 So, we begin with a new chapter homology groups. This is one single chapter but for the reasons of proper presentation in the slides in this Beamer presentation, we will divide it in two parts. First part is just mostly algebra and the definition and basic property of singular homology. In the second part, we will compare it with other homology theories and some assorted topics, applications and such things. Homology as well as homology, they go hand in hand like many other branches of modern mathematics, they have their roots in analysis, more precisely complex analysis of one variable. When you started integration theory of line integral that is of complex valued functions, you find on parameterized piecewise differentiable curve, you realize that the integration does not really require continuity of this function all over. So, piecewise continuity is good enough but differentiable is necessary there. That idea of finite limited discontinuity is used as to what are called as later chains and inside a certain domain, if you look at all the chains which has property that their integrations take the same value, when integration is of a complex rate function, those such chains are defined as homologous. So, that word homologous was there which gives rise to the general theory of homology. Something which is happening inside the domain of the form of function. So, the topological aspect of that is brought out by what is called as homology. The integration itself gives rise to homology on the other hand. Integration of the forms, first this where you can think of this integration of functions as integrations of one forms there on curves, then two forms on surfaces, three forms on cubical things and so on. So, that was that gave rise to what is called as homology. The very first of that kind was Darang's homology. Now, for us we would like to give a more elaborate treatment of these things. So, it is better to separate them out in the beginning homology first and homology later. So, that is what we are going to do. Unfortunately, because of the time constraint, we will never come to any serious study of homology in this course. So, we will just end up studying homology a little more elaborately than what you might have met in analysis or elsewhere. So, as I told you homology, homology at least have the rules in the analysis, but it took several years of various and various people namely Poincare for example, he was the one who came up with the homology of you know what are called as piecewise PL spaces or slightly more general than what are called as simple spatial complexes. So, that was the kind of things one could try to understand in the beginning. So, after that it took several years to actually establish the homology as well as homology theory as as such the way we see it now. So, that can be attributed to Eilenberg and Stainrod around 1952. So, I repeat once again that we would like to keep the homology a little away right in the beginning, though they seem to be more natural in some sense and more structural. Homology is you know on the other hand easy to understand and as far as we are concerned that itself is a full time job here. So, we will begin with the unavoidable kind of algebra that is needed for us. It is definitely unavoidable whether you like it or not that is why one thing which we do is we will keep the algebra to a minimal to begin with and we will keep picking it up as and when we need more and more algebra ok. So, right now what we would like to do is modules over commutative rings, but if that is too much we would just restrict ourselves to just what are called as principal ideal domains. If even that is a bit out of the way for you all that you can do is this principal ideal domain whatever it just think of this as the ring of integer and then the modules are nothing but abelian groups. However, I will give you the treatment just I will keep saying modules over the ring R ok. So, you can just keep thinking about R being integers and this module is nothing but an abelian group ok. So, that is the starting point later on we will see whether we need more algebra. So, the first thing is that we have bunch of abelian groups or modules this bunch has been indexed by integers and then we will take the direct sum of all these infinitely many abelian groups. So, C n denotes one single abelian group there indexed over n indexed over integers and then I have taken direct sum and that I will denote either C suffix star or C dot because both these notations are used in the literature it is better to get familiar with both of them ok. The nth group here will be called graded component graded nth component. So, such a thing is called a gradation also a graded module we can call it as graded module also ok. And elements of C n are also called as homogeneous elements of degree n C star can still be just think thought of as just one single abelian group ok. It is an abelian group it is a module over R whatever you want to say, but there are submodules namely all these C n's are submodules of it and C is made up of these copies in a very nice way name it is a direct sum. So, copies these subgroups C n's are called graded components. So, these are just terminologies there is nothing more deeper there, but then I want to take R module homomorphisms here between two such graded modules C star to C star prime. So, what is that? This is a homomorphism this is a R module linear map, but I may insist on some more properties namely suppose this has C n going into C n plus d for every n where d is a fixed integer then I would like to call it as graded homomorphism of degree d. So, C of R is contained this is C prime of R plus 1 that is what R plus d is. So, this will be graded homogenous of degree d ok. Now, if