Hello and welcome back to the craft box. My name is Brent and today we are doing some Companion at promotion in here in para 101. We are working on Jane Canary today if you have seen One of the videos that I believe was on my second channel where we did Who was it one of these guys here? Someone it's exactly the same course, but that was on my second channel not my main channel No, hero real heroes only come from aquila. You heard me my dear husband is willing to take this line down But I ain't the eagle mentioned some kind of games the contest a champions I mean to enter this contest and show these toga wearing knit wits What a real hero is your game. Let's ask that philosopher fellow how to join in Right, so that's exciting. This is basically sets it up For all of us and I will be here I'll be back in just a second when we get to hawk reties, right? I am back and we're at hawk teeth. Hello there Well, hello again. What can I do for you today existential dilemma unanswerable enigma shall we debate the contest of champions? Ambitious, I don't think they'll let you compete. You're no eagle after all. They're far too proud of themselves But as I always say the important thing is not to live but to live well Go talk to pindar by the coliseum. Maybe he'll make an exception Okie-dokie they better make an exception for my girl Jane canary. She wants to fight and we got a letter fight But yeah, this is quite a long quest not gonna lie On my second channel which you should totally subscribe to I think it was like over an hour long Luckily, I'm gonna be cutting out a bit of this stuff. So you don't have to see all of that Why I'm not sure a foreigner has ever competed at all such things are kids the rules. I expect Well, there's a first time for everything or what you afraid of having all your heroes shown up by little old canary I'm certain exceptions could be made a test our victory wreaths were plundered by vulture pirates Bring them back and you will compete Okie-dokie, I am gonna skip this because everybody's seen me fight vulture pirates a million times More than a million times in fact. So yeah, I'll be back once we have the victory wreaths, right? I am back That was absolutely painful Well done. You may have the makings of a hero after all the contest has already begun Are you sure you want to begin at a disadvantage? You waste any more time and I reckon I might be at a disadvantage So be it all heroes must excel at defeating monsters the first contest is the great hunt report to Orion in Caledonia Okie-dokie, let's head over to Orion who's an akiyo. We are just gonna mark here because I think we have to come back But also marking three so might as well. We are just gonna get some fuel Really quickly here Just as we run past because we are out as always as always we are out and we need more We need more as always also just ignore the lag that may or may not occur when I'm changing areas because You know, I think having this many quests is not a good idea I have come to that conclusion and it is quite sad quite sad because I would love all of the quests But unfortunately that ain't how it work here. We just just know just know it's quite sad But anyway, let's head over to Akiya, which is not too far away Man, it's been so long since we've quested in since I've quested in Aquila. Oh, I miss it so much So much, but yeah that quest for defeating the vulture pirates was not fun. It was a lot of work I think I defeated Like ten or eleven ships just to get four Yeah, it was nasty. It was really nasty did not like it at all. It was terrible terrible terrible terrible zero out of ten Zero out of ten, but anyway, we've finished it, which is good Hopefully there's not too many more defeating colleagues, but there probably will be because you know that makes sense that makes sense I guess right Here we go. That's run run run as fast as we can Can't catch me because I'm gonna talk to Orion Because I'm the man Where is he? I think he's just down here most likely probably maybe No, he is right actually all the way over here somewhere Is he actually in that gear itself? He is he is he is oh my gosh I did not expect that I thought he would have been on the dock area been like hey come and speak to me here Not all the way up here. Jeez Man talk about exercise. I can't be doing this. I can't be seen doing this. What is this? Here he is. You are here for the contest be aware For only the mightiest can hope to succeed the monsters that haunt these hills are fearsome Do I look scared to you? You are bold outlander. I like that Very well. The first challenge is to hunt down and defeat a dread manticore They layer in caves When you've defeated one take its claws to Pindar as token of your success I feel you're doomed, but the immortals favor the foolish look here I think those are manticore prints shall we track them to a cave? Yep We should definitely do that we should definitely we're gonna get this chest because every time we're on this character We have to get this chest because it is very important to us It has sentimental meaning if you watch the privateer walkthrough you would know what I mean Anyway, let's head over to this cave here and see what's inside. I'm sure you know a dread manticore They sound perfectly fine. They sound like they would not kill me Most likely right they they're not gonna do that right Right Yeah, that's probably a good idea right Let's see How this goes? Hopefully this does not go terribly terribly wrong, but you never know it could oh, it's 2v3. That's um fun That is perfectly fine. We can we can deal with this. I guess we'll do big guns. Um You see the problem is I don't really use Jane canary much