 So, today we continue the investigation of the relation with the fundamental group and we will come to what are called as regular cover lines. Before that, we have to consolidate the result that we have proved last time and improve upon them a little bit. Let us go back a little bit here. This theorem it says that induced homomorphism P check is injective. The second part says that there is a surjection theta from pi 1 of x to the fiber P check P inverse of x which is a constant on each right cosets of this k which is subgroup P check of pi 1 of x bar and hence defines a bijection of the right cosets of k with the set f. So, this is purely algebra now. Only hypothesis topology is that P from x bar to x is a covering projection, x bar is path connected, x is path connected, I have chosen a point x bar on top of x. Once you fix this notation, there is a homomorphism P check from the fundamental group of the top space to the fundamental group of the bottom space and that homomorphism is injective, it is a monomorphism. So, the subgroup can be identified under P check with the sub with the group pi 1 of x bar x. Of course, identification has to take place under P check, this map homomorphism is important. Moreover, if you take the right cosets of the subgroup, they correspond in a nice fashion to the points of the fiber. So, this is the statement. Proof is not very difficult, once you have done lemma 1 most important, lemma 2 is purely geometry, everything follows very easily. So, let us look at this one. Let omega bar be a loop at x bar, I am proving the injectivity. Let H be homotopy of P composite omega bar when you go down to a constant loop, it just means that P check of this point, this element in the fundamental group is the trivial element. I want to show that this is service trivial in pi 1 of x bar x, which means that omega bar must be homotopy to constant path as a loop in x bar. When you go down in x, it is homotopy to constant loop. All these homotopy must be relative to end points. So, start with a homotopy such that this homotopy is the starting point is P composite omega bar and there is a constant path. The homotopy is remember it is relative to the end points, these two end points 0 and 1 are constant all the time. Now H bar with a lift of H such that the starting path is omega bar. I do not know anything about that one, this is homotopy lifting property of P I have used that is all for everything. Then if you look at the H bar 0 as just like as what we have seen before, it is inside P inverse of x because this is a constant path below. So, this is inside the fiber therefore as seen above by discreteness of the fiber and connectivity of the interval H bar of 0 x 0 cross i, it follows that this entire thing is one single x bar because that is where you have lifted this whole thing. So, same reason it follows that H bar of 1, S is x bar because it is a loop right. So, all of them are at this point the H bar of T 1 for everything T and S belong to S. Therefore, H bar itself is a homotopy of the of the what of omega bar relative to 0 and 1 to a constant loop. The problem is constant loop the top will be inside the fiber but the fiber is what discrete all the time I am using the same thing therefore it is constant ok. So, P check is injective by this one. The next thing I have defined a set theoretic function from the right cosets of what of k, k is the image of this fundamental group ok to the fiber of P at the point x. So, this is done by defining a function on the group itself and then restricted to cosets observe that it is a constant. So, it can you get a after all cosets set on the set of cosets is a quotient set of of the group by the group by the subgroup right. So, this is what I am going to given a loop omega which represent an element here after fundamental group has to be represented by loops ok. Lift this omega to a path at x bar this x bar is fixed for all this investigation ok and let the function theta be defined theta of omega equal to omega 1 the end point of 1. Any lift this this end point is independent of what class I what loop I take in the class it depends only on the class therefore theta of the class omega is well defined ok. By the first lemma that we proved theta is well defined well defined means what if I change omega to omega prime here omega prime 1 will be also equal to 1. Now take any z inside this fiber x bar is path connected join this one you will like tau be a path from x bar to z in x bar ok. Then see that p composite tau is a loop at x see all these we have seen already I am just repeating it and theta of p composite tau how is got by this you have to lift it you have to lift it at x bar and look at the end point but the end point is there it is started like that there is only one lift remember and it is already a path there and you have taken the image of that path there will be given path. So it is tau itself and end point of tau is z so theta of this one will be z theta of p composite tau will be z. So this shows that this theta is surjective now I have to show that this theta takes same value on the right cosets of k ok. Now I want to say that if it only so that automatically it will be an injective mapping and surjectivity we have already proved. So theta omega is equal to theta lambda what are omega and lambda they are elements of pi 1 of xx ok some loops at x and they represent some class if this happens if and only if the lifts of omega and lambda at x bar at the same end point that is the definition of theta ok. So this is if and only this one but this is the same thing if and only if omega star lambda inverse ok lifts to a loop at x bar. So this also you have seen right this was lambda 2 actually if and only if omega class of omega into class of omega inverse is inside k because this is a loop at x bar ok this what is this k remember k is pi check of pi 1 of xx bar. So this is if and only if this is an element of k k omega is k lambda or k lambda inverse belongs to k it is right cosets. So this is just group theory. So this complete the proof completely analyzing ok whatever happens to the geometry of the slopes lifts and so on converting it into algebra ok. So we shall now investigate the effect of changing the base point in x bar ok of course without changing the base point at x that means you look at fibers of x fiber of x and change the base point there what happens is what we are we are we are interested in now ok and once again x bar is path connected that is important ok. The isomorphism class of pi 1 of x bar being x bar path connected is not affected this part we have seen long back ok change the base point above. But the isomorphism class changes does not mean that p check is the same ok. So can you also say the subgroup does not change that is not clear right. In fact it does supposed to be not true also and this is where whatever we did in the previous theorem comes into picture again. So that has the answer already there. So the regular coverings is the title of this one. For various points x bar in p inverse of x not there may be finite there may be infinite you do not know p inverse of x not is a discrete set right this make we know. Look at all the subgroups p check of pi 1 of x bar x bar ok they are all subgroups that is what we have proved subgroups