Continuing with the applications of various theoretical results that we have proved, today let me first begin with recalling the definition of Euler characteristic which we have defined for a simplicial complex. Namely, if you denote the number of vertices by Vx, number of edges by Ax and so on, in a simplicial complex we can denote f0, f1, f2, f3 and so on, the number of k simplicis by fk, then the alternating sum f0 minus f1 plus f2 minus f3, etc. was called as Euler characteristic. The exactly same way we can define it for pseudograph also, number of vertices denoted by Vx, number of edges denoted by Ex, there is no more two cells and so on in a pseudograph. So, Euler characteristic of a pseudograph is defined as Vx minus Ex. It will coincide with the definition of the ordinary pseudograph, if this pseudograph happens to be a simplicial complex, that is one dimension of the complex. Now, what it has to with whatever we have done, last time we did one important result about the fundamental group of any pseudograph. Namely, if you start the connected pseudograph, the fundamental group is a free group and its rank, the rank of free group is equal to number of edges outside any maximal tree T inside X. So, this is what we are going to apply right now and get a nice formula in terms of Euler characteristic, rate it to Euler characteristic. This is the corollary, this one line corollary, but it is very important and easy to remember. For a finite connected pseudograph, the rank R of the fundamental group is given by 1 minus chi X, the rank is given by 1 minus chi X. So, how do you do this? So, very easy. The number of edges in any maximal tree is equal to V X minus 1. What is V X? V X is number of vertices. If there is only one vertex, there are no edges at all in a maximal tree. If there are two vertices, they are connected by one edge. You cannot connect it by two edge, then it will not be a maximal tree, it will not be a tree at all. Like this, you can see if there are n vertices, you exactly need n minus 1 edges to connect them. So, if there are V X vertices, V X minus 1 is number of edges. Therefore, the rank of pi 1 of X is equal to the total number of edges minus the number of edges inside a maximal tree which are those which are not in the maximal tree.  So, the formula is proved. So, now, we will give application of this corollary. So, this itself was a corollary to this corollary. So, now, we will give an application to one of the very, very important result in group theory. It is called Nissan Trier. Both of them were actually done lot of group theory also, but they were all motivated by topology. They have done lot of topology also. Every subgroup of a free group is free. That means it is a free group. Remember a free group has a basis. The subgroup may not have basis coming from this base. It is need not be a subset of the basis. Only thing what we say is every subgroup of a free group is free. It is just like every subgroup of free, abelian group, free abelian which you may have known already. So, there is a further class here. Indeed, if F is a free group of finite rank R and F prime is a subgroup of finite index, then the ranks can be related by the following formula. Namely, R prime is the rank of the subgroup is equal to 1 minus k where k is the index plus k times R where R is the rank of the free group of the larger group. So, this now we can derive by using Euler characteristic and something more that we have done everything in the course itself. Namely, we have given a simplicial structure to a covering. The simplicial structure coming from a simplicial complex of the given simplicial complex in a very particular way. So, that we are going to use here now. So, let us start with the following. A free group has a basis. Let me call that basis as i, any set. This set is the basis for free group. Then I take as many copies as a number of elements in i, the boundary of the two simplex, the standard two simplex delta R2. The boundary of the standard two simplex you remember is homeomorphic to the S1, the circle. So, it has fundamental group infinite cycling. So, what I will do is I will take as many copies as in i this thing. Then I perform this bouquet one point union, namely identify one of the vertices that E0 from each of them to a single point. Select one point, one vertex from each of these copies of the boundary of delta R2 and identify them to a single point. So, that is the meaning of this wedge of these things. A wedge of topological space is in general is defined by that method. And it is given the usual CW topology here. So, you take this one, boundary of i delta R2, i for each of them is a copy of boundary of delta R2 which is homeomorphic to S1. Then we know that this is being a bouquet of circles we have computed. The fundamental group of this k mod k is nothing but that mod k I am putting delta x is nothing but the free group over i and that is f. So, f is isomorphic to pi 1 of x. So, starting with a abstract group free group, we have realized it as a fundamental group of a space. The space is constructed in a very particular manner. So, it is a pseudograph force. Now, given any subgroup of the fundamental group, we can construct a covering corresponding to this. This is our general covering space theory. Let P from x prime to x be the connected covering corresponding to the subgroup f prime. What is the meaning of this? That x prime, let us say little x prime as a base point sitting over this x naught here. The x naught is a common base point that I am taking here. Then pi 1 of x prime comma x prime is isomorphic to the subgroup via this the covering map P check. Covering map P induces homomorphism. P check is injective map. The image is precisely f prime. So, this is what the general theory of covering space is that I am going to use it here now. But now, x prime itself has a simple shell complex structure, one-dimensional simple shell complex structure because this is a one-dimensional simple shell complex. So, what in the structure that also we know exactly, where k prime is a one-dimensional complex, in particular it follows that it is fundamental group because this means it is free. So, the first part of the theorem is over. Now, for the more elaborate part namely, suppose f is the field group of finite rank R, then what does it mean? The set I must be chosen such that it has exactly R elements. Then this bouquet will also have exactly R copies of boundary of delta 2. R empty triangle, triangles will be there. How many R of them? All of them having one single point as common. One vertex is common. So, therefore you can count the number of edges and number of vertices here. Then for each vertex in k, in k prime, there will be how many vertices exactly depending upon the number of sheet. So, how does the number of sheet is determined? The index of this group, index of this group is what we have taken as k. So, there will be k many vertices for each vertex below. So, that is what I have to write down now and in the counting. So, let me repeat for the more elaborate second part, let I be a finite set with coordinate f i equal to R and let f prime be a subgroup of index k in f. Then, p is a k sheet recovery. The p is a covering projection. It follows that the number of vertices in k prime is k times the number of vertices in k. But number of vertices in k is precisely equal to 1 plus 2 R. In each triangle, there are 3, but one of them is a common. So, that should not be counted too many times. So, that is only once. I mean for other two, for each of them, R of them are there. So, 2 R will come. So, k times 2 R. Similarly, number of edges inside k prime is precisely k times E R. So, k times E k. What is E k? E k is 3 times. Each triangle has as many 3 of them as edges. So, there will be 3 times R. So, it is k times 3 times R. So, if you take vertices, it is k times edges also k times. So, Euler characteristics just k prime is just nothing but k times k of k. You do not need all these elaborate things, but I have just written down. But then R prime by the earlier formula is equal to 1 minus chi of k prime, which is equal to 1 minus k times chi of k, which is equal to 1 minus k times chi of k, it says it is 1 minus R. So, I get 