 So, we have done some algebra, a preparatory work before studying the homology of topological spaces. We shall do one more algebraic preparation, though it is not necessary immediately, okay. This fits into the kind of preparation that we are making. This is, the title of this talk is about Euler characteristic which is very much topological, but here we are going to give it a completely algebraic treatment. The Euler characteristic will be discussed again and again in the, in this course. So, we are only initiating a discussion now. So, here is a definition, take the category of R modules where R is a commutative ring, preferably a PID and if you have difficulty as I have told you right in the beginning, you can just think R is a, R is the ring of integers and modules are just every universe, no problem. An additive function on this category, we mean an integral integer value set functional on the isomorphism classes of R modules, the set of R, the collection of all R modules is not a set, but isomorphism classes of R modules is a set by axiom of choice, whatever. So, on that set you take a set, a set pretty function which takes values in the actually non-negative integers, it should have the following property. What is the property? Whenever you have a short exact sequence of R modules m prime, m and m developed prime, you should have the L of m, L is the function, this L is the function, L of m must be equal to L of m prime plus L of m developed prime, so you can call it a length function which happens to be studied in algebra at various levels. What I want to say is by simple induction, any additive function will have this property, namely whenever you have a finite sequence, not necessarily short exact sequence, a finite short exact sequence m1, m2, mn, this is exact sequence, then the alternate sum of L of m i's, the L of m i I have taken L of m1 minus sign, next one plus sign, minus sign plus sign and so on, minus 1 raise to i I am taking, alternate sum that is always 0. So, this could have been here written in that form, namely L of m prime on this side minus plus L of m which is second term which is with a plus sign, again minus L of m double prime which is third term, so it should 0. So, there is no other way of writing this one either in the general case, that is the correct way to write down this additivity. So, additive functions I have this property on general things summation, alternate summation is there. So, how to prove this one? I have told you that long exact sequences like this can be always split up into short exact sequences and then whatever short exact sequence you know, you can use that inductively. So, here is a chance for that to illustrate that principle. So, let us name this map, the first map m1 to m2 as alpha 1 just for the sake of it. So, now I have m1 to m2 to m3 etc, I do not care what they are, this is the first one is alpha 1. So, what I will do? I will split up this one, I do not go directly to m3, but I go to m2 modulo image of alpha 1. So, there is a co-kernel of alpha 1. So, this map is surjective. So, if you look at the 0 to m1 to m2 to m2 by m3 to 0, this is a short exact sequence. Therefore, L of this plus L of minus minus L of this plus L of this plus L of that minus L of this is 0. So, I know this much. Now, I want to go here m2 by image of alpha 1, this map whatever this map is on image of alpha 1 it is 0. Therefore, it factors down to m2 by L r and gives you this map here and that map because this sequence is suppose this is alpha 2 now just for cycle. Then, kernel of alpha 2 is precisely equal to alpha 1, I am killing the kernel of alpha 1 which is its immediate alpha 1 therefore, it becomes injective. This means that at this point it is exact from here onwards the image of this alpha 2 is same thing as image of this alpha 2 bar whatever beyond that I have not changed. Now, look at the length of this sequence it has come down by 1. So, this you can call it as m2 m1 is absent now 0 to m2 to m3 and so on. So, apply induction on this one, alternate sum of this one will be 0. Add this one you will what you get is precisely the statement of this theorem alternate sum of this one. So, this term cancels out the extra term that you have got is then take alternate sum it will cancel out. So, an additive function has this property whenever you have finitely many terms and exact sequence then this alternate sum is 0. So, this is this should be a very good measure to measure the deviation of a chain complex from being in exact sequence. You see at least in the case when you have only finitely many terms to deal with. So, we go towards that namely on the category of finitely generated R modules over a ring R that R must be a principal ideal domain then there is a structure theorem just like in the case of Abelian groups finitely generated Abelian groups modules over the integers. Every module finite generated module can be written as direct sum of its torsion module which corresponds to elements of finite order direct sum with a free module which is of finite rank. So, for every finitely generated R module over a ring the rank function is one of the most important functions and that function is additive function direct sum of two modules when you take the rank it will be the sum of these two that is why it is called additive from that you can produce a little more namely if you have exact short exact sequence of R modules okay over a PID the rank will be such that alternate sum of the ranks is 0. The simplest case is when R is a field then what are finitely generated modules over a field they are vector spaces. So, if you have short exact sequence of vector spaces the M1 to M2 to M3 the rank nullity theorem is nothing but the dimensions are additive functions namely alternate sum of them is 0. So, this is a very nice name this is called rank nullity theorem rank of a linear map is the image of just the image sorry is the dimension of the image and nullity is the dimension of the kernel. So, this is precisely what it is if you have three terms if you take this map alpha here M double prime is the image of alpha and M prime is a kernel. So, if you put L as the dimension of these things then L of M is something L of M prime plus L of M that is a rank nullity theorem the same thing as dimension is an additive function okay. So, this dimension is called rank in the case