So I really don't know how good she is. I don't even know what she has to be honest I'm assuming she has some musketeer talents like burst wire. Maybe that would be cool She has overwatch, that's a good start and double tap. That's even better fantastic. She's not terrible then She is not terrible. She has overwatch three good to know good to know Right another overwatch love to see it Love to see it. Does she have burst? I think she has true grit, which is not going to be helpful here I think that's one that she starts with from memory, but I could be wrong there It just feels like she would have true grit She just seems like that kind of she does overwatch three double tap true grit. Okie dokie fair enough Let's go for this. We are gonna go for this. Um, and yeah, that's good. That sounds good Here we go Let's give her some accuracy and dodge just in case but mainly accuracy is what we're needing here Um, she gets a double tap which is fan tabulous absolutely fantabulous Um, and then what is I get repel borders? Lovely. Love to see it. I'm gonna have to do a lot of heavy hitting here This is gonna be interesting This is gonna be very interesting. Oh, no. Thank you for the block. Thank you for the block though Repel borders. Love to see it. Love to see it Don't hit me. That's rude. Don't hit me. That's rude. We don't like hitting me Um, can we bomb that'd be so good. Yes, we can lovely Lovely, lovely, lovely. We will bomb here like so. Um, you can do this Done sounds good. Get this man's out real fast Real fast just because it's low health already. You never know what could happen You never know what could happen. Okay. This is not good relentless. Oh, no Oh, no, this is not going well Yep, Jane is about dead. Yep, Jane is dead Nearly nearly dead. I'm gonna have to I think I might have to va. I think I have one in my hand I really don't want her to die at this current moment. Um But it might happen. We're gonna summon the mark of death. I Kept that there just in case we needed it and I think we need it to be honest Think we need it. It's kind of getting to that point where it's a little bit sketchy a little bit sketchy But hey, that's fine. I guess Jane just doesn't like critting here I mean, you know, not everybody has to correct, right? It's not like a necessary thing I guess but it would be handy would be very handy if Jane could just create, you know Would be pretty useful. Anyway, that was a quick, um buff Let's go for one of these And you can hopefully finish it off. How much does this do 600?  Love to see it. Love to see it. I forgot I have bladestorm I forgot I have bladestorm. I haven't played on my privateer in so long like actually played. I've just been farming Doing very low level quests at the moment. So I don't really know what I'm doing Don't really know how to fight for like long periods of time because I haven't done it in so long Um Right, we'll do this. I really want to heal Jane But you know what who needs to heal when you can just hit all the time and win, you know It makes sense. Hopefully they just come at me Come at me, bro. Just just me don't come at anybody else. Just me epic It love to see it double tap again. Maybe you know just one that's unfortunate, but hey, it's fine repel borders. Lovely Lovely lovely lovely attack. No, no, okay note Okay, how much absorb does she have left? I'm gonna have to check how much absorb she has left I might have to heal might have to heal here. Um Yeah, 400 that is a definite heal movement time there. Lovely Definitely a heal moment Definitely a heal moment. That is good. She now has 800 health Um, a double tap super or even better. Love to see it. Fantastic.  Right. Let's um Discard this and this and I guess we could just go Um this this I guess and see hopefully this could be good. We'll see maybe possibly focus fire So that increases damage to it. Um, which is fantastic if that crit that would be fantastic Um Cheap shot with the focus fire is fantastic and attack. Please walk Okay, please. Okay Okay, that is not good. That is not good. Oh well Gigi, we did it. We did it. Whoo. And Jane didn't even die. Fantastic. Nice work. Nice work. I did so well So well, let's go claim our prize Yeah, let's go claim our price. That sounds like a fantastic idea. Let's go back to pinda Um, see what we have to do next Um, I believe We have to go back to akea after this as well. So that's fun. That is fun Oh, you've returned hard day. Oh, you did it unexpected But I must warn you there are two contests left and they will grow more difficult Well, I do appreciate the warning, but I reckon I'll just muddle through So do get to the point As you wish for the second contest bring back the golden laurel wreath from sparta It's an archery contest. You're very late Thank you for telling us that we are very late. I love knowing that I am late for things Definitely my favorite part of the day. Right. Let's head back to akea, which is where the archery range is um Yeah, it just takes me a while to get there. I didn't think I bound to a life fountain and I could have just teleported back to the life fountain But hey, whatever whatever.   