1 minus k plus k R part of the formula is written here. For example, if your group is say free group of rank 2 and you have an index again 2, then what will be the rank of the subgroup? It will be 1 minus 2 plus 4. It is quite a good formula. So, it will be 1 minus 2 plus 4 is 3. It is readily, both of them are 2, 2, but the subgroup has order, subgroup has rank 3. So, this is used in several differential geometry, different algebraic geometry contests to prove many other results. It is extended to what are called Riemann-Hurwitz formula and so on. So, now I will give you another important application of this G covering that we have done. So, here let X and Y be connected locally contractible spaces or you may assume less things like locally, semi locally, one connected, semi locally simply connected, some such thing you can assume. So, locally contractible it is easy. Let F from X to Y be any map such that let us you know C F, you know the mapping cone of F. Then the inclusion homomorphism eta from pi 1 of Y to pi 1 of C F is surjective and the kernel is a normal subgroup N generated by the image of F check in pi 1 of X to pi 1 of F. F is any map. So, if you want to be very careful here, you have to select a base point here, say X naught and then you have to take F of X naught here as the base point for Y and then you have to write down this. Since those things are obvious and also always have to be remembered what is more essential and simple I have stated this. Okay, instead of writing all the base points and so on. So, those things will be done in the proof carefully. Moreover, the entire statement is independent of what base point you take. If you change suppose somebody takes X 1, X 2, X 1 and X 1, X 1, X 2 it will be still true. Therefore, what you have to remember is this statement. But a more elaborate statement has to be also remembered always whenever fundamental group is involved, you if you want to be very careful, finally you have to put the base points. Okay, so that is the summary of this one. So, I have deliberately omitted mentioning the base point so that it will be easy to remember. Not that you should forget it. You should make a practice not to mention it yet. Remembering it. Okay, so let us recall because this is a long time we have done these things. Let us recall what is the mapping cone and mapping cylinder and so on so that we write down a proof carefully. So, given any topological space X, C X is defined as the quotient of X cross unit interval wherein we identify all the points X comma 0 to a single point. The mapping cone is defined similarly but little more elaborately you take X cross interval disjoint union with Y and then you identify X cross 0 to a single point. Also you identify X comma 1 to F X for every X in X prime. So, this way F is involved in this way. Okay, then the mapping cylinder is defined in a different way wherein this part of the identification is not taking place at all. Namely, it is X cross unit interval disjoint union Y and X comma 1 is identified as F X for every X in X. Okay, I am just recalling this I cannot run the whole thing again this theory. There is a standard notation that all the points X comma 0 are identified single point that point is denoted by star. Okay, and that star is called the apex of the cone. The whole C X is star shaped at this star. Remember all this. Therefore, the C X is always contractible. Moreover, if you throw away that point, then the remaining portion is nothing but X cross open interval 0 to 1. Therefore, it is strong differentiated tracks to X. You can bring it back to X comma 1. So that is X. X can be thought of as a subspace of X comma T where T is fixed for any T other than that 0. At 0, there is a not the whole of X, but only single point is there. Okay, there is also an obvious inclusion map. From C X, you throw away the base. That is an open subset anyway. X comma 1 is a base. The whole thing is contained, C X itself is contained inside C F. Okay, you can see that C X X cross I disjoint even why I have taken whatever identification is there that those those things are also there here. Right? Right? Except that X cross I is fully not identified here. X cross 1 is not identified here at all, but there is some identification here. Therefore, if I throw away X cross 1, the open part, then that is contained inside this one. Okay, so C X cross C X minus X cross 1, there are identification here. If you throw away that part, that is included in C F. So let us put, let us just denote it by V. This is temporary notation. That is an open subset. Also look at the open subset obtained by throwing away the star from C F. If you throw away that, that is same thing as mapping cylinder minus X cross 0. If you identify the whole X cross 0 single point that is C F. So you are throwing away here, you are throwing away this part also that is M F. Okay? So U and V are both opens of sets of C F. And C F is U union V. Okay? So, so far what I have done is setting up notations for application of cypher and companse theorem. Like you had a sphere, then you removed North Pole, South Pole and then you took the intersection and so on. That is the kind of thing we are doing here. Okay? So intersection, let me denote it by A. This is all just convenient temporary notation. That is nothing but X cross 0 1. Okay? So let us have these notations also, eta 1 from A to U, eta 2 from A to B. These are the inclusion maps. All right? So let me show you the picture first. So picture is here. This is the entire mapping cone. So this is the star. So this is X cross, let us say X cross half, half way. This whole thing is 0 to 1 here. So at 0 there is one identification. At X 1 also there is identification. Here X will be identified with F of X. So I have taken X naught as a base point for X. But this is actually, I have put it X cross half, half way. I could have put it at any level t, not equal to 0 or not equal to 1. Anywhere I can put it, same thing. Okay? So from here to here, this entire thing, this is my V and from here to the rest of them, this is my U. The union is the whole space. See here, intersection will be, remove this point, remove this part Y and this open part, which is nothing but X cross open interval 0, 1. All right? So we will come back to this picture again. So right now this is the, this is what I have said up the notation. Therefore, so one more, namely I have chosen X naught as a base point, whatever X naught, does not matter. Y naught is F of X naught. But X naught is X where it is, it is X cross half I am taking. Therefore, I would put it inside X naught cross comma half, that is X naught hat. Okay? So that belongs to inside A, X cross 0, 1. Okay? X itself is nowhere there, X cross, some t is there after all. They have everywhere. So you can identify X as this one. Okay? Inside A. Once you have this, you have the one compass theorem. The C F is written as union of U and V. It is a common base point X naught belonging to all of them. I take pi 1. Intersection of these two is A. So pi 1 of A X naught, eta 1 check, eta 2 check. And then these are also inclusion maps. I am not writing them, separate notation is not necessary. Inclusion induced maps here. Okay? The statement of one compass theorem is that pi 1 of C S is the amalgamated free product of this path. Namely, it is free product of this group with this group, modulo, the normal subgroup generated by the image of this, minus this and so on. Right? So that is what we have written down here. It is pi 1 of U star pi 1 of V. Modulo, the normal N is the normal subgroup generated by eta 1 check G multiplied by eta 2 check G inverse. That is one single element as G belongs to pi 1 of A. Take all of them, then take the normal subgroup generated by that. The quotient of this is pi 1 of C F. I can put the X naught hat here. Okay? X naught hat, X naught hat here always. The base point can be always written down. It is written down here. Okay? That is one compass theorem. But now we know all these things. So we have to figure out what these things are. Okay? Now use the fact that V is star shaped at the star. Every star shaped set is contractable. In particular, it is fundamental group is trivial. Therefore, this statement here, whatever