of arbitrary ranks for arbitrary ranks of commutative ranks this is a difficult notion sometimes the rank may not be defined properly okay in general it is not defined alright, but for a PID it is well defined there are other cases also wherein it is defined not necessarily PID, but we do not want to get into that kind of algebra here okay. So, let L be an additive function on some category F of R module a chain complex C dot of R module is said to be finite type finite type means what for a chain complex finite type with respect to L L is fixed if all CN's are objects in this F okay some sub some sub some sub category of R module I have taken and L of CN must be 0 for almost all n that means only finite linear terms are non-zero. So, I am cooking up this one because I want to take alternate some if it is infinite some it will not make sense okay they are integers there is no question of convergence convergence means that after finite state it must be 0 okay. So, that is what I have put in this condition L of CN must be 0 for almost all means what finite limiting except for finite limiting it must be 0 in that case alternate some is defined that is the whole idea okay. So, such a C will be called finite type with respect to L if you change L this may not be finite type okay the given C may not be finite type with respect to one L and with respect to some other L some other edge function it may be so there are such cases also you have to be careful about that. So, what is of very prime importance for us is if R is a PID and L is the rank function okay this is the same as saying that all CN's are of finite rank and most of them have rank equal to 0 because I am taking L as a rank function now. We then merely refer to C dot as finite type when you say it is finite type we are interested in R as a PID and L is the rank function in particular R could be integers and and this R is what rank L is rank. Then we do not say rank with finite type with respect to what and so on finite type means finite type otherwise I have to mention what is this L okay. So, this is just a convention thus for example when R is Z a finite type chain complex of abelian rules need not be finite type with respect to some other additive function other than the rank function. We can give you other additive functions also you can think about it now here is the theorem that we are interested in okay what is this the rank function defined for the chain complex now gives you something interesting on the homology okay. Let C dot be a chain complex of R modules of finite type with respect to an additive function L then the alternate sum of L n for L L of C n is alternate sum of L of H n of C okay indexing must be carefully chosen okay you cannot change the indexing otherwise there will be a sign change here if you change the index just shifting one by one or something then the whole thing will change sign right so you have to be careful about so from the chain complex from the homology you can go if the chain complex is actually an exact one then what happens here alternate sum would have been 0 so that will say that alternate sum of all the homologies 0 is a very weak statement you know that if C n is exact all the h n's are 0 therefore L of h n's will be also 0 an additive function will always take 0 on the 0 module that is easy to verify okay so minus 1 power n summation will be 0 will be term will be 0 so this is a very weak thing but it's very very useful okay it's a much weaker statement but it's very very useful okay so here is a proof once again splitting you know long thing the whole C n I have to do but we will do it by splitting and then some kind of induction okay for each end we have a short exact step in concentrated C n some function some some operator comes here some operator leaves here okay so C n 2 C n minus 1 I have daba n don't take the whole of C n minus 1 but take just the image here then you get a surjective map okay this C n 2 image of daba n this map itself is daba n and that is a surjective map what is the kernel kernel of daba n and that is an injective map so you have a short exact sequence like this okay therefore L of C n must be L of daba n plus L of image of daba n L of kernel of daba n plus L of image of daba n okay now what have definition of h n of C star what is it it is kernel of daba n divided by image of daba n plus 1 so you have an exact sequence there namely image of daba n plus 1 to kernel of daba n and then the quotient is h n of C n therefore L of this big middle thing is equal to L of this one plus L of this one right now substitute this in the left hand side of the equation here for h n h n it's all this okay then you can rewrite this then you get this one alternatively what you can do is you can write this minus this plus this is 0 sorry yeah minus this plus this minus is 0 and then take the summation then here also take that take the summation so minus 1 raised to n L of image of daba n is equal to minus of minus 1 raised to n L of image of daba n plus 1 so plus 1 okay see this plus 1 if you shift it through n then this will be this minus sign will come out so you should cancel out that's why these two are equal right that's all you have to say the same terms I have shifted so that's a minus sign so use this these two terms on the left hand right hand side they will cancel out so what I am done substitute for h n of this one by this this thing and then use this property okay so additive functions from chain complex to homology it works namely they will be additive functions of homology also so now we make a definition with this great observation okay let RBA PID M a graded module of finite type I recall what is the meaning of this namely the most of them okay L of that the rank of them are 0 and finite many of them are there and they are of rank finite okay that's the meaning of finite type with respect to this one this implies that the rank of m n is finite for all n and 0 except for finite community okay that is the that's the recalling what is the meaning of finite type we define the Euler characteristic by the formula this is the standard notation chi m okay m is a graded module now not just one single model chi m is I range from minus into plus in material but it is a finite sum okay minus 1 raised to i the rank of m i okay this is special case not no l l is replaced by the rank rank