 Hello everybody, welcome to the course on Algebraic Topology Part 2 on the NPTEL portal. I am Anand R. Shastri, Retired Emeritus Fellow, Department of Mathematics, IIT Bombay. My teammates are Piyanka Magar from IIT Bombay, Ankur Sarkar, IMSC, Vinay Sipani, Sagar Sawan, Ilan Shiran, Kaninathan, all from IIT M. This course will be presented to you in 60 modules of approximately 30 minutes each over a period of 12 weeks. This is a sequel to a similar course I have given on this very portal, namely Algebraic Topology Part 1. Therefore, it assumes that the learner has attended that course or has gone through the material independently and understands them or has acquired familiarity with contents of that course through other books or other courses or something. In particular, we presume that the learner is familiar with a good amount of points at topology that has reached a certain level of mathematical maturity so that he can attend this course comfortably. You are asked to take this course especially if you have done part 1 so that you will arrive within Algebraic Topology somewhere. The aim of this course is quite modest. We shall carry on from wherever we ended in part 1 and proceed to impart some basic knowledge of Algebraic Topology. The topics covered here are broadly classified under five main chapters, speed of the complexes, categories and functors, homological algebra and singular homology groups, other homology groups and the last chapter is Topology of Manifolds. While doing homology groups, we postpone a lot of proofs in order to concentrate on concepts and applications. For completeness, all these proofs are then collected together in a separate chapter called Assorted Topics. Some of the salient topics covered in this course and somewhat rare to find elsewhere are full discussion on CW complexes, especially product of CW complexes, partition of infinity on CW complexes, CW homology, computation of homology groups of length spaces, hands on proof of equivalence of various homology groups and topological classification of Manifolds of dimension less than or equal to 2. Basic reference is my own book, CRC Press, Boca Raton in which you can find other references for the duration of the course. You may stick to the notes.pdf that will be made available to you on the NPTEL website. The notes.pdf will have everything that is covered in these lectures and a full reference to further study. We hope you will find this course useful for you. A comprehensive bibliography is also included at the end of this module for your radio reference. Throughout these lectures, we shall use the word space to mean a topological space. Similarly, we shall use the word map to mean a continuous function between topological spaces. Here is a list of standard notations followed throughout this course. Let us get these Euler fonts, all these Euler fonts, real numbers, complex numbers, space of rational numbers, integer set of rational numbers and so on. I for always closed in terms of dn closed unit space in Rn, Sn with closed units sphere in Rn plus 1, Pn is n-dimensional real projective space and Cpn is n-dimensional complex projective space. As a ready reference for you, I will now recall some of the main results that we have proved in part 1 and are going to use them in this course. In the brackets, I have included the module numbers where they appear in part 1. So module number 2, theorem 1.1 in part 1 is the following theorem. So I am just recalling them. The following conditions on a space X are equivalent. X is homotopy equivalent to a singleton space that is X is contractable, that is the definition of contractability, the identity map of X is null homotopy. So this can also be taken as definition because they are all equivalent for every space Y, every map H from Y to X is null homotopy. So every space Z and every map H from X to Z is also null homotopy. So these four conditions are very useful to whichever way you want, you can use them when X is contractable, that is the point. Then we have this degree of a map from S1 to S1 which classifies the fundamental group of S1. The function degree from pi 1 of S1, 1 result is an isomorphism. So then there is the simplest version of Van Kampans theorem, module 10, theorem 2.4 in part 1. Let X be union of two open subsets U and V and U intersection V is path connected. Suppose further that for some X0, that is the base point in U intersection V, the inclusion map eta from U to X, free from V to X both induce homomorphisms, eta check and phi check which are trivial, then pi 1 of X, X0 itself is trivial. So the emphasis is here, the maps are actual trivial homomorphisms, this will happen if pi 1 of U, X0 is selfish trivial and pi 1 of U, X0 is trivial, that is a special case. So the next theorem that result is module 16, theorem 3.6, any topological space X is contractable if and only if it is retract of the cone, module 20 producing 4.1, let A be any close of space of X, then XA has homotopy extension property with respect to every space, that is A to X a co-vibration, this is the definition of co-vibration, if and only if the subspace Z which is A cross I union X cross 0 is a retract