It's fine. You know, it's perfectly fine Don't worry about it. Don't worry about it. We'll just zoom over Because everybody likes to zoom zoom over everywhere. So it'll be fine Probably maybe most likely, right? Right. It'll be fine. Maybe Luckily, we have so much fuel now. We just have all of the fuel Just so much of it. It is fantastic Fantastic, um, we just need dodged the harpies which may or may not be in my way and they're not which is lovely Absolutely, lovely. Let's dock in akea And exit our ship Load please please please load. Please. Thank you And run over to the archery range, which is just over here I know exactly where it is from last time also because I was so interested to know what this is for Um, and you know, finally know what it is for Let's explore the archery contest and see what is there. I'm sure nothing bad could happen there, right? Well, we will find out. I actually might want to fill up on health Real fast here for the archery contest You're too late. I've just won it win the next contest and perhaps you'll have a chance Um, okay. Oh, I see how it is probably for the best. This ain't really my kind of shooting anyway Still, it's a shame. I never got to see you in action or you me I'd like to know which of us is the better fighter too bad. We'll never know You doubt my prowess Very well, since you are incapable of archery, I won't give you a chance.  I thought you'd never ask Right, let's just heal up here heal up and fight diana Diana seems a little bit cocky and we don't like that here. You know, we don't like that We really do not like that at all. Right.  So we're just gonna let you let you hold for a moment. Please hold Please hold don't do anything it would be bad And she's gonna get all of the fire damages, which is great. We love to see it love to see it We could go and charge her or we could go for a gunnery, which I think we're gonna do You can shoot which is fantastic next round. I might do a epic With jane that would be pretty cool. I think Um, yeah, we'll do that. We'll do that sounds good We will do that Fantabulous farm tabulous. We are going to Go for this and this and you're just going to actually discard um This and This for now And we'll go for that done Done done done done done Hopefully I told jane to do something. I think I did good. Good. I got a slightly worried there, but that's fine That's fine. I don't need to be slightly worried. So it's okay because I did well. Uh-oh True grit. Yeah, here's where true grit comes in Here's where true grit comes in good job Good job. Love to see it. Right. Let's go for a Um, we could go for the big guns Or we could just hit I think we might just we're gonna discard these heels I don't think the heels are gonna be too needed. So we're just gonna hit Um, I think that's probably the best bit here Um, because we do do a lot and if we crit and relentless It's even better even better because we get a blade storm as well. So this is gonna be gg. Nice. That was very very easy Very very easy. I forgot how over how would I am slightly? It's amazing.  You know nothing aboes. This was your intent from the start to tricked me And you walked right into it. You eagles really do think too much of yourselves Wait, I'm off to win the third contest. Bye now Mm-hmm. All right. Let's head back to pinda in nova aquila and then we have to go all the way to ellio So we're gonna cut there because I ship in all the way over there Um for you because that's a long way. That is a long long way. What is this the golden wreath? I feel you are disqualified. You did not win the archery contest Now just a cotton pick in minute. You puffed up pre madonna.  Right. I am back Another runner you're very late. I fear most of the apples have been taken. In fact, there aren't any left in the regular course at all Greetings stranger You are bold and cunning a worthy opponent, but I fear the race has run I have three apples the only way to gather more would be to take them from the etins and yon cave The brutes have wandered off with so many Etins easy Fighting a bunch of them sounds a lot more manageable than a foot race. Let's get to it Uh-huh Right, it looks like we are going to be defeating etins and they're a wee way away. Um, so I'll be back when I am there Listen up. Etins. I'm here for the guy. What do you say? Crush little ones Crush them all Well, I tried Right, um, we need to defeat four of these so I'll be back in just a bit I do not believe it. The deed is done This year the contest of champions is won by an outlander What display of might guile and pertu you have struggled and persevered strange one And now you have earned the prize Which is The right to climb the peaks of acacia and sacrifice yourself to type on the terror Your virtuous death will keep aquila safe for another year What?  Typhon is invincible. I may be many things, but I ain't no coward We'll see just how invincible this typhon is Maybe there's a legend about how to defeat typhon. Let's go ask Daedalus Um, right. So I am back. Obviously that was a very painful painful painful defeating collect quest I defeated so many ittans So many they just did not want to give me the golden apple Um, but eventually I got four. Um, and obviously we're here about to sacrifice ourselves to typhon Maybe possibly I don't know though. We're talking to Daedalus to see if if we cannot sacrifice ourselves You know because that sounds like a better idea To be honest a better idea to not sacrifice ourselves But hey, you don't even know you don't even know it might happen. Anyway, we will see Daedalus. Hello, my friend. How are you today? Typhon where oh, I see you mean to fight him, eh? They say the hot blood of a titan flows in his veins Indeed typhon is so hot that nobody can face him without fainting from the heat or bursting into flame There's only one thing that might help the aegis an enchanted amulet forged by heffestus coppersmith himself It was lost with general cadmus who led his legion into doom in ileos Cadmus and his legion went into the caves of ruin hunting ophidians They never came back find him and you'll find the aegis All right, I'm going to leave it mark here. We're gonna go to ileos doc since that's closest Um, yeah, that