 Thanks to you all for coming back for the second installment about algebraic k-theory and trace methods. So today we're going to focus our discussion on topological cyclic homology and cycletomic spectra. But since I feel like a lot of math has happened between when we met yesterday and today, I'd like to begin by recapping some things that we talked about yesterday. So let's begin with a short recap of some ideas from yesterday that will lead into our discussion today. So what was the goal of the overview we talked about yesterday? Well it was the following. We said that for a ring or a ring spectrum A, we wanted to study the algebraic k-theory of that ring. Okay so that was our goal. And yesterday we talked about the trace method approach to studying this k-theory. So we said in particular there are maps which are called trace maps that relate algebraic k-theory to something that's supposed to be more computable. And we went through that yesterday and we found that there were trace maps from algebraic k-theory factoring through what's called topological cyclic homology and then to topological hawk shield homology. Now I want to remember a few key facts about some of these objects that we'll play into what we want to discuss today. So let's just remember a few things. So let's start with THH. What were some key properties of topological hawk shield homology? Well the first thing to remember is that this was defined as a topological analog of classical hawk shield homology of a ring. So we developed that yesterday. And when you do that what you get out of that is an S1 spectrum. And in particular we said that it's this kind of S1 spectrum which was called cyclotomic which we're going to talk much more about today. Now from topological hawk shield homology we then are supposed to be able to define topological cyclic homology. So this thing is defined from THH using that cyclotomic structure. And the point the whole thing that makes trace methods go is that topological cyclic homology is a good approximation to algebraic k-theory. So in nice situations this is a good approximation to k-theory and we can recover information about k-theory by studying topological cyclic homology. Okay so we talked about that last time but the question that we left yesterday on addressed is well this method only works if we can actually compute Tc right? I mean the claim is that that's supposed to be more computable but if we can't compute it you know this method fails us. So the question for today is how do we compute how do we understand topological cyclic homology? So that's the big question and I want to mention right up front that there are sort of two approaches to this question. We'll discuss both of them today. We're going to start from the classical approach. So the classical approach comes out of the classical definition of topological cyclic homology which is due to Boxstead, Scheng and Madsen and I'll recall that for you in a moment. But there's also a new approach to topological cyclic homology due to Thomas Nikolaus and Peter Schultze from work that was published in 2018 and so we'll get to that as well. And part of what I want to do today is talk about these two approaches, how they're related and what are the some advantages of this new Nikolaus Schultze perspective computationally. Okay but to get there we really need to understand the classical approach first and so let's start from there. So this is also recalling something that we discussed last time. So the Boxstead, Scheng, Madsen definition of topological cyclic homology as we gave it yesterday was the following. The topological cyclic homology of a ring A at the prime P was what we get when we look at fixed points of topological Hawkshield homology. So we said topological Hawkshield homology has this S1 action. We look at this cyclic group sitting inside S1 and we can take fixed points. And then we were supposed to take a limit over some maps, r and f. And let me just remind you that this map f, it was easy enough to say what it is. The map f from the Cp fixed points to the Cp to the n minus 1 fixed points, that map was just inclusion of fixed points. And I could do that for any aqua variance factor and that wasn't really special to THH. But the other map are the restriction map. This was the map that I said yesterday uses the cyclotomic structure. Okay, so that's where we left things yesterday in terms of our overview of trace methods. And now today we're going to understand topological cyclic homology. We're going to have to really unpack some of these things. So looking at this, I've immediately got some questions. Here's a question I have. My first question is, what do these fixed points even mean? So what do these fixed points mean? And my second question is, what is this map r? If I'm using it so fundamentally in the definition of topological cyclic homology, we need to know what it is. So those are the two related questions that we're going to start from today. Okay. So to get started, I want to maybe recall one last thing from last time. Sorry, so that was done recalling. I'm not quite. I may recall one more thing, which is that we said topological Hawkshield homology is an S1 