function rank of m i where rank denotes the rank function so this is called the Euler characteristic you could have defined chi l also with respect to l then you take minus 1 raised to i l of i m so you can call that as a Euler characteristic with respect to l but without any qualifier Euler characteristic just means minus 1 to the i summation rank of m i s so you have this whatever you have proved it we have this one namely if she is a finite type chain complex of R modules over a PID then the Euler characteristic of the homology is equal to Euler characteristic of the chain complex itself so this is the algebra that we needed later on so we have established that one the above theorem tells that the Euler characteristic of a finitely generated chain complex is the same as the Euler characteristic of its homology groups in our context R is either the ring of integers or a field we could have taken R C Q we do that kind of thing then all these things would be just the dimension of those vector spaces which is same thing as a rank of S model so it should not cause you any worry even if you do not know what are PIDs and what are models for PID the notion of Euler characteristic plays a very important role in the development of algebraic topology differential geometry and etc it manifests itself in a variety of way from the simple observation namely number of vertices minus number of edges plus number of faces equal to two this is the so-called the famous formula of Euler for what for a planar graph number of vertices minus number of edges plus the number of domains faces is equal to always equal to two so this was the observation of Euler we can say Euler was the great grandfather of topology but this simple thing has now become so great you know it manifests in so many other ways there are at least half a dozen different definitions and then you prove for okay this is equal to that that is equal to this this is equal to that that is equal to this and so on at various places and then you take one of them as generalize it and so on so you get one RTS in the index theorem of elliptic operators and all sort of cell work trace formula so many other things they are all interrelated so you take one aspect of which generalize it and do something and so on so this is this has created just one single thing of Euler has created a lot of mathematics so let us do one more slight variation of this one if not all that iterations are and so on so these are called less than number this will also be used in the sequel in this course okay so we fixed ourselves with PID R is a PID M is finitely generated module over R this Tor M denotes the submodule of all torsion elements in M namely R times X equal to 0 for some R non-zero then X is called torsion element it is just like finite elements of finite order n times X is 0 means what X is the finite order right in in an abelian group so that is generated that such things are called torsion instead of finite elements they are not finite they are not called finite order but torsion element this Tor M denotes the submodule of all torsion elements the quotient module M by Tor M is a free module of finite rank so this is not trivial result it is true for PIDs okay which is equal to the rank of M itself the rank of M is the same thing as the rank of the the free part for any endomorphism from M to M there is an induced endomorphism F bar from M by Tor M to M by Tor M because torsion element always goes to torsion element so you can quotient out by that so you get a morphism on a free module if you fix a basis for a free module means there is always a basis right the finite basis will be there now if you fix a basis then this linear endomorphism can be written in terms of a matrix so if the rank is R then it will be an R cross R matrix okay the trace of this F is defined to be the trace of the corresponding matrix for F bar okay and one can verify that it is independent of a choice of the basis this is elementary linear algebra okay trace function of a matrix A B A inverse trace of this the same thing as trace of B because trace of A B equal to trace of B A that is what you you have verified okay so same thing works here the trace will be independent of what basis you choose okay so we can define that trace of endomorphism F itself all right now we are in business so now instead of M we are looking at endomorphisms instead of the module we are looking at endomorphisms so what we do is a simple exercise here okay similar to what we have exact sequences the additivity of the rank function rank of B was equal to rank of C plus rank of A right so we generalize this here take a exact sequence here and an endomorphism of the exact sequence okay so suppose this is a commutative diagram okay the trace of this can be defined trace of this can be defined trace of this can be defined right these are endomorphisms okay it is commutative diagram what the exercise says is trace of the center one G is equal to trace of H pressure itself okay so look out for vector spaces prove that for vector spaces okay linear algebra and proof for general case is exactly the same because you are doing matrix theory here okay that is the solution for exercise but you can write down the details next let us take see a graded module of finite type having done for short exact sequence you want to go for channel then graded module of finite type or any chain map f from C to C dot C dot same self map okay define the left change number LF as alternate sum of the traces okay this is only chain complex here it was exact sequence then this is the this is alternate sum is zero here okay further assume that C dot is a chain complex this is for graded module you can define for C dot a chain complex and f is a chain map if f star is induced on the homology okay suppose it is induced map then the claim is LF star is equal to LF exactly same as Euler characteristic of H star is same thing as Euler characteristic of C star from a chain complex level to the homology level you can come okay so try out these exercises if you do not get it we will be there to help this is this is just straightforward exercise having done it for chain complexes this is not a difficult thing okay so thank you we will stop here next time we are going to introduce the most serious part of this one namely the construction of singular homology groups thank you