we have to, this gives you this pi 1 of V, there is no need to write, this is a trivial group. So it is only pi 1 of U and eta 2 of G, there is no need to write. They are all trivial elements. It is normal subgroup generated by eta 1 of G. G varies over. It is the same thing as normal subgroup generated by the image of pi 1 of A. So image of pi 1 of A under eta 1 check. So this is the neat expression that we have got. Pi 1 of CfA is the quotient of pi 1 of U by the normal subgroup generated by eta 1 check. We are not yet over. So what is this U, what is this eta 1, etc., you have to figure out. Okay? Because these were not in the beginning, our statement in the statement is only x and y. So everything you have to convert it to x and y, what is that? So this just means the inclusion means homomorphism, pi 1 of U to pi 1 of Cf. Remember this is giving this isomorphism, it is not 50 T as it is given by the inclusion of the homomorphism that is subjective and its kernel is the normal subgroup generated by pi 1 of A. So if you go back to this part, you can ignore the upper part here. This part is an exact sequence. This is subjective. Kernel of this map is this one. That is the meaning of this. Okay? So we have to do one more step here to understand in terms of x and y. That is the that is the major part of the thing is over. We have to figure out that the correcting is arrived. Okay? So this is the picture going back to the mapping cone. X cross half is sitting somewhere in between with X naught as base point. This is a deformation retract of A. Therefore the inclusion induced map is an isomorphism here. A itself is contained inside U. Okay? So this is pi 1 of U here and this is that is an identification here. Right? So that is Q check. The quotient inclusion followed by the quotient restricted to the subspace that is the inclusion map. So here it is just forget X comma half going to just X. This is the projection map. So this is a homeomorphism. So this is again an isomorphism in pi 1. So now it is pi 1 of X X naught. So this is nothing but a copy of this here. All right? From here I have f check to pi 1 of Y. So what is the map here? Remember the mapping cone whole mapping cone there is a diffusion but here I have removed just the one open part here that is all. So mapping cone there is a map any bracket X comma T going to fX. This is a very defined map and that is what I have written as f hat. So this is f hat check and f hat itself is a strong deformation retract onto Y. So this is an isomorphism here. So what we know is this is the CF here we are sitting here. There is an inclusion map here but here I have taken different map here. The inclusion map is factored by this way. Okay, goes into Y and then comes here. So this is a commutative diagram here. This is, okay, it is surjective and it is kernally precisely this one is what we have seen from pi 1 of A. Pi 1 of A is isomorph to this, isomorph to this one. All right? So the problem here is you see this is X naught twiddle here. Okay, this f check this is Y naught. So what is this base points? Where do they go? This is what we may wonder. If you don't have any such troubles in your mind you have completed a proof. This part is over now. Okay, the normal this is surjective, this is surjective. The normal subordinate by this one is isomorph to this one is over. However, I have taken trouble to exactly what happens to this map here because this is some other map here f twiddle of that comes to Y naught. That is fine. So you have to understand that this map is a homotopy equivalent. So it is an isomorphism. But why it should be the same thing? Why these diagrams are commutative etc? You will have to learn. Okay, so here I have elaborated this. Let us go through this one. The unlabeled arrows indicate homomorphism induced by inclusion maps here. What are they? This one, it is the only one. This one here. This is something I will explain it. Okay, and q is the restriction of the quotient map q x comma t is our notation is x comma t bracket, round bracket change to square bracket. p of x t is just x, f hat of x t is fx. All these I have told you. Since p is the restriction of the projection, it is a homeomorphism and hence p check is an isomorphism. As seen in our theorem, namely about the homotopy properties of the mapping cylinder, if you do not remember, this is what it is, namely f hat composite J's identity of Y. And f hat is a strong deformation attraction. So that J I have not included here. This J is from Y to the mapping cylinder. So Y to the mapping cylinder up above. Okay, so J and then f hat that is the identity map. Okay, and this is the strong deformation attraction. This is what we have used. I am going to use that one now. Okay, so but I will be write down what is that strong deformation attraction here. Let us take h from mf cross i to mf given by h of x t s equal to x comma 1 minus t times 1 minus s times t plus s. Okay, this defines homotopy of identity map of f with f hat relative to Y. What are the points here when t equal to 1? Okay, that is a point of Y. So on Y it is identity map. You see the meaning of this. Okay, when t equal to 1, okay, t equal to 1, it is 1 minus s plus s which is just 1, x 1. So you can extend it by identity on Y. Okay, the rest of the time it is homotopy. When t equal to 0, what is this one? x 0 s is t 0s x comma s is identity map of the mapping cylinder itself. Okay, so when s is 0, s is 0, this is x comma t plus s. Yes, s is 0, x comma t, x comma t goes to x comma t. So in particular, f hat check is an isomorphism. Okay, so I have repeated it three times at least. Commutativity of the left side rectangle is obvious at the topological level itself. Look at this, what is it? x comma half goes to x. Here x comma t, bracket x comma t goes to x. Here x comma open, x comma half open, there is no identification, goes to a bracket here. So this is obvious. So this f of that. Okay, once some diagram is commutative at the topological level here is there. Induce map diagram will be commutative at the homotopy groups, at the pi 1 level. Okay, only commutativity of the triangle on the right side involving the dash arrow has to be justified. What is this one? So this is remember h tau, what is this tau? So I will explain that one now. Okay, which changes the base point x naught to y naught, x naught hat to y naught hat. Okay, remember this is subspace of that, but this is not an inclusion map here. Base point has been changed. So this is what it is. So x naught hat is here. Okay, and y naught is there, f of that. And you have the arc here from half to one. You trace it this way, Uta, the other way around. That is tau. That is a path. That tau is a path. Whenever you have a path, you have h of that path, h tau, define how take a loop here, start with the base point here now, go via tau, trace the loop, come back by the tau inverse. So that is the definition of h tau. If you remember that, that is fine. Otherwise I have just recalled it. Okay, so what I am going to do, use this thing and homotopy here to push this x naught by a homotopy to this point. So that is precisely what I am going to do. Namely, as if the base point is slowly moved along this point to this point. That is the homotopy I am going to write it down. And then you are confused. These two are the same thing. Okay, so otherwise this map is there something else changing from base point here. We know it is some isomorphism. But why the isomorphism should give you the same normal subgroup etc. That kind of doubt can be there. So that is why I am taking elaborate time to explain this part. Now let us look at what we have written down here. Okay, so check. For 0 less than s less than 1, put tau s of t equal to x naught comma 1 minus 1 minus s times t by 2. Okay, so I have taken various tau s now. Let us look at s equal to 0. Tau 0 of t is x naught 1 minus, this is completely s is 0. So 1 minus t is 1 minus t by 2. So it is 1 minus t by 2. That is precisely this trace delta from 1 to half. That is tau naught. What is tau 1 when s equal to s equal to 1? This 1 minus s is 0. It is x naught 1, which means it is just this point. So I have taken, first I take this full part and slowly I take only trace up till here, up