spectrum. And I mentioned last time that THH is actually can be constructed as what's called a genuine S1 spectrum. So I want to talk a little bit today to start with about what that means. So what I'm going to do now is I'm going to take a little detour. I'm going to set aside THH for the moment. That's our motivation for these things. But I want to take a little detour where I'm going to say some basic things about aqua variant stable homotopy theory. So let's have a little detour into some basics of aqua variant stable homotopy theory. So topological Hawkshield homology is our motivation, certainly for considering the constructions I'm about to talk about. But this is going to hold in a more general setting. So in particular, I'm going to let G be a compact lead group. And I first want to talk about what does it mean to be a G spectrum or a genuine G spectrum. So a G spectrum, I'll call it X, is built out of G spaces. Okay, so a G spectrum is built out of G spaces, which I'll call X of V. And V here is a finite dimensional G representation. So when we think about ordinary spectra, our ordinary spectra, we think of as being built out of spaces, you know, X1, X2, X3, etc. And now I'm saying these G spectra also built out of spaces, but now they're G spaces. And now I have one attached to these finite dimensional representations of my group, living in some what we call universe, some collection of representations that I'm interested in. Now, these also have some structure maps with them. So we have together with G maps. And what do these structure maps look like? Well, now my structure maps are going to take X, V, and they're going to be maps from, well, now, with my ordinary spectra, what do you have? You have your structure maps where you smash with say, S1. Now I'm going to be able to smash with representation spheres. And I land then in X, V plus W. So representation spheres are one point compactifications of representations. And one way to think about what just happened in this definition is that, you know, with ordinary spectra, we can invert suspension, sort of the idea of what's happening with ordinary spectra. And with these equivariance spectra, you can also invert suspension, but you can invert suspension by representation spheres. Tina, sorry to have interrupted you. Do you consider orthogonal representations over here? Yeah, so I am working in, so what I'm saying right now is like sort of a genuine comment about what G spectra are. But you pick some points at model for these categories. So most of what I'm saying implicitly, I'm living in an orthogonal G spectra. But I'm not going to really dive too deep into what that means. Okay, so, right. So this is what I mean when I talk about these equivariance spectra. And now one of my questions that I wanted to address is what are these fixed points that I've been claiming that we need to understand. So let's let X be one of these genuine G spectra. So X is a genuine G spectrum. And H in G is a subgroup. And I want to ask, well, what are these fixed points? What does it mean to take a fixed, the fixed points of such a spectrum? So I'm going to write the fixed points as X superscript H. Okay, so what, what are fixed points here? So I want to tell you what this spectrum is, this fixed point spectrum. And so I want to tell you what is the V space of this. And the V space of the spectrum is, it's kind of the thing that you might guess it to be, which is that I take my spectrum X, I take the V space of that, and then I take the H fixed points of that space. Now this only makes sense if this representation V is a G representation that was fixed by H. So in other words, what you get out of this when you take the fixed points of a G spectrum is you get a G mod H spectrum. Okay, so that's nice. These are the kinds of fixed points that we're talking about when I say we're taking fixed points of THH. And on the surface, it looks like that was a nice thing to do. It turns out that it was actually not such a nice thing to do because these fixed points are really badly behaved. So maybe that's like a theme in mathematics, things that are easy to define or badly behave. So what do I mean when I say they're not behaved well? Well, for instance, a thing that's true in spaces that you would hope would be true in spectra is the following. I'd like it to be true that when I take X smash Y, I had two G spectra and I smashed them together and take the H fixed points of that. I would like that to be the same as the H fixed points of X smash the H fixed points of Y. And sadly, for this notion of fixed points, those are not the same. Another sort of nice thing that we would want to be true is the following. If I have a G space and I take the suspension spectrum of it, I would hope that when I take the fixed points of that, I get the suspension spectrum of the fixed points. And again, that is not the case. So this notion of fixed points is easy to sort of characterize but is badly behaved in some sense. And I'll say maybe more about that in a few minutes. So there's another notion of fixed points that's better behaved, which is called geometric fixed points. So when we talk about the geometric fixed points of a spectrum, we