sounds like a fantastic idea to get an amulet that will probably be good Maybe we'll see we're gonna have to defeat a couple things Um And stuff and things and yeah, it's gonna be great probably not but you never know It may not be great. Um, yeah, but that quest. Oh my gosh. They really need to fix that. That is, um An extremely low drop rate extremely low. I think I did about six or seven fights just to get four things Um, and that just ain't okay. That is not okay. Just saying not okay Not okay at all just as well. I cut all of it out for you Because man, it was terrible. It was so bad so bad Right, what is in the caves of ruin? Let us find out Um, right, we need to defeat these. Um, I'll be back once I have defeated these Um, because it seems like we're not going to get all of them in a fight at once. So that's fun, right Let's go get that amulet. I'm back and we collected those We defeated those eagles.  I hate them so much I never ever ever ever want to fight them again, but we have to anyway. Um, but yeah, let's defeat cadmus He is boss boy. Um, surely this won't be terrible. There are more eagles I don't like them as I have previously stated. They are the worst things ever. Um, what we're gonna do is we're gonna do that We're gonna do this um, we're gonna do This we gonna do this and we're gonna just chill here for but these guys heal like no tomorrow Um, like as soon as they're low, they just take themselves right back up to full. It's really frustrating Um, so you basically have to kill them like twice, which is not fun at all And it's very very annoying Very very very annoying, but oh well. We're here. Um, we try We try I guess and we'll see what happens Um, yeah, right double tap is lovely burst fire is also lovely Um burst fire again on the actual boss is fantastic Absolutely fantastic. Um, I'm really loving jack russell at the moment. I haven't I don't really use him much, but he is Pretty epic. He's pretty epic. He crits a lot on these guys because he boosts off will Um, and these guys have like no will at all. So Yeah, he does quite well very very well. I'm very proud honestly. It's like a proud father moment. Um Yeah, and he just pops off like that like Just no stopping him. No stopping him. It's amazing. It's Fantastic really absolutely fantastic. Um dodge. Yeah, nice nice nice nice nice Um overwatch again, lovely jane canary with the overwatch is actually pretty decent pretty decent could be better But hey could be worse jane still also through this whole quest has not died yet Um, which honestly, I think is an achievement um I don't I don't want to like jinx anything, but that's pretty big achievement the the worst thing that we had was probably Um, one of the fight the mint the what were they called the menta core fight? Um that we had right at the start that was probably the closest that jane has come to actually Not being with us much longer Um, yeah, that was not good, but hey, she has not done that for a long time. So i'm happy I'm really happy can't be mad at that. We're gonna boost our damage Because damage is good.  Hopefully fingers crossed Um, please double tap onto it. Okay, that's fine. He's gonna heal go back to full health But whatever I guess it'd be like that. I guess it'd be like that, you know It'd be like that. Unfortunately repelled borders. Can you do it? No, I want vengeance strike, but oh well It's fine. Whatever whatever it is fine. I guess Um, bladestorm a lovely relentless. Oh, I had a crit. Oh, even better even better. He's gone nearly Nearly gone nearly gone another bladestorm is good lucky jack has so many bladestorms and this is why This is why he has all of the bladestorm because he just crits so much for no reason I don't even buff as well and he just crits. He's like, yes, friend. I will crit for you I will crit for you every time all the time. I will absolutely obliterate everything that comes and they're gonna heal Okay, he didn't heal. That's good At least this is a vengeance strike, which is great. Another bladestorm possibly maybe not. Um, I doubt it No, no more bladestorms, but hey that is pretty good. This guy over in the back there. He's gonna heal No doubt. Is he gonna heal a hundred percent bear Hundred percent bear. He's gonna heal Yep, told you and he's gonna go like right back up to full health nearly Yep Actually, that one wasn't too bad because he was very very low to start with but hey, it's so annoying those heals They have each of them have one and it is the worst thing ever. It is honestly terrible. I hate it I really really hate it without lucky jack here This would not be gone as quick as as I would have hoped. Um But you're actually not doing too badly Um time-wise, but I have been doing this for quite a while now quite a while Um, I think over an hour Um, but hey that it happens. It happens. I guess Right double tap lovely lovely lovely Burst fight the one time you don't want norcica to just pop off on one enemy. She does Every other time she doesn't At least she got double tap here, which is pretty which could be another double tap. The burst fire is good I'll take that I will take that. Okay, gg. Nice work. That was easy.  So, you know, she is the queen I got the aegis Right, this is I believe in akea. Yes, it is. So I will be back once I am in akea Right, I am back It's full of typhons victims. This was never going to be the easy part. Let's go. We ain't waiting on me Um, and we need to feed skeleton plot spots. Wow. Why did they make this quest so terrible? Um We made it as a long way away. Um, it's like right at the end of the akea area Um, really annoying and now we have to defeat these guys. So I'll be back when we've defeated six of these guys. Oh my gosh Right. I am back.  