till here, up till here. Finally, this is the constant factor. Why not? So that is tau s family of arcs. Okay, and h s of x tau, you just write h of x t that whatever you have written earlier this h, h this h, you take this one, that definition. h s of x tau is h of x to s s. Check that tau naught is a constant part at y naught equal to x naught comma 1. And tau 1 is tau in the segment x naught cross half 1. Trace in the opposite direction. Now given any loop omega in M f like this picture, this is omega here at base point x naught, okay, inside the whole of C f. That will be converted into, by pushing it at base point here in the right in the beginning. Finally, it will be pushed into a path inside omega loop inside y itself. Okay, there are two of them, two things you have to do here. So to begin with what we have, now start it, okay. Now given a loop omega in M f, okay, at the base point x naught hat, you want to prove that h tau omega is homotopic to f hat composite omega. F hat composite omega is a path, is a loop in y at the base point y naught. So if you show this one, then this diagram commutativity of this path follows, this h tau. Okay, when you push it, this loop here, this loop here, inclusion map followed by this one, that is fine. Okay, so this is what we have to prove. And here is the formula, that is all. Consider f of T s equal to, see remember the first thing is tau star, okay, tau inverse of omega itself. So h s starting the point will be just identity. So that is what you have to write, tau s star h s composite omega star tau s inverse T. Check that f we say homotopy as required for h tau of omega to f hat of omega. So it follows that the inclusion is homomorphism from y to pi 1 of c f y naught is subjective with its kernel equal to the normal suffrage generated by this one, once you have established so we shall come to the last reverent of this series now. As a special case of this general theorem, take the space x to be a circle and f be any map into y, that is a special case. When you take the cone over that, it is nothing but the entire space is obtained by attaching a two cell to y, okay, because the cone over s 1 is nothing but the disc d 2, right. So that is a two cell being attached via the map f to y. So that is a special case. So we shall slightly generalize this one, not just one and that is the next result here. So let z be a space obtained by attaching two cells d 2 alpha, you know not one but several of them indexed over lambda, capital lambda, okay, to a path connected space y via the attaching maps f alpha from s 1 to y, okay, these are f alphas. Let tau alpha be a path in y from the base point y naught to f alpha of 1, this 1 is indicating the base point in s 1, this complex number 1 there. Let lambda alpha, you know the element in pi 1 of y y naught represented by this tau alpha, tau alpha is a path, right, I have chosen from y naught to f alpha 1. So we take f alpha tau alpha, now trace f alpha, f alpha is a loop after all, it is a map from s 1 to y, right. So f alpha composite tau alpha involves, okay. Then the fundamental group pi 1 of z y naught is isomorphic to the quotient group of the quotient of pi 1 of y y naught by the normal subgroup generated by all the elements lambda alpha which you have defined here as alpha runs over lambda. So take the normal subgroup generated by this one and go by go by quotient out by that one that is pi 1 of z. So the missing thing here if you look at is that in the previous theorem we had x is connected, y is also connected and we have chosen a base point there. So pi 1 of x naught to pi 1 of x naught to pi 1 of y y naught could have taken, right, connected spaces. But now here d alphas by definition they will be disjoint, they will not be connected. So except that it is similar to the previous theorem. So we have to be careful how this will this can be done, we cannot directly apply that. If d alpha consists of just one family, one single element, one element in this family say lambda is a singleton, then it is direct application of the previous theorem, okay. Then it is image of image of this pi 1 of s 1, right, pi 1 of s 1 which is infinite cycling. The normal subgroup generated by that one that is precisely what it is, okay. Now I want to question you here namely we shall give the proof of this only when f alpha is a finite family that means lambda is finite. The general case follows by what is called as a direct limit argument which we shall teach you in the second part. Not only that in the second part the attaching cells etc itself will be formalized into what is called as CW complex. Not only just attaching one cells and two cells you will do attaching k cells in general and then we study those things very elaborately, okay. So you may think that this last result here is a motivation for the second part, all right. So let us work out this one when f alpha, the family f alpha is finite family, okay. The first of all I want to repeat the post lambda is a singleton. Then we can apply the above theorem by choosing the base point y naught to be equal to f alpha of 1. Choose that one because we have, we feel to choose the base point. Here x will be s 1 and z can be viewed, z is mapping, mapping cone now. I mean you can think of this mapping cone of this map f alpha. You can call it as obtained from y by attaching a two cell yi f alpha. These two pictures are the same, right. Since phi 1 of s 1 is fine infinite cyclic group generated by just the identity inclusion map s 1 to s 1 to take the identity map that will be denoting an event in the phi 1 of, that loop phi 1 of s 1 and that is generator. It follows that its image and f alpha shape is nothing but the class f alpha as a loop. It is generated by f alpha, the loop f alpha because f alpha composite identity I have to wait. So, it is just f alpha in phi 1 of y y naught. Therefore, phi 1 of z with the base point f alpha of 1 is the coefficient of phi 1 of y f alpha 1 by the normal subgroup generated by f alpha, okay. You do not have to take all the elements because all other elements are powers of f alpha, okay. So, case lambda is just singleton is over, okay. Now, what we are going to do? We do a simple induction. The only problem is each time the base point will have to be kept changing and therefore it is important to realize our earlier theorem was base point free. It is true for even if you change any other base point, okay. That is why I had taken so much of trouble in introducing that one, okay. Let us do it. How we do it? Let us see. So, now consider the case when f alpha 1 may not be equal to y naught, okay. We then appeal to the isomorphism h tau alpha, okay. What is tau alpha? Tau alpha is any map from y naught to f alpha 1. Any path from y naught to f alpha 1. Then h tau alpha from phi 1 of y f alpha 1 a loop here get converted into loop here and this is an isomorphism which we have, okay. Once this is an isomorphism and normal subgroup will go to normal subgroup here and what is this normal subgroup? It is lambda alpha which we have defined elaborately here in the statement here.  So, f alpha will come to this one. The normal subgroup generated by f alpha will come to the normal subgroup generated by this one, okay. Under isomorphism normal subgroup will go to normal subgroups, okay. So, this will give you now the statement for y naught where y naught is a fixed point, base point for y independent of what I have called. But now this is applicable for all the other alpha solves when lambda is not a singleton. So, you can apply one by one. First you get identify say alpha 1, alpha 2, alpha n. First attach alpha 1 through alpha 1. Next attach alpha through alpha 2. So, what do you get? The quotient of the second one is quotient of the normal subgroup generated by f alpha 2.  So, this is a finite case is got by a simple induction, okay. The infinite case won't conclude like this one. Even the countable case you won't conclude like this one. So, for that you need what is called as direct limit arguments. So, that will be part of the next course, second course in this one. So, you are welcome to attend that. Thank you. I have enjoyed teaching you people. Hope you have learned something from this course. Thank you.