write it like this, phi H of X, that's one notation for it anyway. And let me tell you how to define geometric fixed points. So the first thing I need, if I'm going to define these geometric fixed points, is I need to define, I'm going to let F sub H be the family of subgroups of my group G, which do not contain H. So family is not just like, I'm not just choosing the word family, it has a technical meaning here. So there's some technical meaning of what it means to be a family, which has to do with it being closed under subconjugacy. I'm not going to worry too much about that right now. But when you have such a family, you're able to attach a space to it, which is called the universal F H space. So I'm going to write E F sub H for the universal F H space. And what this thing is, is that it's a G space. And it's characterized by the following property. So it's characterized by the following. If you have your E F H, and maybe let's add a disjoint base point. So the lower plus is a disjoint base point. When I take fixed points of that for some other subgroup K that lives in my group G, one of two things happens. Either this is equivalent to S zero, if K is in the family, or it's a point, if K is not in the family. Okay, so that's the property that characterizes this universal F of H space. And maybe this is a little bit clearer if I give you an example. And this example is going to be important for us, because this is going to be really, the example might seem silly when I write it down, but this is actually going to be the example that we're going to use going forward. So let's say that my subgroup H is CP, and my big group is CP to the end. Okay, so I'm thinking of CP sitting inside CP to the end. And I want to know what is this family F sub CP. Okay, it's supposed to be the family of subgroups of CP to the end, which do not contain CP. Well, there's not much in that family. So the only thing in that is the trivial subgroup. So what does that mean for us? Well, if you compare this property characterizing E F H to what we have characterizing sort of our usual universal spaces, E G, you see that E F CP is just E CP in the usual sense of contractable space with a free CP action. Oh, I see a question. What is the plus above the example? Oh, right, this plus here. Yeah, sorry. This plus here is and we'll see that throughout is just notation for adding a disjoint base point to make this space a pointed space. Okay, so, so in this, this nice example, what we're recovering is our usual notion of like a universal space E G. Okay, so why do I mention these universal spaces? Well, I meant to be defining what the geometric fixed points are. And here's how we define them. So I have this space E F H plus. And there's a map from that space to S naught, which is just projection onto the non base point. So it sends everything except the base point to the non base point of S naught. And that map has a co fiber. And I'm just going to name the co fiber of that map to be E F H tilde. Okay, so that's just notation. And what are the geometric fixed points then, well, the geometric fixed points of my space are what you get. Sorry, my, my spectrum X are what you get when you take this E F H tilde, smash it with X, and take the H fixed points in the sense that we talked about above. So that might look bad. I mean, maybe it's not so intuitive why we just did that. But I claim that this is a much nicer in many ways notion of fixed points. So now those properties that we wanted before, like that it behave nicely with respect to smash product and suspension specter of spaces, those all now hold for geometric fixed points. So this is a more well behaved notion of fixed points. Let me note something that's important about these geometric fixed points. I'm going to rewrite this sequence that I have here one more time, space it out a little bit differently. Well, if I let me say what I'm doing. So if I take this whole sequence and I smash it with X, what do I get I get a sequence that looks like the following I smash with X there as not smashed with X is just X. And then I get E F H tilde smashed with X, I just smashed my sequence with X. And now I'm going to H fix the whole sequence. So I'm going to take H fixed points of the whole thing. And what do I see when I do that? Well, if you look at this right hand object, well, that was just the definition of the H geometric fixed points. And these are the fixed points that we had above. So what we've just learned is that you always have a map from the fixed points. Sometimes those are called the categorical fixed points to the geometric fixed points. So you always have a map relating the two. Okay, so geometric fixed points, I've just told you are sort of some nicer, maybe notion of fixed points in some ways. But why do they enter our story? So why are we interested in geometric fixed points in the context of what we're talking about today? Well, it comes into this idea of a spectrum being cyclotomic. So we're going to see two definitions of cyclotomic spectra today, a classical one. And then we'll see this or Niklaus Schultz a characterization a bit later. So I'm going to write this as the classical definition. So what does