 So, today we will start the last topic of the course, Apology of Manifolds. As I have told you several times, Manifolds are the central objects of study in topology. So the idea of Manifold can be traced back to Riemann. The formal definition as we use today is perhaps due to Hermann-Weil. It studies a must for any kind of higher mathematics and theoretical physics. Our aim is here quite modest dealing with only few salient topological features of the topological Manifolds. Normally as we keep putting more and more structures like differentiable Manifolds and PL Manifolds, League Groups and so on, so its study becomes more and more interesting and more and more concentrated also. We are not going to do all that. Usually the topological Manifolds are quite difficult to handle because you do not have much structure there. On the other hand, some of these things are neglected in the other studies, suppose we are studying League Groups, you just assume whatever is happening to all of our Manifolds quite often. I mean that is what happens with the books and teachers. So I thought of giving a good treatment for just topological Manifolds here. Earlier we have studied Simplicial Complexes in Part 1 and then CW Complexes in the very first part of this series. They are in some sense a slight generalization of Manifolds and help us in understanding Manifolds better. So we will bring them in study of topological Manifolds also, the CW Structure as well as Simplicial Structures. First we shall study purely topological and little homotopical aspects of general topological Manifold. We shall then take up classification of one-dimensional Manifolds. And we shall discuss Triangulability of Manifolds in general, okay. Finally we will end up this talks with classification of compact surfaces, surface means two-dimensional Manifolds assuming that they are triangulable. Of course that is triangulability because of lack of time we are not able to present the triangulability of surfaces. So this is just the gist of what is going to come in another 15 modules, okay. So today we will just see some basic working definitions and examples. Fixed integer, positive integer n, let X be a non-empty topological space by an n-dimension chart at X, for X we mean a pair u, psi consisting of an open subset u, okay, of a point X of a neighborhood at a point X inside X and a homeomorphism psi from u to Rn on to an open subset of Rn, okay. So such things are called charts. By an atlas we mean a collection of charts u, psi, i, for X such that the domains of these charts will cover the role of X, so X is union of uj. If there is an atlas for X we say X is locally Euclidean. The locally Euclidean just means that each point as a neighborhood which just looks like, what it looks like homeomorphic to an open subset of Rn, okay. In fact one can assume it is on to a whole of Rn also no problem. A chart u, psi is called a chart at X if psi of X0 is going to 0, okay. This is a special word I am just using, this may be temporary in the literature you may not find this one, this is my own convention you may say chart at X0 means just saving the time to say that psi of X0 is 0. Writing psi at psi 1, psi 2, psi n there are n coordinate functions for any psi here, right. Taking function values inside Rn. These n component functions psi i are called local coordinate functions for X at X. That is the reason why I am putting this X0 going to X0 this at X0 or X whatever. As soon as you coordinate X that psi of X would have gone to 0, that is the meaning. So 0, kx corresponds to the origin and these functions will be now like X1 coordinate, X2 coordinate and all, f1 these are psi 1, psi 2, psi n functions are coordinate functions. If n is greater than or equal to 1, okay, an axis topological space, we say a topological manifold of dimension n, okay, if first condition is the definition here locally Euclidean, that means there must be a chart like this, there must be an atlas like this, okay, the n is all the time fixed remember that for each of ui the same Rn is there, okay, that is the first condition. Then we put two topological conditions namely AX must be a horse door space and second countable. The second countable means at each, it has the topology has a countable base, okay. So that finishes the definition for n greater than or equal to 1, for n equal to 0 also you would like to make a definition, any countable discrete space is called a zero dimensional topological manifold. Local Euclidean follows because it is discrete, horse doorness follows because it is discrete. Second countability is taken care by taking only countable discrete set, okay. So this zero dimensional manifold also falls into this category, only thing what are homeomorphisms there that is why I do not want to discuss zero to Rn, what is R0 and so on. So it is a discrete space that is the meaning of this. So you would like to include empty space also, first I started with non-empty space, so it is convenient to include this empty space as a topological manifold. However, there will be always a problem what n we should assign, n equal to 1, 2, 3 or 0, what should be assigned. It makes sense to have n equal to minus 1, just next one, okay, like in combinatorics where some authors at some places algebraic geometry and so on, it is convenient to assign minus infinity. But you will see that the best way to do is do not assign any degree, any specific dimension to 0, just like the zero degree polynomial, the zero polynomial can be taken as having all the degrees, similarly the empty manifold can be given any dimension you like, that could be a quite satisfactory answer also, okay. So these things do not matter right now for