PSU women hosting Valley City State on Saturday, second half Evers trailing Sydney Arrington though. The hoop and the harm, free throw good deficit is four. Then later freshman Brooklyn Bachman, coast to coast.  Then time winding down Maria Apachele. Give her three of those. They outscore Mavo by 12 and the fourth. Bachman with another 20 point game. Evers complete the comeback and hang on to win 63 to 62. If you've enjoyed this segment of Lakeland News, please consider making a tax deductible contribution to Lakeland Public Television.


Hey survivors and frivers, in this video I am going to be discussing how and why life is so much easier without narcissists. Please like, comment, share and subscribe. If you are interested in a one on one coaching session with me please send me an email. My email address is in the video description. If you would like to make a donation you can do this through the live chat or you can donate for my paypal. The link is Life is so much easier without narcissists. When you are around narcissists life can be so difficult they affect every aspect of your life. The longer you are around them you will notice everything seems to fall apart. You start thinking, feeling and behaving the way they want. You change your personality and character to make them feel more comfortable. You forget about the person you used to be before you met them. You stop doing the things you once enjoyed doing such as hobbies or interests. Go into certain places or talk into certain people. You don't do that stuff anymore because the narcissists trained you to lower your vibration, to make them feel more comfortable and to prevent any narcissistic injuries. Being around narcissists you are moving through life feeling intense negative emotions such as anger, desire, fear, grief, apathy, guilt and shame. This is no fun.  This is not how you are supposed to be living. Life is meant to be enjoyed, not endured. Life doesn't have to be so hard. It can be easy. It only seems so difficult because you are around difficult people. You are around people who believe that life is supposed to be difficult. I prefer to avoid these types of people.  I prefer to live a life which is more enjoyable.  They claim that you are not good enough or something is wrong with you. Funny how you never dealt with any of this before you met them. Before you met these people you are likely vibrating at quite a high level. This is what attracted them to you. They love high energy people. They do not have that. They also do not experience the high emotional vibrational frequencies which you were probably experiencing daily before you met them. Emotional vibrational frequencies such as peace, joy, love, reason, acceptance and willingness Narcissists are attracted to those frequencies initially but these frequencies are not compatible with any narcissist. Narcissists are shame based individuals.  This can never be compatible with any high emotional vibrational frequency. So by default all they can do is bring you down. All they can do is cause you to feel intense low level frequencies such as anger, desire, fear, grief, apathy, guilt and shame. The narcissist will use manipulative tactics such as denial, projection, blame shifting and gaslighting to make you believe that those are your natural frequencies. They are not. It is the narcissist who brought you down to that level but of course they will not accept responsibility or accountability for anything that they do. They will deny any responsibility and instead project their insecurities, flaws or imperfections onto you. They will shift the blame onto you and they will gaslight you into believing that there is something wrong with you and you are crazy. So each day you feel all of these intense negative emotions, low vibrational frequencies. You are likely struggling in your life, feeling completely dissatisfied and unfulfilled with your life and worst of all you have been made to believe that this is all your fault. This negative low vibrational person is who you are or at least that's what they want you to believe. So you can continue living your life in this way, experiencing constant difficulties, isolation and loneliness without ever questioning the narcissist as being the source of your problems. Problems are boring. They are no fun. They do the same boring things every day. Never experiencing any real joy or happiness with anything that they do. They have all of these perceived limitations and they have put themselves in the box because of these perceived limitations. They feel that they need to limit you. They need to put you in a box where you cannot talk to your friends anymore. You cannot have a social circle. You cannot have hobbies or interests. In some cases they won't even let you have a job. They just expect you to stay home all day. All of this to prevent any possibilities of them having a narcissistic injury. But even with all these precautions you still experience daily arguments or disagreements with the narcissist. Every day is filled with a hatred, anger, envy and jealousy. It's like you can't do anything without setting them off and over time the narcissist has led you to believe that this is normal. The truth is it's not normal at all. Let me wake you up to the truth of these types of people. They are trying to enslave you, trying to put you into a box, trying to prevent you from doing all of the things you would otherwise be doing if you were not with them. Because any sense of happiness, satisfaction or fulfillment you experience in your life reflects on how miserable, dissatisfied and unfulfilled they are in their lives. That's why they are always so hateful, angry, envious and jealous towards you. That's why they will constantly argue or disagree with you. Life can seem so hard when you are around these people. Everything seems so difficult. Even the most basic, simplest things can be turned into something complex and confusing when you are around them. And although they will say and do whatever they can to make you believe that this is normal, it's not normal at all. It's normal for them, in their world where they feel compelled to have all of this abuse, manipulation and reality distortion to deny or manage any kind of reflection on their suppressed emotions or whatever else they are trying to deny. Everything seems so complicated and complex around these types of people. Because of this, from childhood they have learned to suppress these intense negative emotions. They were not allowed to have them around their narcissistic parent so they had to suppress them. They were taught by their narcissistic parent to prevent the action or expression of a feeling or idea. They were not allowed to express these actions or expressions of their feelings or ideas. They also never learned how to self reflect or look within themselves to deal with this. By default, their only other option is to project these actions or expressions onto you. They want you to express them or act them out. So it isn't long until your daily experience is to live in misery and suffering, experiencing constant arguments and disagreements, which are often completely meaningless and insignificant, worst of all you are blamed for this and made to believe that this is just how life is supposed to be. Let me wake you up to the truth, survivors and frivers. Life is meant to be enjoyed, not endured. All of these arguments and disagreements most often have nothing to do with you and everything to do with a narcissist. They are not personal at all. The narcissist is acting on their dysfunctional thoughts which then produce these intense negative emotions which they have to project onto you. Just think about this for a moment. If we