Let's defeat typhon. This is the final boss of this lovely fantastic Thing we are just gonna go for a full regular normal team here. Um We we get everybody and jane. So that's even better.  Right. Let's um Let's do this. I don't know if he has like too many Cheat things, but we're gonna just pretend that he doesn't Um, so yeah, we'll do that. We'll do that. I slightly worried here, but I think we'll be fine Think we'll be fine untamed inferno Okay, so he does do something of the sort Good to know every time he gets hit he seems to hit his minions. That's fantastic I don't mind that. I don't mind those sort of cheats. They're fun to deal with um Especially with norse could just absolutely obliterating the minions. It's fantastic. I love to see it Absolutely love it burst fire again.  Thank you so much Thank you so much much appreciations much appreciations Right, what is this? This is damaging. Okay. They pick it nice Um, okay, that's just everybody that is not good. That is just everybody around Fair enough. Right. Can we just um not die to this? That would be great Jack Russell's gonna absolutely Pop off here. Hopefully. Um, can we get some more crits? That would be fantastic. Not from you, Mr. Typhon but from um Everybody else would be great. Um, there's an overwatch nice Um, can we get another overwatch? No, that is a shame not quite not quite Um, overwatch here is good. Is that a oh so close so close That's a block lucky lucky lucky lucky. Um, right. This is norsica. That's fine. Whatever Whatever you would wouldn't you you would You would hit norsica you would right um, what we're gonna do is we're gonna do this We're gonna do this on typhon here.  So You know, that's a thing that is a thing we need to worry about Um, right cheap shot is not cool. We don't like cheap shots cheap shots are not available here Not allowed repel borders. No, thank you No, thank you Not today sir Okay, there's the minion gone. Lovely. Love to see it. That was not where I wanted you to double tap, but whatever This burst fire however, that's fantastic.  No, no, please. Please stop. Please stop. No, no, no, no today Um, there's a chest back there as well. We should get that after this after this fight We should really get that I love being in fights where you can just see a chest afterwards. It's great. Um, right This should hopefully do a bit just a bit just a bit there. Can you just stop? I really hate this un Untamed inferno. I really just like it. I I'd really just I just like that it can crit I really do but at least we're quitting on him. So he's dead now. We're just fantastic. Love to see it That's basically the whole fight over really now because no one ever gonna destroy us anymore Um, so yeah, that's good. These guys are really easy. The skeleton hoplites. We've defeated them quite a few times now Uh, we had to do two fights and jack just absolutely obliterates them It is quite sad for them, but not really for me because You know, it just makes it even better Even better. Jane canary is also very good with jack because of the will debuff that her epic gives It's fantastic really really good. So, um, it goes well. It goes well together very synchronized team Um, but yeah, there we go. Bonnie Ann with the final shot there. Lovely. I love to see it We did it boy. Howdy. He was a rough one. Well time to go show these eagles what a hero looks like Yeah, let's um quickly get this chest here because chest is cool. We've got a croaking egg Ooh, we got a pet from that fight. Lovely. Um, let's head back to pinda in nervous aquila We have to walk just a tad bit and then we'll go and finish this off and finally It's been so long since I started recording this holy moly nearly like one and a half to two hours. I would say of just, um Fighting and stuff because yes, I am becoming the contest of champions Jane canary will become a hero once and for all makes sense. So that's fun. Right. Let's quickly run quickly run over to Um, what's his face do do dad pinda thing my bob do hickor. What's it? Where is he? He's over here somewhere He's just up on by that tree right there. I see him. I see him. He's in sight. We can do this Oh so close. It's so far away still. Oh Yes, please. Oh, lovely Typhon is defeated amazing. You are more than a champion friend outlander.  His name was Wyatt Well captain, I reckon it's time to celebrate. Let's head to a tavern cellar Right, let's head down to a tavern cellar. Luckily. I don't have to do this. Um, again Actually, I have to do this again a few times, but I don't have to do this for bill peacock Um, who's privateers also get because he's just like nah, I don't really care Whereas jane was like, yes, it is very important to me Very very important. Um, but yeah, that's cool. I I for those of you don't know I am trying to get all of the companions on this um character So I will be buying a lot of companions and I will have to do this again I've already done this quest once and that was with duck holiday. Um, and this is the second time We've done it and it was still painful both times have been painful That'll teach me to join a contest without knowing what the prize is Still it was worth it to show up those stuffy eagles. That was fun Right, there we go. Jane canary is now a canary Hero look at that. She looks fantastic now Absolutely fantastic. Nice. We coat there. Lovely. We boots. Oh amazing, right? We can give her some effects Not some yeah, we can give her some effects Um, let's give her double two because doubles fun and then we'll give her As well lovely and she can go back to brawling. Good job. Um, but yeah, that is all for today. Thank you everybody for watching I hope you enjoyed. Um, we'll be possibly doing some more of these. I don't know how many more we have Maybe a rat bed one. I don't know. We have already done rat bed though Quite a few times. Would you like to see things done again? I don't know. Leave it down in the comments below Otherwise, thank you for watching and remember to leave a like and subscribe if you subscribe press the little bell I don't know what I said there and remember to craft outside the box