it mean? Oh, sorry, I see a question in the the chat, which is, is X a space or a spectrum so far? My X is a spectrum. These constructions, right, X is a spectrum talking about a genuine G spectrum. So spectra so far. So the classical definition of what it means to be cyclotomic is the following. A cyclotomic spectrum is a genuine S one spectrum X together with compatible equivalences as follows. So the equivalences are supposed to be from the CN geometric fixed points of X back to X for all N. So I'll call this little R. You have to be maybe a tiny bit more careful, or maybe it's just a bit sloppy to write it this way, because the thing that I have on the left here, these fixed points, well, what is this? Well, this is now an S one mod CN spectrum. So you first identify that back to an S one spectrum using the isomorphism given by the end through to consider that as an S one spectrum. So it's a cyclotomic spectrum is a spectrum like this. It's like saying that when you take the geometric fixed points, you somehow get something back that's equivalent to the original spectrum, which seems, I mean, sort of like a strange condition, like why would you ever expect that to be the case? But it turns out that there are examples of these kinds of things. And in particular, topological Hawkshield homology is cyclotomic. Tina, sorry to have interrupted you. We received a question. Are the categorical fixed points Lex or Lex monoidal? Right. Yeah, let me return to that when we talk a little bit later about some properties of these different kinds of fixed points. So the, okay, so topological Hawkshield homology is cyclotomic. And I claimed earlier that that restriction map that our map between fixed points was supposed to depend on that cyclotomic structure. So now we're ready to answer the question. So what is the map are from the CP to the end fixed points of topological Hawkshield homology to the CP to the end minus one fixed points? Well, here's how that map is defined. So I'm starting with the CP to the end fixed points. And I can identify that as taking the CP fixed points, and then the CP to the end minus one fixed points. So I've just rewritten it in that way. And now I have a map. So we said that we always have a map from the fixed points to the geometric fixed points. So I'm going to take these CP fixed points inside and map them to the CP geometric fixed points of topological Hawkshield homology. And I've still got those CP to the end minus one fixed points on the outside. But now topological Hawkshield homology is cyclotomic. So I have a map from the geometric fixed points back to the original spectrum topological Hawkshield homology. And that composite is that restriction map for THH. Okay, so now we've talked a little bit about what are those fixed points in the definition of TC and what is this restriction map also in the definition of TC. And so the question now really becomes, well, how do we compute these fixed points, we're going to recover topological cyclotomology from this kind of perspective, we need to be able to compute these fixed points. Now a sad thing in homotopy theory, or maybe it's not sad, it produces a lot of interesting mathematics, but a thing in homotopy theory is that these these categorical fixed points are in general very difficult to compute. So there's one method that has long been used when you need to understand fixed points, which is to try to relate them to something more computable, which is yet another notion of fixed points. So I apologize, but I'm now going to introduce a third notion of fixed points, which are called homotopy fixed points. Tina, there is a question about the second map used to define the capital R. Is it an isomorphism? Yeah, so that's an interesting point that you've got there, which is that yes, I mean, the condition of being cyclotomic tells you that this map, that last map in that composite has to be an equivalence. But we're not actually using that in this definition of the restriction map because it's going in the correct direction for us. So maybe in to define this restriction map, we actually need a bit less. But yes, it is the case for topological Hawkshield homology or any cyclotomic spectra that that map is indeed an equivalence. Okay, so, right, homotopy fixed points. So fixed points in general are hard to compute. And I claim that this idea of homotopy fixed points is going to be more computationally accessible. But first I have to tell you what it is. So what are the homotopy fixed points? So for X, a G spectrum, the homotopy fixed points of the spectrum are what you get when you take maps function spectra from EG plus to X, and then take the G fixed points of that. Okay, so one question you could ask is like, well, how is that related to the regular fixed points that you take? And here's an answer to that. So you have a projection map from EG plus to S naught. We've already used that same map. That's the map that then projects onto the non-base point. And that map induces a map from X to this function spectrum, functions from EG plus to X. So if you G fix both sides of that, what do you get? Well, on G fixed points, that gives you a map, which I'll call gamma, from the fixed