had a conversation together, you and me and we had a disagreement, it would not go on for long periods of time, it would be resolved within seconds and we would reach a mutual understanding or resolution. It would also not be brought up again the next day or the day after that. There would be no grudges or resentment. You cannot do this with a narcissist. How can you have a mutual understanding or resolution with the person who has an inner dialogue or inner critic which is programmed to block or deny any form of mutual understanding or resolution. And believe me, there is nothing normal about this. If we were talking and we had a disagreement about something, it would be over in seconds. It would not go on for days, weeks, months or years like with a narcissist. You will not agree with someone all of the time. Disagreements are a natural part of the relationship but that does not mean that they have to turn into arguments. We can compromise. We can learn to be in the same proximity of understanding as emotionally mature adults. We naturally have good conflict resolution skills so a disagreement will never turn into grudges or resentment that lasts a lifetime. All of these intense negative emotions you have to deal with from them. All of the arguments and disagreements you can't go out anywhere. You can't talk to anyone or do anything and you have been taught to believe that this is normal. This is how your life is supposed to be. Every aspect of your life, every aspect of your character and personality has been changed or adjusted to make the narcissist feel more comfortable and now you are not even living. You are not even being you so it should be no surprise that life is miserable when you are around these people. You can't even be yourself.  You have been trained to live within the bounds of the perceived limitations which the narcissist has created in their own minds and then apply to you and your life to prevent the sense of injustice or unfairness they would otherwise feel. This is how your life will be. When you are around self-hating insecure and emotionally unhealthy people but there is nothing normal about this. If we were in a relationship together your life would not be this way. You would not be stuck at home every day. You would not be afraid to go out and develop social circles or have hobbies and interests. You would not feel as though you are doing something wrong when you talk to certain people or go to certain places whether we were going together or if you choose to go on your own or with your friends. In a healthy relationship none of that would be an issue. There would be no envy or jealousy because in a healthy relationship there is a mutual understanding that we have our own separate lives. There is no need to control every aspect of your life or set perceived limitations. You can move through your life freely without constant fear, stress or anxiety. Life is so much easier without a narcissist. Everything comes together naturally. There are no constant or repeating arguments or disagreements. None that can be resolved within a few seconds. But you don't get that with a narcissist. Everything is a problem. Everything has to be controlled. Your daily living experience has to be limited or restricted in some way and it's no fun. It's not living. You are basically an autopilot living the way that the narcissist wants you to live. Not having a life of your own, not having a mind of your own and that is why you feel so drained and lifeless as though there is no happiness or joy in your life. Leave the narcissist with all of the arguments and disagreements, grudges and resentment which they have held on to. Leave them with all of the negative energy. It clearly resonates more with them than it does with you once you leave the narcissist. You will discover how much easier life can be. Everything that was once limited or restricted in your life will soon develop on its own. You will find social circles, have healthy relationships, good careers, hobbies and interests. All of this is so easy to achieve and obtain without these people in your life. It's only difficult because you have a narcissist in your life. Controlling everything you do, their life is difficult because they lack the capabilities or motivation to create what they want. They have limited or restricted beliefs. Because of this, they feel as though they have to limit or restrict you in some way but once you leave these people, you will find that life is so much easier. You will realize that you had the capabilities and motivation all along. It's just that they were holding you back. They were preventing you from being your best self because it hurt them. They didn't want you to know that so they made you believe that something was wrong with you or you were not good enough. That was not true. You are good enough. You are more than enough. More than they will ever be and they knew this all along which is why they felt so compelled to limit or restrict you. Prevent you from being your true self.  Thank you for watching. I hope this video resonated with you. Please like, comment, share and subscribe and I will talk to you soon.


Live from Las Vegas, it's theCUBE, covering Veritas Vision 2017, brought to you by Veritas. Welcome back to Veritas Vision, everybody. My name is Dave Vellante, I'm here with Stu Miniman. This is theCUBE, the leader in live tech coverage. This is day one wrap of the Veritas Vision conference. Veritas, as we said earlier, is a company that has gone through a number of changes, Stu. I remember when the company launched in 1983, it was sort of, you know, it was focused on backup. It was Veritas and Legato, and they kind of grew through the PC era and the client server era, really trying to take off. And then they exploded in the internet era. Their evaluation went through the roof. They ended up buying, you know, Seagate's backup business. They really drove that and then got, you know, purchased by Symantec for a big number. I mean, the number was, you know, 15 billion. It was in the teens, as I recall. Really never did much under Symantec, or Symantec never did much with Veritas. I think they had a vision of information management and that never really panned out. Spun the company back out, divested it, sailed to Carlisle and some other investors for, I thought the number was 8 billion, somebody told me 7 billion today, that must be net of cash. 2.3 billion dollars in revenue. My understanding from sources is that valuation is way, way up, nearly double from 2015. Now, maybe that's an inflated number, but I'm not surprised, the market's been booming. So that's sort of the inside sort of organizational issues. We're here at the Aria, what do you say, Stu? A couple thousand people, 2,500? Yeah, it's about 2,000, David, it's interesting.  We got to talk to the Vox community people. They've got 10,000 people online contributing to their forums, participating, launched a VIP program for some of the super users they have here. Definitely, you know, good crowd in the keynote.  You know, the number one topic conversation is GDPR. As you said in our last interview, you know, we're going down the deep abyss of how you're going to get, you know, elitigated out of all of your money if you don't follow this. It's like way worse than Y2K, ah, some stuff's going to break and you know, maybe turn off for a bit. Well, you know what I liked about the GDPR discussions though? They had answers.  