 So, today we shall take care of few side remarks and certain left out things and so on. So, this is not exactly a topic module, so I have called it as assorted topics. To carry on with what we are doing, namely study of Lenspaces, let me first give you a few more information on Lenspaces. So far considered only finite dimensional Lenspaces, namely from an odd dimensional sphere we made a group action and then take the quotient, whereas we can do this in a more general fashion, namely we can keep on doing it because each time the action that we take in the next stage is an extension of the action that we take for the previous sphere. So, S3 contained inside S5, S5 contained inside S7 and so on, we have been extending the action. Therefore, you can go all the way to infinite dimension sphere, ok. So, what will be the starting data there? Instead of finitely many q1, q2, qn numbers which are co-prime to p, we have sequence of numbers which are co-prime to p, ok. Then all that you do is take this infinite sequence of complex numbers such that after a finite stage they are all 0, that is the meaning of this one, z0, z1, zn, etc. Sitting inside S infinity, thought of as the unit sphere in C infinity. Remember C infinity is nothing but the direct sum of infinitely many copies of C, namely countably many copies of C. So, zeta is a pth root of infinity operating upon an element here z0, zn, etc., will be first coordinate will be multiplied by zeta, second one by zeta power q1 and so on. Verification that this is fixed point free is to be just it is already done because what you have to do at each time you have to verify it at the finite level. So, all these verifications work out, this will be a quotient space, the quotient map will be a p fold covering from S infinity to whatever space we get we are denoting it as Lp q1, q2 dot dot dot. So, that is called infinite length space, infinite dimension length space, whatever. It has some theoretical importance, I will let you know about it a little later. Right now, for coming back to finite dimensional complex, finite length spaces, there are some easy observations you can make. Lpq, remember this, this depends upon both p and q. But there are some relations namely Lpq is homeomorphic to Lp minus q. See, we never use that qis are positive, the negative numbers are also allowed. Only thing is they should be co-prime to p. The same way you can define instead of theta raised to some power q, it is minus q no problem. So, you take the same thing. But Lpq and Lp minus q are isomorphic, are homeomorphic actually diffeomorphic. How to see this? Consider there are many different ways of seeing, I am telling you one way. Consider the map S3 to S3 given by z0, z1 going to z0, z0 raised to t times z1, where t is some integer. It is immediately a smooth map and its inverse is z0, z0 raised to minus tz1. So, it is a diffeomorphism, anyway it is a homeomorphism if you do not know what it is. Now, for a special t, namely t equal to minus 2 q, you can verify that f actually becomes equivalent map from Lpq to Lp minus q. That is the meaning of the equivalent map. You take the phiq, the q action on S3. So, compare it with f, it is same thing a phi of minus q of f. So, this S3 is equivalent of this map, which means that f boils down, factors down to a map from Lpq to Lp minus q. Exactly similarly, its inverse will also boil down. There will be inverses of each other. So, that defines a diffeomorphism of Lpq to Lp minus q. All that you have to do is you have to verify that f component phiq is phi of minus q component f, which I have done here. So, z0 z1 under phiq would go to zeta z0 zeta power q z1. Under f, it will go to zeta z0, then I have to multiply zeta z0 raised to minus q, right, times zeta q z1. So, minus q, zeta raised to minus q, z0 raised to minus, anyway. So, if there is some typos, you better change it. So, you can directly verify it. Under pf, first this goes to z0, z0 raised to minus 2, 2 times z1. And under minus q, it will go to zeta raised to minus q, zeta not raised to minus 2, 2 times z. Both of them are equal. That is what I have verified anyway. So, that is one thing. The second thing is if q and q prime are such that one is inverse of the other modulo p in the ring z by zp, these two numbers must be inverse sufficient, namely q, q prime is congruent to one mod p, okay. Then again, Lpq will be isomorphic to or homeomorphic, difumomorphic to Lpq prime. So, for this, you look at the interchange in the factor, permuting the factors, t z0, z1 equal to z1, z0. That is difumorphism. So, what we can verify is t of phiq is same thing as phiq operated q prime times composite t. Now, phiq operated q prime times or phiq operating upon any other square q, whatever it is, they will give you the same orbit space as the action of phiq because you have to take all the powers. The only thing is that the zeta that you have chosen should be primitive and through the primitive p through the infinity. Then all the powers will repeat no matter what power you take, okay. This means t and