and k plus 2 and all of that. Same thing. Okay. So, 0 is no problem. For 1, 1 means what? It is a, it is a, like an edge. Okay. So, what you have to do? You have to look at the 0 cells of that, namely the two end points and itself. The end points will never get divided. Only the simplest can get divided depending upon what you choose. And that is all. There is no match, patching up and so on. So, for 0 and 1, there is nothing to prove. Okay. Here, sigma f is the same as the pseudo structure coming from p of f sigma. Whatever you have chosen, f sigma, that same thing will happen for entire. Okay. Now, assume the claim is true for k. Then we will do it for k plus 1. Okay. So, now sigma is a k plus 1 phase of some member of pr. Okay. Let tau belong to f sigma. Okay. Yeah, sigma itself, yeah, you can assume k, tau is k plus 1 phase by assuming f sigma to be of some higher dimension thing, no problem. So, let us look at it first. The first thing to note is that the CW structure, say rho f on all the k phase is rho of tau. Okay. Tau has already struck because it is a lower dimension thing. Right. All k phases, k phases of tau will patch up to define a CW structure on the boundary of tau. For, suppose rho 1 and rho 2 be any two k phases of the same tau. Okay. Their intersection may be empty or may be a single point or a j phase rho for some 1 less than or equal to j less than k. Because 2 rho 1 and rho 2 are k phases are intersecting. The phase have to be of lower dimension. In the last case, only we need to check. A 0 empty 1 case, there is no problem. In the last case, because of condition 2 above is 2 above, the structure on the intersection is finer. Rho f is finer than the 2 induced structures, CW structures coming on rho, coming from rho 1 and rho 2. Okay. It follows that this patched up structure on boundary of tau itself is finer than the CW structure on tau because rho f itself is finer. Below that of lower dimension, it may be even finer and finer. So therefore, the structure on boundary of tau is finer than the CW structure coming from P of f tau. So this implies that we can put them together to obtain a CW structure on tau. Okay. So this comes with the proof of the inductive step. So k plus 1 phase also you have proved. Okay. So we shall carry on these things. Remember these things. We will use them and we will try to give now examples of when we are taking infinite families of these f's. Okay. Thank you.