So welcome back to day two of Hager-Florhomology. So what did we do last time? Last time we defined Hager diagrams, and we proved that if you have two different Hager diagrams for the same three manifolds, then they're related by a sequence of isotopes, handle slides, and stabilizations. And so while we pointed, we noted that if you wanted to define a three manifold invariant, well, you can just define one and you can define some invariant of a Hager diagram. And then as long as that invariant doesn't change under isotopes, handle slides, and stabilizations, well, then you've got yourself a three manifold invariant. So that's what we're gonna do today. So goal, so from a Hager diagram H, build a chain complex, we'll actually build two different chain complexes. So the first will be called C of hat of H. This is gonna be a finitely generated, graded chain complex over the field of two elements. I'll also build a chain complex CF minus, which is a finitely generated, graded three chain complex over the ring F of join U.  And then while it turns out these chain complexes, the chain homotopy type is gonna be unchanged under Hager moves, so in particular, the homology is gonna be invariant of the three manifold described by H, so, homology. So H of hat, that'll be the homotopy type. So H of hat, that'll be the homotopy type. The homology of the chain complex CF hat, or H of minus, which will be the homology of the chain complex CF minus is an invariant of Y. So here H is a Hager diagram for Y. Oh, and so all of this, basically everything that I'm gonna say today is due to Algevof Sabo. So I'll write that once right here, but then I probably won't say it every time. Basically every statement I say is due to Algevof Sabo today. They owned, right, so there's these two different flavors, and if I'm gonna, sometimes I'll talk about sort of, I'll say some fact that's true for either of them, and so sometimes I'll write this little circle where a circle can mean hat or minus. There's some other flavors plus and infinity, but they can be determined from, actually from H of minus, so sort of knowing H of minus is sort of, I'm gonna tell you everything that you're gonna wanna know. All right, so before I give you the definition of these chain complexes, I wanna tell you some properties and sort of what structure they take. So let's talk about some properties of these invariants. Great, so the first is that we have the following splitting. So they split over spin C structures on Y. So it just means that, well, it's a direct sum of pieces like this. Okay, and if you don't know what a spin C structure is, well, it's good enough for our purposes to think of them, they're in one-to-one correspondence with second cohomology classes, or if you like first homology classes. So C of minus, C of hat is a finitely generated graded chain complex over F, so in particular, it's homology is also going to be finitely generated, graded vector space. Great, and so we'll know what a finitely generated graded, finitely generated vector space looks like. It just looks like F to the N for some N, and so then in each grading, this just looks like F to the N for some N. So that's what that looks like. Say that again. Do I mean, oh, so the homology splits as a direct sum, and in fact, the splitting comes from the splitting on the chain level. Yeah, so it comes from a splitting on the chain level. So I could have made a statement for either HF or CF. That's right, wait, okay, and then HF minus, right? So it's the homology of a finitely generated graded free chain complex over F, R, and U. So that means that, well, the homology is gonna be a finitely generated graded module over F, R, and U. F, R, and U is a PID, so there's a fundamental theorem of finitely generated graded modules over the PID, which is in flavor exactly the same as the fundamental theorem of finitely generated abelian groups, which says that every such group is a direct sum of free parts and torsion parts. And so you have the same statement here that every finitely generated module of a PID is a direct sum of free parts and torsion parts. We're here by torsion, we mean U-torusion. So what am I saying? I'm just saying that any finitely generated graded module over F, R, and U is of the following form. It's a direct sum of some free parts plus some torsion parts. So the torsion parts are all gonna look like F, R, and U, mod some polynomial, and that polynomial has to be homogeneous in degree because this thing is graded, and so since U is degree minus two, every homogeneously graded polynomial is just a monomial. So it's just U to the nj for some j. So just algebraically, this is what hf minus looks like. I guess there are some gradings everywhere. So I'll put some subscripts to denote the gradings. So here, this denotes the grading of element one in this ring, and then U lowers graded by two. Great, okay. So that's sort of the algebraic structure of these invariants. If y is a rational homology sphere, then we can actually say even more. So if y has the same rational homology as S3, then Ajavar Sabo tell us that there's exactly one free summand. So what do I mean by that?  