What I liked about what I'm hearing from Veritas is they've got at least a quasi-prescription as to what to do, so that's good. But the more interesting part to me Stu is you've got this enterprise backup legacy, I'll say it, legacy backup install base, enormous. A leader, they've mentioned many times, 15 years in a row, a leader in the magic quadrant, and I believe it, you talk to customers, what are you running, net backup, everybody's running net backup. But how they're transitioning into this vision of multi-cloud data protection, resilience across the enterprise, across clouds, hyperscale. What I'm not fully clear on yet is how they get customers from point A to point B. We heard from the keynote this morning in Bill Campbell, we invested a bunch of dough in R&D, we're writing stuff that's cloud-native, container-based, microservices, so sort of all the right application development buzzwords, and I believe that they're developing there, but I don't understand how they migrate that install base. Is there some kind of abstraction layer? Is there some kind of new UI? We heard them jokingly say today in the keynotes that we hear you, customers, we know our UI sucked, we're working on that. I didn't see any announcements on that, but that's something that presumably is a promise they're putting forth, but I wasn't clear, maybe you could help clear it up. On how you get from point A of legacy-installed backup software to this nirvana of multi-cloud hyperscale microservices. Yeah, I mean, Dave, when we talk to Bill Coleman, his three Vs, value, vision, and that values of the company itself clearly has got a compelling vision. He said, not only 10 years from now, but probably five years from now, every product I'm selling is going to be obsolete, and it's an interesting thing to hear because 15 years of experience, we're trusted, but we know that every product that you bought from Veritas in the past is going to be replaced by new things. And right, how do we get, say, okay, I've been buying that backup for a decade, do I get this visualization product enough I'm looking at AWS? Is Veritas the company I turned to? So it really gets down to that blocking and tackling. Talk about the consultants, the partners, both on the go-to-market side as well as the technology side. Can Veritas get in there? Can they have compelling differentiated products that solve a need in the market? We've talked to a number of companies this last year where I've looked at is this hybrid multi-cloud world. If you're software, how do you play in this market? Because isn't Microsoft, Google, Amazon, aren't they going to just do this? Information governance invisibility? Absolutely, Amazon has a solution for you. Google has a solution for you. Microsoft has solutions, but if I'm going to speak across those environments, we haven't had a solution that goes across all of those environments, so there's a hole in the ecosystem and Veritas along with many other companies are trying to put that big elephant on the table and eat pieces out of it, so it's interesting. And Coleman's background from BEA, started BEA, I think he took the thing up to half a billion dollars, sold at the Oracle for a big number. But you look at what BEA did, they were sort of the application integration glue. And that's a lot of here, a lot of similar messaging modernized around multi-cloud, around hyperscale, around microservices and the like. So Coleman obviously has experience doing that, I thought he was a very clear thinker, I had not met him before, Furrier knows him pretty well from his VC days, but I thought he laid out a pretty clear direction, so he's got street cred on this. Seamcom done it before, I think this company has decent balance sheet, they seem to have some patient capital in Carlisle, it doesn't appear that Carlisle is trying to suck all the money out, they don't have the 90 day shot clock, he basically, Bill Coleman basically said, look, we're fine shrinking to grow, we're shifting from a upfront license model, perpetual license model to a rateable model, we could never do that as a public company. So it's going to be very interesting to see if and when they emerge as a public company, what that looks like and where they come from. And Dave, one of the things I've been poking at is, where do they sell to? If this was the backup administrator, that's not somebody that's going to help them with the transformation, it's digital transformation, it's my cloud strategy, it's things like GDPR, where I'm going to need to get up the stack to the CIO, to the C-suite, prove the value that Veritas has, and therefore they can then get all these new products in where everything, the 360 data management, really at the core of what they're doing, and whole lot of other products. I mean, we didn't even dig into some of the object and file storage pieces that are in here. I know we've got their chief product officer on theCUBE tomorrow, but a lot of products, pretty broad software suite. And for an infrastructure company, it's always interesting to hear them say, really, infrastructure doesn't matter, the no hardware agenda, but it's your data that matters, and we've got a vision here at Veritas Vision to bring you forward, and lots of plays on the name of the company, Veritas. The show is the truth in information. Yeah, so let's talk about the lineup tomorrow. A lot of product stuff tomorrow, Mike Palmer's coming on. He gave a great keynote this morning. Very funny, he gave a scenario of the world ending because basically people didn't have their data act together, they had these images of Las Vegas hotels and chaos and waterfalls running through the hotels and drones attacking and just total chaos. So we're going to get into a lot of the portfolio stuff, and I think try to answer Stu's, some of those questions that I raised about how do you get customers from point A where they are today to point B? Are you going to, how are you going to transition them? Are there financial incentives? Is there some kind of abstraction layer that you're developing? What is that framework that brings us to that nirvana? So give me the last word here, Stu. Yeah, so looking forward to digging in more with some of the customers, some of the partners. Good energy at the show. It's exciting to be here for the first time, and looking forward to day two. All right, good, good wrap, Stu. Thank you, and thank you for watching. Go to siliconangle.com for all the news. We saw some Oracle news today. Hitachi changed its name, or Pentaho changed its name, we're not really sure about that. With all the news on siliconangle.com, go to wikibon.com for all the research, and of course, thecube.net to see this show, replays, youtube.com slash siliconangle is where we archive all this stuff. Lots of websites. Yeah, make sure to subscribe on our YouTube channel. Yeah, please do subscribe on that YouTube channel and follow us on Twitter, at thecube, at Stu Miniman, at D-Villante. That's a wrap, day one. This is theCUBE, we'll see you tomorrow from Veritas Vision, from Vegas. Take care.


As spiritual nomads with an insatiable desire to delve deeper into the depths of the great mysteries of our universe, we must learn to seek and sense the presence of God. If our faith and indeed our lives are going to flourish, we must discover the ways in which we can best connect to God and prioritize them in our lives. Not everyone will resonate with contemplative spirituality and not everyone will enjoy charismatic worship. But each and every one of us must do our best to seek a deep knowledge of God that goes beyond our intellect and touches the depths of our souls. God has always been reaching out to us with extended arms, ready to reveal God's self to us. But we must find the way that opens our spiritual eyes and allows us to gaze upon God's glory. For when we can finally stand in the presence of God, our lives will be changed. And when we don't have to wait till heaven because God is here with us now.