you compose a degree d map with a degree d prime map what you will get is degree d plus d prime map ok. If you fix the degree then it will not form a category you can compose it will the thing will jump unless d is 0. So, degree 0 means C n goes to C n prime that is very nice it has a special significance for us because then you will think of the graded modules as a what along with these homomorphisms as a category because composite of 2 degree 0 homomorphisms will be degree 0 ok. So, that is of some importance for us. So, now the next thing is we want to put some more structure on these graded modules. So, the structure comes by another morphism Daba. This Daba what is called as the boundary operator this is a self operator C to C C to C star to C star, but this time it is a degree minus 1. So, degree is minus 1 and it has one extra property namely Daba composite Daba must be identically 0 ok. It just means that Daba is a homomorphism of graded modules of degree minus 1 just means that for each C n you have a homomorphism here Daba n you can say you can if you have time you can write as a Daba n otherwise there is no need to write it. That will take C n to C n minus 1 then composite with Daba n minus 1 from C n minus 1 to C n minus 2 that composite is 0 for every n that is the meaning of that Daba square is 0. Such a structure with such a structure a graded module will be called a chain complex ok. So, usually classically even now they were writing it as C n arrow C n minus 1 arrow C n minus 2 dot dot dot that look like a chain. So, the name chain complex is stuck to it. It is nothing special about this one chain. A chain was in as we have seen a chain of edges etcetera was one edge followed by another edge starting at the end point and another as they are starting at new points on. So, it does look like that. So, this were a calls of chain complex. The endomorphism Daba is called differential or the boundary operator. So, there are both the names are there ok. If C star and C star prime are two chain complexes then a chain map F from F star from C star to C star prime we mean a graded module homomorphism of degree 0 that much we have already done. But because of this extra structure on both C star and C star prime which I do not write I just say it is chain map it is just like a now it is not just a set, but it is a topological space it is like that. So, C star to C star prime there is a Daba here there is a Daba prime here let us say it should commute with that which just means that for every n Daba prime n composite F n is F n minus 1 composite Daba n. You can just write it as saying that Daba prime composite F equal to F composite Daba one single without writing any n it does make sense because all these things are fixed and F is of degree 0 and Daba is of degree minus 1 from that without writing n also all these suffices it makes sense the important thing here is now now we have a sub module a sub category of the graded chain common graded morphisms namely their chain complexes the chain complexes along with the degree 0 morphisms forms a category okay all that I have to say is that the composite of the standard composite of two morphisms is again a morphism in the in this category okay. So, this category we can write as chain chains over R so CHR we can just write it as CH when the ring R is understood to be the integers okay we rarely use this terminology but I have just introduced this if you like such an important category you can have a notation that is all there is an obvious way to define several of the operations which are there for abelian groups or for modules like you can take direct sum of two modules direct product also you want to take various things one can take one take home of of modules and so on so one single sub thing is the direct sum of family of chain complexes so now I am indexing them with a superscript alpha and alpha in lambda lambda is indexing set each of them is a chain complex the graded module is taken to be the direct sum of these things which just means that the nth component is the ordinary direct sum of the abelian groups of all the nth components the corresponding nth components n is fixed over all alpha okay and then what is this daba daba is also direct sum of daba alphas at the end stage okay for instance let us just take c1 c upper one and c upper two with daba upper one daba upper two okay nth component will be cn1 direct sum cn2 for every n take an element here take an element here that is c1 direct sum c2 is a generic element of this direct product daba of that is daba one of c1 plus daba two of c2 this plus is direct sum because it has to land into a direct sum of of c1 prime and c2 primes okay of c1 and c2 so daba is defined like this okay so it is straight forward to again when you apply daba compose daba this will be daba this will be daba this will be daba this will be daba so this is again okay so so this is the way direct sums are defined anyway a sequence of r modules I am imitating everything whatever we have for r modules you can imitate them for chain complexes now a sequence of r modules if you do not know this name is called exact so m prime to m to m double prime so there are only three terms here okay it is exact set to be exact if beta composite this alpha composite beta is zero but not only that the kernel of beta should be equal to image of alpha the kernel is equal to the image then we call it as exact okay this word exact forms are this word exact is actually borrowed from calculus exact differentials exact equations and so on okay suppose now we have a sequence of several terms a chain like that it may not be a chain but we have sequence of several terms okay and homomorphisms are there module homomorphism this should be said