Robert Bruner: This is the third part of my demonstration of the use of the exit code here which we'll talk about the computing co cycles so here's the.
Robert Bruner: outline of what we want to do i'm going to start all the Co cycles for him for the cosmology of kho Phi renew using a little script called make all co cycles.
Robert Bruner: I will then use do this to compute the action of.
Robert Bruner: On the call Margie em for collect the data from those lists using collect and then we'll look at some of the things that tells us.
Robert Bruner: Okay, so let's go over here to a so let's go down into so we're in the directory here, corresponding to a modules and we'll go down into the directory for him for, and we will.
Robert Bruner: Make unco cycles.
Robert Bruner: Okay, so that's very yes, if you say make co cycles, instead of make our code cycles it doesn't actually do it let's try that oh that works lovely okay So you see here it's making quite a few co cycles, not a surprise because you look at him for here at the top of page you see.
Robert Bruner: That their their dimension over F tues reasonably large Okay, but it didn't take very long, so now we will do all the lists.
Robert Bruner: Oh, before we do that let's look at what this has done to the maps file.
Robert Bruner: In mirroring the maps file before all we had was a single.
Robert Bruner: map, namely the Co cycle that the one co cycle that defined the extension.
Robert Bruner: And for going into em going to the suspension of them one now we've got all the Eco cycle 001011 month to corresponding to each of these co cycles, we see in the co homology i'm for your 00101112 and 31415202122 etc OK so.
Robert Bruner: Now we'll do the lifts.
Robert Bruner: go all the way to filtration 60 or call much could be 60 and.
Robert Bruner: Now Okay, so now the start maps here are actually computing map applied to differential applied to generator.
Robert Bruner: And then, once it's done all the start maps that land in a particular column logical degree, it goes over to the resolution of F two and writes down sections, the sections that it is computed for the differentials and does all the lists at once, so.
Robert Bruner: It is doing all of the lifts in parallel, essentially in that, in that it does all the sections it competes the section for given by degree once.
Robert Bruner: And then it does all the lists that are needed, using that section, so this actually means it's a good idea if you're going to pollute, a lot of chain maps.
Robert Bruner: To do them all at once, because then each.
Robert Bruner: Each lift is a little faster because it only has to compute the section once.
Robert Bruner: Okay, so this is this takes it a little well another thing you can use this for rather than just doing list of all the code cycles, is to find a minimal set of generators for X, to the module.
Robert Bruner: When how you do, that is, you have it, lift all the code cycles, corresponding to generators gone much reserve then look to see which of the co cycles and calm logical green one are not in the image of.
Robert Bruner: Acting on it there's a little script called missing or missing gems missing, which will tell you this.
Robert Bruner: And it then you add those few and call much for the Green one that we're not already hit by the action of.
Robert Bruner: On the extra module and do the list again, and now you repeat and Cohen much good three two etc now in when you're working over a Of course this is.
Robert Bruner: only going to be finite in a finite range, but if you're working over a tune, this is a genuinely finite process and write down a list of the complete list of.
Robert Bruner: X eight have to F to FT generators for X day of to module with two okay looks like it's done here, yes, very good okay so let's collect that.
Robert Bruner: Results of that into something I will call all that action, the action of.
Robert Bruner: On there and let's look in.
Robert Bruner: All that action.
Robert Bruner: To see what it tells us OK so number here is the.
Robert Bruner: Next, for the module and four and down at the beginning, we see only the sort of obvious things that the identity, acting on the bottom generators the bottom generator.
Robert Bruner: let's see one zero F two that's H not acting on each not enough to acting on generator on the bottom generator.
Robert Bruner: Is one zero here okay so that's telling it's the same as what we see from the vertical line in the chart here, so the H I actions, of course, are not so evident, but you can see.
Robert Bruner: Other things that may be less evident let's just scroll forward a tiny bit see there's something looking for this.
Robert Bruner: generator three four tone much for the grade three in the eight stem that should be three three times of if to acting on the bottom class I bet there it is yes, very good Okay, you see that this Class three four here is three three let's see not in.
Robert Bruner: Actually, on the bottom class okay so that's shows you that in here the class C naught is actually divisible by each one.
Robert Bruner: Now of course that's already tuned cool fiber have to as well Okay, the most interesting thing I wanted to show you here, there are two things I wanted to show you first if we jump ahead to filtration six generation numbers 13 and 14 Okay, that will be up here.
Robert Bruner: generator Adams filtration or colon logical degree six generators number 13 and 14 are sitting here in the 30 stem.
Robert Bruner: And we have to such generators so it's.
Robert Bruner: Up to those here okay So here we see two lines for each is i'm.
Robert Bruner: The first line here 613 00 613 simply says the identity element in the collemaggio to acting on 613 is 613 okay that's not that's not news.