q and q prime or co-prime to one, both that is the important thing here. This means t is an equivalent difumorphism of the phiq action and phiq prime raise to q action which is equivalent to phi of q prime with the choice of the primitive root zeta raise to q instead of zeta because q is co-prime to p, zeta raise to q is also a primitive root of infinity. Hence, lpq is homomorphic to lpq prime. If you combine these two, you will get lpq prime, okay. I can put a minus sign that is also homomorphic to lpq, okay. So, second factor, second number, p is the same first number, second number could be negative or inverse of, I mean, multiplicative inverse, additive inverse or multiplicative inverse. You can both combine. So, that is what these classes will be the same of difumorphism, they are the same, okay. What rademaster has done is he has complete classification of lpq's in the three-dimensional length spaces. So, it is a very classical radar. lpq is homomorphic to lpq prime. If filled only if q plus minus q prime is 0 or q, q prime is plus minus 1 modulo p. Not only I will just treat it for q, q prime equal to 1 here, but minus 1 we will also do. So, like that we did it, see q equal to q prime, there is no problem. q equal to minus q prime we did is this part and here there is one more part and the whole thing is if filled only if, okay. If this happens, then we have more or less in the proof. The only part is beyond the scope of this course. So, that leads you to define what are called rademaster torsions and so on. So, it is a deeper theory, okay. So, similarly there is a theorem of whited which gives you homotopy classification. So, let me state that one. In any case, all these things are in a nice article by in 1973 by Cohen. So, if you are interested in you can read that one that is all. So, lpq is homotopy equivalent to lpq prime if filled only if q, q prime is a square modulo p, okay. So, homotopy equivalence automatically takes care of homeomorphisms also. So, this is a much larger result. So, this is whited result, okay. So, I will let you know about infinite lens spaces. The the strangeness with infinite lens spaces is it depends only on the first integer p. All of them whether finite or infinite, the fundamental group of the lpq, q1, q2 or whatever, they all have fundamental group isomorphism to z by pz, the cycle group of order p. That is true for infinite lens space also. But what happens is the covering the p-fold covering s infinity is contractible. Even though it is a sphere, the infinite dimension sphere is contractible. From this one one can deduce that no matter what q1, q2, qn you take all of them have the same homotopy type. Only the first p matters. lp any sequence is homotopy equivalent to lp any other sequence, okay. And these are important because they are what are called island berg-macklain spaces, model for island berg-macklain spaces of type z by pz, 1, okay. So, because these things are new to you they may be only just an information sake, okay. So, let us go to another topic here namely Euler characteristic revisited. Remember in part one we defined Euler characteristic of a finite simplicial complex. A finite simplicial complex has a finite sequence f0, f1, f2, f3, okay, etc., fn where fis are the number of i-dimensional simplexes in a k. So, this f i, f1, f2, this n-tuple, n plus 1-tuple is called the phase vector in combinatorics. What we are interested in is the summation sign which is missing here, chi x, okay, is equal to summation minus 1 to the i fi of k. This is what we have done as the Euler characteristics of a finite simplicial complex k. For the simplicial chain complex c dot of k, we have another definition of the chi of ck for any chain complex which is finitely generated. We have another definition and we have also seen that the other definition namely alternative sum of the ranks of this c dot of k is equal to the alternating sum of the ranks of the corresponding homology modules, okay, that also we have seen, okay. So, because of that and because of our study of the simplicial chain complex here, so what are the nth homology, so what are the nth chain group in c dot k, what is cnk? It is just the free abelian group over the number of n cells, so it is just snk, so fnk. Therefore, these two definitions are the same. We are going to use the simplicial chain complex to define the Euler characteristic or directly do the way we have done it in part one, okay. So, these two things are same, that is important. Since we have also proved that Euler characteristic of any chain complex is equal to Euler characteristics of its homology, okay and since this one is canonically isomorphic to H star of mod k, now you see the Euler characteristic is independent of what triangulation you take, okay. So, this leads us to define the Euler characteristic for topological spaces also of course with some restriction, okay. Anytime it has a simplicial complex structure and if it is finite, namely if you start with a compact space and then you give a simplicial structure, immediately