points we had earlier, those categorical fixed points, to the homotopy fixed points. So you always have a map from the actual fixed points to the homotopy fixed points. And in our case that, just to make that more explicit, in our case, that means we're going to have a map from the Cp to the n fixed points of Thh to the homotopy Cp to the n fixed points of Thh. And so what is the idea then? Well, the idea is that if I want to study these actual fixed points, what is the method? So the method is going to be, well, your fixed points are hard. I don't have good tools to access the fixed points. Like just computationally, they're just really difficult to access. But hopefully the homotopy fixed points should be more accessible. So the idea is to study the homotopy fixed points and this map gamma, and hope that we can get information about the actual fixed points. Now, why should we believe that the homotopy fixed points will be better? I mean, frankly, the definition in some ways looks worse. Well, the thing, one key thing that makes homotopy fixed points so much more accessible than the actual fixed points is the existence of nice spectral sequences. So in particular, you have a spectral sequence that computes the homotopy fixed points, it computes the homotopy groups of the homotopy fixed points. And what does that look like? Well, I want to compute, let's say pi star of the Cp to the n homotopy fixed, homotopy fixed points. Let me make that more clear because this really is for homotopy fixed points. And there's a spectral sequence that does that, which you get by filtering eg, for instance. And what is the e2 term of that spectral sequence? Well, it's group cohomology, like classical group cohomology from homological algebra of the group Cp to the n with coefficients in the homotopy groups of topological cohomology. Tina, sorry, we have received a couple of questions for cyclotomic. Can cyclotomicity be expressed as a relation between the G spectra X and X mesh EFCM without taking fixed points? On the top of my head, I don't know. I mean, from this perspective, cyclotomicity, I can't say that word so easily, is really something about fixed points. I mean, the motivating example for why you would ever think about cyclotomic spectrum in the first place, like why this idea of fixing something and getting back the original thing is meant to have any tuition is supposed to come from free loop spaces. So free loop spaces have this property that when you they have an S1 action by rotation, and when you take fixed points, Cn fixed points of that thing, you get back the free loop space. And so I don't I don't know. I would have to think through whether it makes sense to talk about a cyclotomic spectrum if you remove the fixed points completely. Oh, right. So there's a question about the in the definition of small. When you say that the map are small in the definition of cyclotomic spectrum is an equivalence, is it a level of equivalence? Yeah, right. So I was vague here about what I mean for this. I said that these were compatible equivalences. Oh, that reminds me that I also didn't say anything about what I meant by compatible. So let me unpack that a little bit more. So these are S1 spectra and you want you want them to be equivalent in some sense as S1 spectra. Sometimes people require that they require an equivalence of S1 spectra. Sometimes people require what's called an F equivalence, which is a bit weaker. So when you have equivalences of equivalent spectra, you really ask for equivalences on homotopy groups for all fixed points. And in this case for cyclotomic, you can ask for that only for the fixed points with respect to finite subgroups of S1. And that's also a reasonable notion of what it means to be cyclotomic here. So that's the equivalence part of it. The compatible part is you do this for all N and then there's some commutative diagrams I didn't write down that tell you if I look at this map little r for CN or CM or CNM, how are those things all related? And there's some diagrams that that one can write down that I'm just omitting. One remaining question. Yeah, right. Okay, so I can see this question, which is if we're just talking about homotopy fixed points now, what were those other things? The spirit of that question, I'm re-interpreting the spirit of the question, but the spirit of that question as I interpret it is that so these categorical fixed points that I defined earlier, these fixed points up here. So I mentioned that they're badly behaved and I gave you some examples, but they're even more badly behaved than that. If you're, if you don't do a vibrant replacement or if you don't know something special about your spectrum, they're not even really homotopy invariant. So you can have maps that are, you know, you can have an equivariant map that's an underlying week equivalence and it will not necessarily induce an equivalence on fixed points, which is homotopically like a really bad property. It's a fluky thing or I mean, it's not, that's a bad way to say it. It's a true thing that for topological Hawkshield homology, these fixed points actually are well behaved and we'll maybe see a little bit of glimpse of why that's true