exact if it is exact at each mn the exact at here this was exact at m so here this will be exact at each mn then the whole thing will be called an exact chain exact sequence okay the chain complex is not an exact sequence it is just false sort of that what is it it is composites are zero which means the image of the previous arrow is contained in the kernel of the next arrow if it is equal then it is called exact okay so exactness is much more stronger than being a chain complex now we can talk about these are the terms that I am just recalling you may be familiar with them take a sequence of r modules like this there are five terms the two terms starting and the ending are zero so essentially there are only three terms okay a short such a thing is called a short exact sequence okay the short exact sequence why I would say it is this one only three terms and nothing is told about the and other other ends but usually the name for short exact sequence is this five-term sequence which the end points are end modules are zero okay suppose this is an exact sequence that means kernel of beta is equal to image of alpha it is called a split exact sequence sorry not only that beta to m double prime to zero means is this exact means what kernel of this is the whole m m double prime image of beta is equal to m double prime means beta is surjective similarly this just means that alpha is injective okay alpha is injective image alpha equal to kernel beta and beta is surjective that is the data here this is an exact sequence it is called split exact sequence if you have a map from m double prime to m such that say that is called that s that is called splitting of beta that means first s then followed with beta is the identity of m double prime okay so beta is a right inverse so that I think or such a thing is called a split exact sequence I am just recalling this if you don't know this you will have to work out these things they are not very difficult once you have that you can have a map from say tau or something is here s there may be r from m to m prime say r here such that alpha composite r is identity of m prime the third thing is that m you can write as the direct sum of a copy of this alpha of m prime here m alpha of m prime sub module and s of m double prime which is another sub module so alpha of m prime plus s of m double prime they will give you the entire m and they will be intersection will be 0 so they will be direct sum m can be written as a direct sum of these two sub modules so having a splitting is equivalent to all this okay so you can you can start with a direct sum of m prime and m double prime take the inclusion here and take the quotient map here that will give you short exact sequence giving a short exact sequence is not the same as giving a direct sum because direct sum requires one more name which there must be splitting okay so last condition is the same as saying that there is an earlier map tau t t composite alpha is m prime that is what i i explained here okay and also it is alpha m prime plus s of double prime this direct sum if you put instead of m you put this this thing here and take the map alpha and beta here that will be exact sequence so direct sum of two things we always gives you short exact sequence but not commercially okay now we want to imitate that for chain complexes now suppose we have short exact sequence of chain complexes where f and gr of degree zero morphisms okay unless stated otherwise chain maps will be always degree zero okay because that is the one which forms a category all right so take c dot c prime c double prime short exact sequence like this this will be called exact if for each end there is a splitting so this is a slightly unconventional definition but this is the definition i would like to say that you are you are having a category of modules and you would like to have a splitting which is also categorical no but this word splitting exact or even sorry just i will first we are talking about the exactness the splitting comes here exactness for each end it is exact that is that is fine okay that is easy but it is called split if for each end this is split this splitting path you know what you demanded from cn uh cn double prime to c you have a map morphism this morphism may not commute with a boundary operator that is possible okay if it commutes with boundary operator then it would have become the categorical splitting namely of in the category of chain complexes okay so that is the difference so splitting always this definition is taken a weaker definition here because what happens is in practice that is what you get and that itself is useful to some extent so there is a subcategory of short exact sequences of chain complexes for modules of a of category of all chain complexes this is a full subcategory okay defined in an obvious way this these will be used to split up the study of longer chain complexes suppose you have a chain complex it is very long you can always you know rewrite these terms into a large number of short exact sequences and then study each of them we will come to that later on how to do that and so on okay I just want to at this stage I just want to say why the short exact sequences are considered okay so these are a stopgap it is like inductive step one at a time kind of thing okay since you can't take isolated single module there you have to take the previous one and the the follow one so that is the that is the minimal thing you have to take that is the best thing you can do so one by one we will study short exact sequences a lot of them and that will give you information on chain complexes so that is the idea okay now I want to introduce maybe I should stop here if it is time for today we will take this one next time okay thank you