Euler characteristic defined, no dependence on the, on the, what simplicial complex, simplicial structure you choose. Therefore, you could directly define it from alternate sum of the ranks of H star of mod k, okay, of the underlying space. Therefore, it immediately begs for a definition. So, by the way, these things were done earlier time, you know, almost even slightly before than Poincare. So, the rank of H i of X k, this becomes an important invariant, it is what beta i of X. No matter what, you know, whatever the choice of mod k or whatever choice of a triangulation, the rank of H i of X does not depend upon that. So, this I can always define, this may be finite, this may be infinite and so on. For all spaces, we can define this, beta i of X is rank of reason and this is called i th petty number of X. The alternate sum of i th petty numbers, if it is defined, that means what? If these things are all finite, then you call it as chi X. Even if one of them is infinite, then it is, this is not defined. So, because of topological invariance of homology groups, petty numbers can be computed using any triangulation of a space and the corresponding simple shell homology provide, of course, X is triangulated. We can go one more step ahead because now we know what is C w complex instead of a simple shell complex. Exactly same way we have studied the simple shell, the C w chain complex. Therefore, what we get is chi X, which is minus 1 raised to i beta i of X. This formula can be computed using any C w structure also and using the C w chain complex also. So, this is valid for finite C w complexes, which is slightly larger category than finite simple shell complexes. All these comments will apply to the left change number also. Recall what is the left change number of a continuous function from one topological space to another topological space into the same topological space, f from X to X. Now, suppose X is such that all its homology modules are finitely generated. Then take the induced homomorphisms f stars at each chi level. Look at the trace of those. These are linear maps. You can talk about the trace of this. Take the alternate sum, that is the definition of left change number. So, this can be computed either using the C w structure or using simple shell structure, etc. Provided that map f is cellular or simple shell and so on. But final result says that it is independent of all that. It can define for any map from any X to X provided X as finite regenerated homology. Okay, here is another kind of homology, cellular singular homology. We now come to this homology which is between C w homology and singular homology of the underlying space. This is similar to the case of simple shell homology and singular simple shell homology of a simple shell complex. So, take a C w complex. I am going to define a subgroup. What is this subgroup? The C cell N, like C C W N, I am writing C cell N with a free abelian group generated by the set of all continuous maps, sigma from delta n to X, which are cellular. If you take all continuous maps, you get s dot, s n. If you take only cellular, then you get C n cell. So, obviously, this free abelian group is a subgroup of s n X. This forms a subgroup of s n X and the boundary operator is taken exactly the same way as in s n X. Therefore, not only that, if you take a cellular map, boundary of a cellular map is a cellular map and that is why you can take the boundary, boundary defined as in the s n can be taken as the definition of boundary here also. Okay, what happens is if you take C dot cell N as direct sum of C n cells along with this daba, it will be a subchain complex of s dot of X. Okay, now look at what was C C w. C w of X was defined as a free abelian group only on the what are the maps? The characteristic maps, one single characteristic map will define the one single n cell. I am taking one single n cell as a generator as an element in the basis. So, you can represent it by the one single, what is this the characteristic map, which is automatically cellular. Therefore, C C w is a sub of C cell. Okay, so that is what we have. There is an inclusion, sorry, this inclusion map is a chain equivalence is a statement. Okay, but what we have is this C cell is a subgroup is a subchain complex between C C w and s x. So, the statement here is that inclusion map is a chain equivalence. Therefore, when you pass to the homology, it will be an isomorphism. Okay, as usual, we will postpone the proof of this one. Okay, so this is what I had, C C w sitting inside C C n sitting inside another group. Okay, and you can always take relative portion of these things. We are not much going to use this one here, but understanding this one will be immersed when you go to, you know, higher algebraic topology, like understanding spare, a relay spare spectral sequence and so on. The spectral sequence idea actually you know has roots in this kind of the C w complexes and this kind of substructures here on that. Okay, so let us stop here and next time we will start doing applications of homology. Thank you.