you, but when you do the cobaltism theory and so on, you will see the importance of this remark. Consider that once a chart u psi exists around a point x naught, okay, then we can choose a chart v psi at x naught such that v phi of v equal to the whole of rn, okay. So it is some open subset of rn, open subsets of rn, okay, at any point whatever point these are gone, around that you can choose a round disc, okay, all points which are less than or equal to less strictly less than some epsilon will be contained inside that one, that ball, right. Any open ball is again homomorphism of the whole of rn, so you can compose another homomorphism and take that as phi, okay. So this is the advantage of topological manifolds, it is true for c infinity also, but when you hold analytic manifolds like complex and so on, this is not true, so you have to be careful about this. By composing with a translation, we can assume that phi of x naught is 0 also, okay. These are not restrictions, these are always possible, but when you make such a choice we will have that now, psi will be called as a chart at x naught, okay, so that is what I, you can choose the ball of radius r and then you use this homomorphism, so you can make it the whole of rn. For an atlas, it is necessary to assume that integer n is the same for all, same for all the charts. Of course, if you assume x is connected, it is an easy consequence of topological invariance of domain, okay, if two non-empties there is intersection and open subset once cannot be homomorphic to open subset of rm and then rn, okay, so two things will not happen. So that is an easy exercise. If you assume connectivity, you can say any mi and then the mi will be, will happen to be the same for the whole connected component. So why bother assume that the dimension n is fixed for all, for all the charts, okay. For a topological space that is uniquely Euclidean, there is a first condition. We have put two other conditions, Osdorff's nests and second countability. The second countability condition is equivalent to many others such that, such as mectrizability or para-compactness plus connectedness, of course, okay, all these are there. So depending upon the favorite of the particular author, you may find different definitions. But second countability is most popular and easier to handle. The point is that under slightly suitable conditions like Osdorff's nests and so on, they are all equivalent. So do not bother about if somebody has taken a different definition. If you ignore Osdorff's nests, of course, there will be chaos. There are people who study locally Euclidean spaces which are not Osdorff's also, okay. So that is what we are not going to do here, okay. So we find second countability most suitable for our purpose. The Eurydense metrization theorem, which is a standard developed in point set topology tells you that every second countable locally compact Osdorff's space is metrizable. You can put it, you can embed it inside the Hilbert queue, okay, and then take the induced metric. Also, a metric space is always para-compact, okay. Because this is not a standard that we shall directly prove that a manifold is para-compact. Instead of proving all this locally compact Osdorff's space is metrizable and so on. If you have gone through those topological, in your topological course, then there is no problem, okay. So here we will see that just assume 1, 2, 3, namely locally Euclidean, second countable and Osdorff's space, we will show that it is para-compact or more suitably what we are interested in that it admits partition affinity. So this proof will be much simpler than the proof of partition affinity for CWD complexes, okay. So that is one of the aim. So that we will do next time. Clearly the differential manifolds inside Euclidean spaces, okay, that you may have studied in your calculus courses like study-wise, studying Stokes' theorem and so on. They are topological manifolds in this sense, okay. Each point has a neighborhood which is difumorific to some open subsets of air in there. So since difumorisms are homomorphisms, automatically our conditions are satisfied, okay. The boundary of the unit square I cross I, okay, and the boundary I am taking as subspace of R2 is a topological one-dimensional manifold, it is a simple example, okay. R itself, Rn itself, they are all manifolds, R is one-dimensional manifold, R2 is one-dimensional manifold, any open subset is a one-dimensional manifold, sorry, any open subset of Rn is an n-dimensional manifold and so on, okay. So those things are easy examples. Other than this, if you get out of R, okay, the circle, okay, is a nice smooth manifold, what I call differential manifold, one-dimensional manifold, okay. But here I cross I, okay, it has corners, yet this is a topological manifold, okay, one-dimensional topological manifold. You may have learned that this is not a smooth sub-manifold of R2 because there are corners, so I just wanted to give this example just to contrast between topological manifolds and smooth manifolds, take the union of two-axis, x-axis and y-axis in R2, okay. This is not a manifold, if it were a manifold, it would have been a one-dimensional manifold, right, because you take any point on R, that is a neighborhood which is homomorphic to the interval, right, except at the point of intersection of these two axis, namely at 0, where I have a problem, everywhere else there is no problem. But now if you take a neighborhood of this origin, okay, so you get x there, a cross, right, no matter how small your neighborhood is, there will be always x, and if you remove the point, then there will be four components, an interval when you remove a point will have at