of pi 1 of x x not and all these subgroups are conjugate to each other verifying relation the best relation that you can assume ok they are conjugate to each other ok. Once again the previous theorem is they contains this proof let us go through that one little bit ok. Take a path which joins x 1 to x 2 x 1 bar to x 2 bar ok its image will be a loop at x. So take alpha going to omega inverse star alpha star omega and then it is class ok this omega is a loop omega is a path from this one but when you go down it is a loop. So this will be a conjugate of alpha right sorry this is this I am taking directly I have not applied p here. So I am in x bar itself I am doing. So this will be a loop starting with alpha if alpha is in x x 2 ok from x alpha is in x 1 it will it will take this one to x 2 is a loop at x 2 start with alpha and x 1 first omega inverse. So omega inverse from x 2 to x 1 then alpha as a loop and then omega from x 1 to x 2. So this will become a loop at x 2 bar. So this defines an isomorphism pi 1 of x bar 2 on to pi 1 this we have seen already. So this is the isomorphism between any two fundamental groups at any two different points of path connected space ok this will depend upon omega of course if I choose another path this may be different this is an isomorphism isomorphism. Observe that when you take p on omega that is a loop. So let tau equal to p composite omega in pi 1 of x when you p of this p of this one becomes tau inverse p omega p alpha something from element into this tau. So tau inverse tau now tau is a loop so its class is an element of pi 1 of x. So when you come down this arbitrary isomorphism through path is becomes a conjugation ok. When you pass on to the base space the above isomorphism becomes a conjugation by an element tau inverse here tau inverse alpha tau ok is that is that clear I mean there any doubt in this one. So pi 1 of x bar x 1 bar p check of this is some k 1 p check of this is some k 2 right take a loop here that will look like p of some alpha where alpha is a loop here look at this this map it will give you a map it will give you isomorphism of this one with another elements of pi 1 of x bar x. But p check of that will be k 2 so but this will give you tau k 1 tau inverse k 1 tau 2 is equal to k 2 when you come down. So let us say the conjugation is tau inverse ok. So various groups k 1 k 2 etcetera all of them their image is of p check when you take the different points inside p inverse of x ok they are all conjugates. So this definition and this theorem automatically leads us to study of normal subgroups because for a normal subgroup all the conjugates are the same conjugating element may be any element from the group right that is by definition right. So this leads to the notion of normal subgroup here. So let us make a definition a covering projection is called a normal covering this normal over it is borrowed from from group theory. I am not sure whether whether group theory has borrowed it from the covering space theories or the other way around ok because group theory was developed by Galois much later than Poincaré ok Poincaré had already studied these things. So one is not sure whether this normal covering was borrowed from group theory too here or the other way around ok. It is also called Galois covering that is definitely for Galois and there is other word regular covering also all three different wordings are used by different authors ok. What is it if the subgroup p check of pi 1 of x bar x is normal in pi 1 of x. This x bar is some point in p inverse of x. If it is normal for one x bar for all other points for all other points in x it will be normal ok alright. So it is immediate from definition that if the group itself is abelian then the conjugation is identity no matter which element you conjugate then every covering projection is normal or you can call it Galois or you can call it regular covering ok. However we shall soon see that there are many interesting spaces with pi 1 of x not abelian. If all pi 1 of x were abelian this would not have been a useless definition because everything is normal after all. So having some topological criteria for a normal covering is quite desirable. This is the definition of normal covering in terms of algebra. So purely in terms of topology what does it mean in terms of geometry what does it mean that is what we want to investigate. But this is already the more or less we have done the basic thing all that I have to do is combine propositions 7.1 and 7.5 immediately ok. So this 7.1 it says this one conjugation and the other one is already there for us alright to this theorem. So we will just we will just put them together ok regular covering ok. So this is the theorem the covering projection is normal if we turn down leaf given any loop attacks inside x all of its loops all loops means what you have to only change the starting point that is all we have no other freedom. If you specify starting point the lift is very different. So starting point you take in any at any point of the fiber all of them will be either loops or none of none of them is a loop. So this is purely in terms of topology if this happens for any one p then p will be a normal covering why can you see why this is true if it is something is normal what does it mean k is a normal subgroup that is the definition right. So what is the meaning of normal subgroup conjugating by any element is again inside that element take a loop omega ok in x if its conjugate some conjugate inside k then only it has a loop as a lift right yes or no. Yes sir. Right if it only if that is what we are seeing if this omega does not belong to no conjugate belongs to this one then none of its lifts will be a loop. So as soon as it is a loop you can conjugate by that element you will get into inside k that is what we are seeing. So because of that theorem this is the precisely the meaning of this one some element some conjugate belongs to k as soon as that happens all conjugates will be also inside k that is a group theory if no if even a one conjugate does not belong then no conjugate will belong that belonging is converted into lifting into a loop or lifting not into not a loop which is does not belong. So group theory part is converted into this one but this is purely in terms of topology ok yeah. So next time I will stop here next time we shall use this one to to illustrate you know not to not to develop any theory but to illustrate that the fundamental group of wedge of two circles this is a figure eight it is fundamental group is non-abelian we will use that non-abelian therefore there must be some subgroup which is not normal everything will be normal if it is so abelian. So we will try to do that first I think and then use a covering ok and then show that some loops will lift to a loop some other loops the same thing some other lift will not be a loop same one single loop below in the covering at different points you lift them at one place it will be a loop another place will not be a loop that will show that the covering is not normal that means the fundamental group the image of pi one of the above space inside the pi one of x is not normal if you have a subgroup which is not normal the group cannot be abelian so that is the way we will do but we will do it next time thank you