of X, it is a very useful result, module 22 theorem 4.4, now let A to X be a co-vibration, we have defined here for the co-vibration that is it has homotopy extension property with respect to any space, let me do a map here. Then there are 3 statements here, A is a weak deformation retract of X, if and only if it is a deformation retract of X, if A is contractable then the quotient map Q to Q from X to X by A where A is collapsed to a single point, that quotient map Q from X to X by A is a homotopy balance, if A naught to A is an inclusion map that is single point and that is a strong deformation retract, that means not only A is contractable A naught is a strong deformation retract that is a stronger thing, then the inclusion map X A naught to X A of the pairs is a homotopy co-balance, as a pair, module 22 again the theorem 4.6, suppose X 2 Z is a co-vibration where X is a close subspace, if we have 2 functions F and G from X to Y which are homotopy then the adjunctions pair, adjunctions space is AF and AG, Y is a subspace of that, AF Y and AG Y as the colloquial pairs are homotopy co-balance. This tells you that if you have a co-vibration a smaller subspace inclusion makes co-vibration over that if you perform adjunction space constructions then that adjunction space is AF and AG are homotopy equivalent provided F and G are homotopy homotopy to each other. Now, module 33 here will come to simbial complexes, if K is a finite simbial complex and U is an open covering of X then there exist n such that for all little and bigger than n the barycentric subdivision iterated n times S D composite, S D composite say n times the barycentric subdivision iterated n times of K that will be finer than U. So, this is possible if K is a finite simbial complex and you should take an open covering this will be finer than U. This finer than U I will recall namely this means that if you take in the simbial complex to take any vertex then take the star open star of the vertex that will itself in open cover that open covering is finer than U. That means each star is contained in some member of show. Once again simbial complex is only a map F from mod K 1 to mod K 2 it is a continuous function map means continuous function remember that admits simbial approximations it should not leave the simbial complex K 1 is finer than the open covering F star F inverse of star V star V and V V V range over vertices of V 2 of this K 2 that is an open cover for K 2 mod K 2 F inverse of that is an open cover of K 1 and K 1 is finer than that that means star of U for any vertex U must be contained inside F inverse of star of V for some way that is the meaning of that. If that happens then there will be or a simbial approximation to F that F from mod K 1 to K 2 be any map and K 1 is a finite then there exist an integer n such that for all n bigger than n there are simbial approximations from parasyntric subdivision iterated n times. So, this little n should be sufficiently large from these simbial approximations we have we have produced this famous theorem Brauer-Schwitzer point theorem for any integer n greater than 1 the following three statements are equivalent to each other and each of them is true. The first statement is Brauer-Schwitzer point theorem every continuous map F from dn to dn has a fixed point. Fixed point means what there is x in dn such that fx equal to x. The second statement says the boundary Sn minus 1 of dn is not retract of dn. Third statement said Sn minus 1 is not contractable. These three statements are equivalent and in the proof of that Brauer-Schwitzer point theorem we use that also and then we just prove that this Sn minus 1 is not contractable. Therefore, all these three statements are proved. For proving this one we use simbial approximation and what is called as perner So, module 62 this is almost at the end there is a lot of covering space theory here which I will recall only when I need them but right now this is one theorem to which we may have to appeal again. So, that just indicates how far we are going we have gone in covering space theory and fundamental group that x and y be connected and locally contractable spaces that f from x to y any map and cf dn the mapping cone of f that x not belong to x y not belong to y equal to f of x not. So, you have chosen base points for that x not close to y not then the inclusion is homomorphism data from pi 1 of y to pi 1 of the cf which is cf is the mapping cone over f this is surjective the kernel is the normal subgroup n generated by the image of this f check pi 1 of x to pi 1 of y f is here x to y take the normal subgroup of of generated by and generated by the image of that the image is a subgroup but may not be a normal subgroup the taken normal subgroup go more below that that will be isomorphic to the pi 1 of cf. So, this is the statement of this theorem. So, that is all for today. So, hope you will enjoy this course we we myself and my teammates are there for you to help out all your troubles all your queries use the the portals which I have been NPTEL has been provided to you carefully and usefully and you know you can learn a lot from this experience. Thank you.