most two components, therefore there cannot be any homomorphism of a neighborhood of this point origin to an interval, okay. So this shows that whenever you have this kind of forking, that kind of graph, they are not manifolds, right, the union of two-axis, any two intersecting lines, they are not parallel. Parallel lines of course, they are manifolds, of course they will not be connected, we have put the connectivity condition here, okay. So here is an example of a locally Euclidean space, which is second countable also, but not half dot, it is very easy to construct such things, namely all that you have to do is take two lines, okay, identify all the points from one line to the other line, origin here and origin there stand out separately, so that is called a line with a double origin. What happens in this, take say the first origin, let us call it as zero prime, the second one is zero double prime, whatever, a neighborhood of zero double prime will be an open interval, that open interval will intersect every open interval containing the other origin, origin and double prime, so it cannot be house dwarf, okay, all other conditions are satisfied here, okay. So you can do many such simple examples wherein house dwarfness does not follow from the local Euclideanness, so it has to be put there, so that is why we have taken care to put the house dwarfness condition, okay. So I have written complete detail here, what is the quotient topology, you take two disjoint union, disjoint union of two lines and then make this identification take the quotient topology, so that is one way of doing it or you can directly declare what are the opens of sets and so on and do all these things, whichever way you like. So I have compared zero and zero twiddler to neighbor two distinct origins except that there is only one line, okay, one single line except the origin is has pattern double, you can make a triple origin thing also like this, but the purpose is perhaps this one is not to study and what kind of things are there and so on, just to illustrate the fact that house dwarfness is not a consequence of the local Euclidean spaces and we are not interested in this kind of spaces, okay, we do not call them manifolds. Similarly, there is something called long line, it has all the features of the real line, but it is too long to be second countable, to understand this one what you need is to understand the ordinal topology, ordinal numbers, ordinals are first of all a well-ordered set, okay and then you take the first uncountable ordinal and stop there, then what you do for each ordinal, you take ordinal and open in a closed interval 0, 1, okay, so what you are doing is ordinals have a directed set has a, you know, it is totally ordinal set, whenever you have an order there is a topology, so you have fixed the topology on the set of ordinals, then you take product with closed interval 0, 1, okay, that is set theoretically, but you have to give a topology to, this topology is not the product topology, okay, so what you are doing is you are giving the lexicographic topology, lexicographic ordering on this one, namely an ordinal, t and another ordinal, t prime, one is less than the other, if ordinal is less than go to the other one, less than other one, if they are equal then t must be less than t prime, so that is the lexicographic order, right, so you do that, okay and identify the top point say this ordinal, comma, 1 with the, its ordinal plus 1, namely the successor, comma 0, okay, if you do not like that you can just take the half closed interval to 1 open, closed interval 0 to 1 open and do not take the end point at all, so that way you can do, okay, so that is what I have done here, omega naught cross 0, 1, 0, 1, on this one you put the lexicographic ordering, alpha t is less than or equal to beta s, I am defining this, if alpha is less than beta fine, if alpha is equal to beta then t must be less than s, this is the order, okay, this will become again a total order and give the ordered topology and that space can easily seem to be a locally Euclidean space of dimension 1 because of these intervals, okay, it is not second countable, that non-second countability follows by the ordinal topology, okay, we have no time to recall the ordinal topology and so on, so you have to assume that one, if you want to read more about it, well there are lot of sources, one source is K. D. Joshi's book on point set topology, okay, this space though it is one dimensional manifold and it is connected also, it is not path connected, there are strange kinds of things happening, okay, it is locally with a few morphotons but X does not have a countable base, it is also not path connected, okay, very strange thing. Another type of non-example is obtained by taking disjoint union of manifolds of different dimension, take a circle and take a discrete point, like we can do that kind of things in a CW complex but this will not be a manifold, take a circle and take a disjoint sphere, here you have to take one, there you have to take two, so the integer changes, so that is not allowed, so that is also not a manifold, okay. So we have quite a few elementary kind of examples and non-examples but now let us come to more serious and more useful and central examples and the first one is of course spheres, okay, parabolas, hyperbolas, all conic sections in the, they are a one dimension manifold, spheres of all dimension, they are also manifold, toerai, product of those things, all these things are standard, right. The one important thing is the projective space which does not come as a sub manifold of R n, it does not occur but it is finally a sub manifold, that is a definition, it is not because