later, but in general, this is a bad notion of fixed points for that reason as well, is that homotopically it's not really well behaved and that's the issue that homotopy fixed points are trying to fix. Homotopy fixed points are nicely behaved when you have, they respect these kinds of week equivalences. It turns out that the actual fixed points for topological Hawkshield homology do as well, but that's sort of something special about THH. Sorry, too many questions tonight. Can you compare your definition of homotopy fixed points with the one used by NDSS? Yeah, so we're going to talk later about some Nicolaus Schultz perspective and Nicolaus Schultz, they're developing a lot of these same tools that we're talking about now, they need them as well, but they're working in an infinity categorical setting and so they give like a more infinity categorical characterization of these kinds of fixed points that we're talking about like homotopy fixed points right now. Let me return to that once we've talked a little bit more about the Nicolaus Schultz work to have some context for that. Okay, so where was I? I was claiming that homotopy fixed points are more understandable than actual fixed points because we have nice spectral sequences that can compute them, they have E2 terms and some homological algebra things and so, you know, spectral sequences don't solve all problems, these spectral sequences can still be hard, but at least there's some approach that we can use. So that's the idea, you know, at the end of the talk last time I gave a lot of examples of algebraic k-theory calculations that have been done with trace methods and this is sort of the idea that went into pretty much all of those calculations, but it turns out this is not quite enough. We need, we need a little bit more than just this map to homotopy fixed points. We're going to need more structure to try to get a handle on this. So what we're going to use is a diagram which comes out of work of Greenleason May and we see in equivariate homotopy theory this kind of diagram appears a lot. So let me develop this diagram for you and bear with me, things right now are going to start to look like really bad and then they're going to look better again. So bear with me as it starts to look bad. So okay, we had this cofiber sequence earlier and let's write it again. So we had a cofiber sequence that was I take es1 plus and map it to s0, projection onto the non-base point and then I'm going to name the cofiber of that es1 tilde. Okay, so now I'm going to take that cofiber sequence and I'm going to smash the whole thing with topological Hock-Schild homology, which I'm just going to call t, just as notation, just to make this look less messy. So I'm smashing this whole thing with t. Let me now, okay, I'm going to do it right down here. So es1 plus, smash t, maps to t, maps to es1 tilde. Okay, and now we had this map that we defined above. It came up when we were studying our homotopy fixed points. We had a map that looked like this. I'm going to use that same map here now for topological Hock-Schild homology. So I'm going to map from topological Hock-Schild homology to this function spectrum. And I'm going to do that everywhere in my diagram. So here, this is why I say it's going to look kind of ugly. So bear with me. This is going to map to es1 plus, smash some function spectrum. This thing is going to map to that function thing. This thing will map to, oh, I forgot to smash that with t, sorry typo, es1 tilde, smash this function. Okay. So I've done that. And now I'm going to cp fix everything. So everywhere in this diagram, I'm going to take the cp fix points of what I have. Okay. And I claim that that's meant to be helpful, although that's not at all obvious from what I've written so far. I mean, it looks like it's gotten much worse, right? So that looks bad. But let's try to unpack what just happened. So I want to start my unpacking by looking at what happened in the middle. So what do we have here in the middle? Well, the top thing is the thing I'm trying to compute. It's those fixed points that I wanted to compute. Now this bottom middle thing, well that is just the definition of the homotopy fixed points. That was my definition of the cp homotopy fixed points of t. And this was that map gamma. That was exactly the map you always have from fixed points to homotopy fixed points. That was the map I said we were going to try to understand. Okay. So what happened on the left over here? Well, if we look at these things on the left, it turns out that there's a theorem due to Adams and comes out of Adams and Lewis and May that says that that thing in the upper left ES1 plus smash t, the cp fixed points of that is actually something which is a familiar object. It's something that's called the homotopy orbit spectrum. So that's not the definition of the homotopy orbit spectrum. That's a theorem saying that what you recover is the homotopy orbit spectrum. So this upper left is the homotopy orbit spectrum. The lower left also it turns out is the homotopy orbit spectrum. And this map is an equivalence. Okay. So that looks better. Now let's move to the lower right. This thing. Well, this thing is by