Okay. And so some of you may have heard of something called the Dn variant, coming from Hager-Full homology, and that's exactly this D right here. Maybe you're worried that this splitting is not canonical, so you can also define D more invariantly as the maximum grading of an element X in hf minus such that U to the n times X is non-zero for all positive n, eight. Since U lowers graded by two, you can check that this D was exactly gonna agree with whatever the grading of one is right here. So another term you may have seen in the literature is something called an L space. An L space is something called an L space, which is something called an L space. So this is something called an L space, and L space is a Hager-Full homology lens space. That is a three-manifold that has the same Hager-Full homology as a lens space. But I guess we can define that in terms of whatever in here. So this U torsion part has their name. It's called hf red, wait. And we say that a rational homology sphere Y is an L space if hf red of Y is zero for every spin state structure. You might be wondering if there's some sort of relationship between the hat flavor and the minus flavor. And there is. So it comes from the following relationship that you have on the chain level. So on the chain level is a short exact sequence, like this, cf minus to cf minus to cf hat. For this map here, so remember, these are chain complexes over f of join U. So there's this action by U, multiplication by U, and this map here is multiplication by U. And so you have this short exact sequence on the chain level, and so this induces an exact triangle on the level of homology. So you have the following exact triangle. So exact just means that the kernel of this, the kernel of any map is the image of this map coming before it. Another fact about Hager-Full homology is that if y is a rational homology sphere, the dimension of Hf hat of y is always greater than or equal to the order of H1 of y. And then it turns out that if you have equality, well, that's equivalent to being an L space. And in fact, you can prove that this if and only a statement are using this exact triangle and the definition over there of an L space. What other properties do we have? You might wonder, how does this invariant behave under connected sum? It satisfies a Cronin's type formula. Cronin's formula. So if you wanna compute, if you wanna know the chain complex of the connected sum, oh, great, so since the chain complex associated to a Hager diagram, well, the chain homotopy type of that is an invariant of the three manifolds. I'm often gonna abuse notation and write the chain complex of a three manifold. But really, I mean, pick a Hager diagram for that three manifold, compute that chain complex. Since this pick any chain complex is chain homotopy to that. Great. So this is homotopy equivalent to the tensor product, Cf hat of y1 and Cf hat of y2. And then you get an analogous statement for Cf minus. And in both cases, the tensor product should be taken over the ground ring. So for Cf hat, that's F, and for Cf minus, that's F adjoin U. All right, another nice property of Hager flow homology is that it behaves nicely with respect to cabortisms, which Chipman alluded to yesterday morning. So if you have a four manifold, W, smooth compact, and the boundary of W is the distant union of y0 and y1. Well, oh, and so let's put some extra structure on here. So let's also equip our cabortism with a spin C structure, S. S induces a map from the Hager flow homology of y0 to the Hager flow homology of y1. So this induces a map from the Hager flow homology of y0. So this denotes the, if you have a spin C structure on the four manifold, well, you can restrict it to a spin C structure on either boundary. And then these cabortism maps behave nicely under composition, if you stack two cabortisms next to each other, lots of composition of the two respective cabortism maps. If you have a product cabortism, that's going to induce the identity map. Another property is that there's a surgery exact triangle. So what's the setup here? The setup is that we have a compact three manifold Z with torus boundary. And now we're going to have three different simple closed curves on the boundary of Z. And these simple closed curves are going to satisfy a certain relationship. So we have gamma naught, gamma one, gamma infinity. These are simple closed curves on the boundary of Z, such that if you look at the algebraic intersection numbers, so let's look at the algebraic intersection of gamma naught and gamma one, that's going to be equal to the algebraic intersection of number of gamma one and gamma infinity. I'm just cyclically permuting them. It's going to equal the algebraic intersection number of gamma infinity and gamma naught. These should all be equal to minus one. So in case you're wondering why did I choose these as my subscripts like zero, one, and then I went to infinity, well, let's look at, let me give you an example of such curves. So here is a torus. Let's say you can check that as I call this gamma naught, this gamma one and this gamma infinity, and then maybe you can choose, I guess the oriented simple closed curves. And then you can check here, will these satisfy this condition? And then while the subscript is just denoting the slope. So that's why I named them that way. And so now, right, so Z has torus boundary, so we can fill that in with this, do a day in filling along each of these curves. So what do I mean by that? Let Y i be Z union of solid torus such that gamma i bounds the disc such that gamma i is equal to a point across D2. Ashroth and Sabo tell us that we get an exact triangle of the Hager flow homology. All right, so