Cause how much time do they sentence you to life without parole life without parole at 20 years of 21 life without parole. Cause how much time do they sentence you to life without parole life without parole at 20 years of 21 life without parole. You going down here. Do you expect to be here today? Yeah, to my out on the ground. Yes. How because I always to like, I never stressed about the time.  Like every day I'm waking up, telling myself, man, you ain't going home. I got life. They say you ain't going home. You got life. It didn't sound right. You know what I'm saying? So I'm like, if I got life, that means all the people to be happening. Everybody puts a walk away from me. I'll put a new body. I won't really feel it. You're and I really want to pray for that like though, like, Lord, if this really what it is, show me a sign is what it is. And I'm not going to come in here. So you could feel you. So I'm like, look, if, if, if this just going to be a chapter in my life here, I write, let it be. I'm ready to put this story together.  Okay. If this my final destination is, I know it's a wrap here. All of that went to take a place for me. Even while I was in there, I feel like it wasn't going to be happening. If that really why I was supposed to be here.   They went in disbelief. I was all of that anger. I was all of that sort of stuff. I was like, when I first, I'm like, oh, that can't be real. They just told me life. I'm like, boy, look, I was laughing in that.  You're because a life sense, but I don't even sound right for me to have here. You watch a movie life. Yeah, I watched that. That's what that reminds me of. That's what that was. I watched that year. And he got life.  How do you get out? All right. This is how I got out. See, like after I got life, my, the, the witness on my case contacted my investor, get my lawyer and when he contacted my lawyer, he was like, man, look, why y'all doing this appeal? I got some information that I want, you know, I want to turn over to y'all want to do an affidavit for him, you know, to help him out. A lot right back.  I'm like, man, I've never talked to that man in a long time. Tell that man, I'm going through the process. You know the man. Yeah, I knew who he was. It was a nigga that was with me. Okay. So, um, he like, man, so look, what are we going to do? We're going to go up there. Go see it and see what he, you know, whatever he's saying that we're going to go from there. I'm not right back. So when they went saying them, what he said was that um, he was out on buying here and he went to jail on a burglary charge and he got out on buying and he missed court and some kind of way to DA's, you know, found a way to contact him cause he was running. They can't contact him. Not since he on the species. Like I want nothing to do with the case here. So once they contact them, telling them, look, well, you know, his trial getting ready to come up. We need you to testify and like, man, I want nothing to do with that. Well, look, if you don't want nothing to do with it, you already missed court on your case, you can go ahead and put the warrant out for your arrest on top of that. Charge you with the murder that he on and that we got that we had to drop on him before you're going to charge you with everything and just revel you. That's what they told them. Oh, they told them. You're like, man, but only you're like, we're not going to do it only if you agree to cooperate with us. You're like doing what? Like, man, his trial coming up. That's what we need you to say. Say this, say that. Like, man, well, look, I got to do all that for y'all. You know, I got a state lawyer. You're like, man, my lawyer not fighting for me or nothing. You're like, man, we're going to get your state. We're going to get your pay law and everything. Once you get go to the trial, get on the stand. Don't tell a people that we gave you no debt. We paid for no lawyer. None of that. You're even just put all that on there. We're going to tell you everything to say. Don't tell them about nothing and we're going to go from there and he agreed to it. So that's how, you know, once, once he put it on file, my lawyer wound up submitting appeal and submitting that affidavit with all in there. And I, I think he went got some, we, we had some type of, some type of paperwork that confirm everything do what we're saying.


So you get you get kind of the idea there so so note if if you were married then you would think everything would kind of double when you're married but it's not right the cap only goes from 43 it phases completely out from 43 492 to 49 266 49 622 so you can imagine like for example if we were maxed out if we had two people maxed out at what did we say like 20,000 20,000 let's say that gets us to 3,733 3,733 if I said that was our starting point 20,000 income at 3,733 and then you got married let's say we got married first and we keep it keep everything else the same no other income involved so now we're going to go married filing joint and if I go back on over so now married filing joint now we still we just have the one individual here and we kept the income the same so obviously the standard deduction has doubled so that's good from a married perspective that does kind of what you what you kind of expect it to do and you've got the the child tax credit that plays in here but but now we're at still with the earned income credit is still at the 3,733 right so from the earned income level you still had 20,000 income with a married couple and it didn't really change that now that's a little bit deceiving because there's some interplay between the child tax credit and that stuff as well but you can imagine situations where people become kind of worse off from a earned income credit standpoint when they get married so for example if you had another the other spouse let's say earned revenue as well so let's say we had ww2 income at another 20,000 and now you're going to be pretty close to the threshold being over the threshold so now we've got a one qualifying child and if I go to here so now we're at the 1,534 because although although the tables are not exact here you get more of a benefit if married they're not like doubled you know so you can imagine a situation where you get married and you lose a substantial part of the earned income tax credit whereas if they were separate in this case the second spouse one spouse would have gotten the earned income credit and the other spouse would have been over the threshold at 20,000 to earn the income tax credit with no children you could have a situation where you have two people that both had a child let's say they were kind of even here they had 20,000 each and they both had one child so the earned income tax credit is 3,333 if they got married they would be jumping then from two people at this level to a married couple two people here and here to to now one couple that has two children and would now be on this side so let's just for the fun of it we'll get into that more next time with that two people but let's just add another dependent just for the fun of it so now we're in a situation if I go to page one we had married couple uh uh mr and mrs handerson two dependents now so they were single before now they got married the income is doubled to 40,000 so they were two individuals completely the same from a tax standpoint of 2020,000 each and one uh and one dependent the the standard deduction goes up substantially which is what you'd kind of expect because it doubled because now you might have doubled the income so now you've got the income but if I look at the just the earned income tax credit it's now at the 3,265 right if I look at that and before so before I would have had if I multiply this times two and I did this individually you'd have 40,000 of income and you would have two people filing head of household getting 3,733 both right so they would have they would have got an earned income tax credit of 7,466 and when they get married the earned income tax credit is 3,265 now that's not like a a perfect comparison because we can also basically look at the the the other things involved here the the child tax credit uh that's going to be involved we could just take a look at the bottom line and of course the standard deduction that got doubled so if this is at 5852 let's say 5852 and go back to the single situation and let's say we get rid of Jill and that sounded bad we got rid of her we're gonna get rid of her she's we're going we're reversing back to where we were before so and then we reverse back not to single but head of household head of household boom so now we're at head of household one and let's bring the income back down to 20,000 20,000 income all right i think i got everything right now so so then we're at the five two three three so five two three three so there's still you know a difference a difference involved and the difference could be substantial because you would have two people filing head of household versus the one married filing joint return so you end up with a substantial difference i did that quite quickly but the general idea is that you'd basically want to do some tax projections uh when getting married if you're in that kind of situation not so that it'll be a determining factor to get married or not but just to basically understand what the taxes will be so so it's not like a like a shock you probably want to do some projections there you know on the lower income side of things with these credits that are going to be involved these refundable credits in particular they they could possibly result in getting married having a tax detriment as opposed to a tax benefit which is often the case uh when when you're not dealing with these refundable credit situations oftentimes now i also just want to point out all the time that this you got to watch out for the scammers there because the scammers in order to maximize everything well we'll often try to use the schedule c right so if you had no income if your income was zero then it's like well you shouldn't you don't even need to file because you have you have no income right and so but uh if you had some income then you might be able to get the the earned income tax credit so you could say well what if we did the the schedule c income right and just also just to point out schedule c income is valid but you also have to be careful of people like scamming the schedule c income because trying to just generate some income that would then increase the earned income tax credit because that's something that you would think the irs would be uh and is you know skeptical on and looking you know possibly looking at so you might get an audit if you do that kind of thing but the schedule c is you know earned income subject to self-employment tax and then could impact of course the earned income tax credit whereas the passive income is not typically earned income thanks from interest and dividends so that wouldn't be included and if you have a lot of passive income then the whole you might lose the earned income tax credit because the idea would be that you got a lot of money in the bank or investments in order to generate that income and therefore don't need the earned income tax credit