Robert Bruner: This one's interesting now this one says 610 in F to acting on 00 is 613 should read this as 613 this read this line is 613 equals.
Robert Bruner: Well, or is contained in perhaps a better way to say it 610 and a half to 900 okay so let's go back and look at the column ology the stinger and algebra briefly 610 here, this is the golem ology.
Robert Bruner: Of the ordinary stinger algebra and much easier Algeria in this 30 stem in degree in college degree six here we have this generator 610 is generally called are or zero.
Robert Bruner: Okay, so that element 610 acting on 00 here.
Robert Bruner: produces 613 well not quite actually if we look down at the next pair of lines.
Robert Bruner: The same line here 614 is showing up here now here's how you interpret that remember this, is the chain map that's being reported here so here's how you look at what's going on.
Robert Bruner: These two lines are telling us that 610, in other words are in the column ology of the steam right algebra X on generator 00.
Robert Bruner: By producing 613 plus 614 now here's how that works if you let SG star be the age generator of ISA s dual to the Co cycle SG.
Robert Bruner: Then here's what we have, we have the Co cycle 00, which is the bottom, corresponding to the bottom, sell them for here we lifted that back to college degrees six at least here, and then we compose the Eco cycle 610 now what is 610 do it evaluates non trivially on the generator 610 dual.
Robert Bruner: Okay, so, then, if I look at the what this chain map is doing is it sending 613 dual and 614 dual here over two 610 dual plus other generators, and when you apply the code cycle six tend to that of course it gets to one.
Robert Bruner: Okay, so looking at it in in this in this fashion, you can see that what you're seeing in the.
Robert Bruner: chain map is of course the duel of what you're seeing the Co cycle, so when you see a line like this in the.
Robert Bruner: All that action here, you have to be careful to look and see that it's I mean that tells you that 613 is a part of the answer, but you look at the following.
Robert Bruner: Entry and you see that six to 14 is another part Okay, so if you simply look at the generators for each sub six and we go back to the chart here, we see that there are two of them in this by degree there.
Robert Bruner: That we just need to see where both of those go under the map, when we apply the code cycles 610 okay so that's how you should actually.
Robert Bruner: You should be careful when you're doing that to make sure you don't leave off one term care or another.
Robert Bruner: and actually it's um This shows, one of the virtues of doing this calculation if we just look at the chart here.
Robert Bruner: We might be tempted to think that the answer would be 614 because it's supports this nice little ah not tower for a ways the same way that our does Nicole Majid esteem right algebra.
Robert Bruner: Okay there's one other thing I wanted to show you in here let's go back to come logical to internal degree not internal very geometric degree 1111 stand here and there's.
Robert Bruner: So we'll be talking about degree seven here in a second let's go over here and look at the summary I prepared in the 11th stem here of X still a of em for there's a class one for down here now, this class one for.
Robert Bruner: If you think about the 11th stem of.
Robert Bruner: The 11 stem for the sphere, you notice that the first class, you see, is in pH to up here in chronological degree five, in particular, is nothing in the filtration one that means this filtration one class here must project to something non trivial on the top so.
Robert Bruner: The S, the S for sitting at the top of coal fire burning.
Robert Bruner: Okay, so that means, then this class must go to a generator.
Robert Bruner: That to the generator Sigma of pile of invest for an average by seven spirit Okay, so this class one for here is actually detecting.
Robert Bruner: Class i'll call Sigma bar a lift of Sigma back to Koh Phi renew now this each night tower here says that pie 11 of co fiber new is cyclical so that's telling us that.
Robert Bruner: Which maps onto the Z 16 and play seven events and then pH to comes in here remember pH tues the periodicity on H2 in Adams filtration five comes in and gets this generator number three up infiltration five here Okay, and if you go back to.
Robert Bruner: Go back a few pages five three there okay five three there, we see that in fact fights three is five to from F to acting on 00 Okay, so what was five two and a half to.
Robert Bruner: That was exactly PhD here Okay, so that confirms what we could guess from just staring at the lines X sequence here.
Robert Bruner: Okay, so this is the other kind of thing you can get from this just knowing the X chart we now see that the lift of Sigma from the top sell back to co fiber new actually generates all of 11 of good fiber new.
Robert Bruner: Okay, well, those are the main things I wanted to show you there's more detail in that archive posting I talked about in that archive posting is also.
Robert Bruner: available with the data set that john rudeness, and I put up on the North shore and.
Robert Bruner: That has this do I here which you can use to refer to it, and you can go download the column ology the mighty stream right algebra that's your degree 184 including all the chain maps and all the differentials and use this in your own exploration of your own.
Robert Bruner: modules you're interested in thanks.