as you'll see in the problem session, well, this can actually be enough. So sometimes, like, in certain special cases, if you know two of these and you know that they fit into an exact triangle, that's enough to deduce what the third group is. You know where these maps come from, right? And so if you notice in this setup, right? So why not and why one differ by how we glued the solid torus into Z. And in particular, there's a two-handle cubortism from Y naught to Y one. And this map is exactly the cubortism map coming from that two-handle. Oh, yes, thank you. So if you noticed over there, for that exact triangle, I stated it in terms of HF hat. You're probably wondering, well, is there also an exact triangle, an analogous exact triangle for HF minus? The answer is basically yes. For some sort of annoying technical reasons, you actually have to, instead of working over the polynomial of F or join U, you have to work over the power series thing. So we have an analogous exact triangle for HF minus, except you need to work over the power series thing instead of the polynomial. All right, maybe I'll give you a more concrete example of a family of Y naught, Y one, and Y infinity that fit into an exact triangle. So for example, if you take Z to be a knot complement, and then you take, you can take your curves to be a meridian, an n-frame longitude, and an n-plus one-frame longitude, you can check that those fit into those satisfy the conditions. So that means that you have an exact triangle like this, where this denotes n-frame surgery, and then you can take your curves to be a meridian, n-frame surgery, okay, as you heard about this morning. So those are the formal properties that I wanted to tell you about before I move on. Are there any questions? Great, so now we're going to dive in and define the chain complex from a Hager diagram, associated to a Hager diagram. Great, so the setup is that we have a Hager diagram, so that's a closed surface of genus G, G-alpha circles, G-beta circles, and a base point. Wait, let's assume that our alphas and betas all intersect transversely. From this, we're gonna build, we're gonna build a space. What space are we gonna build? We'll build a sim-G, a sigma. So this is the G-fold symmetric product. So what do I mean by that? We'll take sigma cross itself G times, and then quotient out by the symmetric group on G elements. Points in this space consist of unordered G-tuples of points in sigma, right? So this, well, here's a G-tuple of points in sigma, and then we quotient out by the symmetric group for saying, well, now it's an unordered G-tuple. So this is a space consisting of unordered G-tuples of points in sigma. Now that since we're taking the symmetric product of a surface, this is actually a smooth manifold. So that's one of your exercises in the problem session. I'm gonna describe some subspaces of sim-G for you. Great, so the first will be T-alpha. So this is just alpha one cross alpha two cross alpha G, and then similarly, you could define T-beta, which will be beta one cross beta two cross beta G, and you'll notice these are both half dimensional subspaces. So define, we haven't done anything with the base point yet. So we'll also, we'll use that now to define a subspace, which we'll call VW. So this is gonna be, well, it's gonna be points in sim-G where at least one of the points is your base point W. So a slightly abused notation, but I'll write that as W cross sim-G minus one. Great, so this is just saying, well, at least one of your points is W. This is a co-dimension to subspace. Let's also look at the subspace T-alpha intersect T-beta. So we assume that the alphas and betas intersected transversely, and then we'll T-alpha and T-beta are both half dimensional subspaces. So this is gonna be zero dimensional, so it's gonna be a collection of points. Right, so these are points in sim-G, but we can see them in the Hager diagram, right? So, well, what are these, if you wanna sort of see these in the Hager diagram, what are they gonna look like? Okay, well, it's gonna be a G tuple of points that are intersections between the alphas and betas. Moreover, each alphas circle has to be used exactly once and each beta circle has to be used exactly once. So, at the end of yesterday's lecture, we ended with this Junus-4-Hager diagram for the trough oil, but if you forget one of those base points, well, it was just a Hager diagram for S3, and so a good exercise to sort of make sure you can understand these points in T-alpha intersect T-beta is to sort of find all of those points in that Junus-4-Hager diagram, right? So, it's a G tuple of intersection points between the alphas and betas, where each alphas curve is used exactly once and each beta curve is used exactly. I'm making such a big point about that particular subspace. The reason I'm doing that is because the chain complex, CF hat, so it's generated over by T-alpha intersect T-beta. Any questions so far? Let's move along with the definition then. So, let X and Y both be intersection points between alpha and T-beta, and D is gonna be the unit disk in the complex plane. We'll make the following definition. Whitney disk, we'll say from X to Y is a continuous map phi from this disk to SIM-G of sigma, satisfying the following four properties. So, maybe I'll draw a picture of this disk. So, here's minus i, here's i, and then let's also label the boundary of the disk. We'll call the part of the boundary the right part, but we'll call this e-alpha, and we'll call the left side e-beta, and then we'll require that phi sends minus i to the intersection point X, that phi sends i to Y, that phi sends e-alpha, that that lands in T-alpha.