it is occurring as a quotient space, right. A real projective space, the foremost one is the n-dimension real projective space, okay, you can take the complex projective space also exactly similarly, it is a quotient space of, you can take it as quotient of Sn by antipodal action, okay, X is identified as minus X. For more details of this one as we have done, like actually you can take R n plus 1 minus 0 and then identify, have an action of, you know, by scalar multiplication and so on. That is another picture of this one which will give you that projective space is actually the set of lines inside R n, R n plus 1 passing through the origin, okay. So, given X belong to S, consider V to be the set of all points in Sn that are at a less distance less than square root of 2 from X. Take a point, look at a ball of radius square root of 2 around that point and intersect with the sphere, okay. If you restrict the quotient map to that one, that will be homomorphism, there is no identifications there, okay. And this open subset is definitely an element of, it is homomorphic to R n, some, some open subset in R n. So, that will tell you that the projective space is a manifold, okay. The only thing that you have to verify is that it is Hausdorff. The point is that this being a compact space, every compact space is second accountable or automatically, okay. So, why this is Hausdorff? That also will follow, take any two points, depend upon their distance, okay. You can choose epsilon neighborhood around that point such that X, Y minus X minus Y, all these four points are separated by an epsilon neighborhood. They are all disjoint, okay. Now, when you go to the quotient, you will see that these two open subset will not intersect. So, there will be copies of these open sets which are disjoint. So, that is what you have, B epsilon plus minus X, intersection B epsilon plus minus Y is empty. So, you can choose such an epsilon, okay. If you draw a picture of S1 and S2 and so on, it is obvious, but there is no need for pictures here. You can argue logically, depending upon starting with X and Y different points, you look at their distance. Choose an epsilon and say those distances are different. Automatically, this will happen. Now, I want to give you some more theoretical examples, namely, when you take a covering projection, okay, I would like to have a picture like this. If X is a manifold and X bar to X is a covering projection, X bar will be also a manifold. Or the other way around, if X bar is a manifold, X will be on it, okay. And if you anticipate it, it happens to be true also, okay, but we have to be careful here. So, let us take, first of all, connected, okay. Let us not bother about too many connected components. Take a connected manifold and take a connected covering space. Since P is a local homomorphism, it follows that X twiddly is locally Euclidean. If and only if X is locally Euclidean. So, if you start with the potoperic homomorphism, X twiddly will be automatically locally Euclidean, okay. And also, we have seen in the first part that if X is horsedor and every covering, covering space over it will be also horsedor. This is not a very difficult exercise, okay. The only problem is why X twiddly is second countable. If X twiddly is second countable, being a quotient of X twiddly, X will be second countable. That is not the difference. But the converse, it is not clear why it should be second countable, okay. So, I said second countability is very easy to handle, but now we have a problem here. If you try to do metrizability or paracomposites, you will have more problems, okay. So, problem is there. How to prove that a connected covering space of a topological manifold is second countable. In fact, it will not be true if this covering is uncountable. Why? Because if you have uncountable covering, means what? The number of sheets is uncountable, means what? The inverse image of a point is an uncountable discrete spec. If you have an uncountable discrete subset of a topological space, it cannot be second countable. So, if I am hoping that the covering space is a manifold, then the fiber, the number of sheets must be countable. Finite is okay. Otherwise, it must be countable. What is the fiber to do when you have connected covering space and so on? The number of points in the fiber is closely related to something, what? Namely, the cardinality of the fundamental group. It will be a quotient of that. Therefore, if the cardinality of the fundamental group is uncountable, then you can take the simply connected covering that will not be second countable. Luckily, we can prove that the fundamental group of a topological manifold is countable. So, that will be one of the central results here. So, once we prove that one, it will follow that the covering spaces of a topological manifold is a manifold. So, this is the fundamental problem. Is the fundamental group of a topological manifold countable? The answer turns out to be yes and that is one of the results here. There are other kinds of problems with covering spaces like if you have differentiable manifold below, whether it will be all those things are easy to handle. This topological aspect was a more difficult one. Automatically, if a manifold exceeds C infinity, the covering space will be C infinity. If it is an analytic manifold, the covering space will be analytic. If it is a complex manifold, the covering will be a complex manifold and so on. If it is a leak group, the covering will be a leak group. So, those things are easier. Whereas, topological aspect is near some proof. Alright. So, I think the Paracom Patness is the first thing that we want to prove. We will do it next time. Thank you.