definition. So this is due to Greenlease and May. This object is just the definition ES1 tilde smash this function thing. And then I take the cp fixed points. That is by definition what's called the tape construction on topological homology. Okay. So this lower right hand side is something called the tape construction. Maybe a word about why it makes sense to call that the tape construction. So let me just note that that object has a spectral sequence that helps compute it, which is called the Tate spectral sequence. And this Tate spectral sequence is a spectral sequence that computes we could just do cp to the end tape construction of a spectrum. And the e2 term of the spectral sequence is classical tape cohomology. So it's the tape cohomology of cp to the end with coefficients in the homotopy groups of T. So we saw earlier a similar spectral sequence that can compute the homotopy fixed points. And now I've got a spectral sequence also for that tape construction. The e2 term now is again in one of these classical homological endurance. This time it's tape cohomology. Okay. And now all that's left in my diagram is what's happening in this upper right. And what is this thing in the upper right? Well, that is by definition the cp geometric fixed points of T. But topological Huxield homology is cyclotomic. And so the cp geometric fixed points just give you back topological Huxield homology. And this map is that restriction map that we defined a minute ago. And this map here is often written gamma hat, at least for now. We're going to rename that in a little bit. Okay. So this is let me box this diagram because this diagram turns out is quite important. And so what is the idea behind all of those algebraic k theory calculations that I mentioned yesterday? What is the idea? Oh, let me say one thing before that. Let me make a note. So this diagram that I drew for you, it related the cp fixed points to the original spectrum THH. But there's an analogous diagram developed the exact same way relating the cp to the n fixed points. So that would be your top middle to the cp to the n minus one fixed points. That would be your top right in the diagram. So you get this diagram more generally. I just did it only for cp for reasons that maybe we'll become clear in a few minutes. Okay, so that's the first note is that we have this analogous diagram in this more general case. And the second note is that the bottom row bottom row is meant to be more computable. So why is that? Well, we've already touched on it. So the bottom row, I have homotopy orbits, homotopy fixed points and the take construction. And all of those things have nice spectral sequences that study them that compute the homotopy groups of those spectra. So we saw the homotopy fixed point spectral sequence I wrote down. I wrote down the take spectral sequence as well. There's also a homotopy orbit spectral sequence. And the E2 term of that one is group homology. So we have these nice spectral sequence tools to understand the bottom row. And so the idea behind all those k theory calculations that I mentioned last time was the following. Well, we want to use spectral sequences for the bottom row. We need to know something about these maps. So we need to know something about gamma and gamma hat. And then usually there's some kind of induction, right? So you work up from the cp fixed points to the cp square to the cp cube, etc. This method has been incredibly powerful in terms of doing algebra k theory calculations. For every different situation that you're working in, for all these different k theory calculations like the ones I mentioned yesterday, this needs to be tweaked. Maybe you need an equivalent version of this that's graded by representations. Maybe you need to adjust it in different ways. So it's not really just executing this dying arm every time. There's a lot of sort of subtlety that goes into it. But this is the idea behind a lot of those constructions. So many, many constructions have been done this way and it's very powerful. I want to emphasize that. But there's a but. So but this approach requires some really serious aquavariant stable homotopy theory. So this approach requires serious aquavariant stuff. And what do I mean by that? Well, in order to study these things, we had to compute actual fixed points, which as I mentioned is always a problem. We needed to go through an understanding of geometric fixed points, homotopy fixed points. We needed to be working with genuine aquavariant spectra in order to make sense, even of some of these constructions. And recent work of Niklaus and Schultz gives a new perspective on what it means to be cyclotomic and the definition of topological cyclic homology that allows us to avoid some of this difficult aquavariant work. So I want to tell you now about the Niklaus-Schultz definition of what it means to be cyclotomic and how it's related to everything that we just did. So as I mentioned, this is coming out of the 2018 work of Thomas Niklaus and Peter Schultz. So what they did is the first thing they did is they redefined what it means to be a cyclotomic spectrum. So they said the following. A cyclotomic spectrum is a spectrum with S1 action and S1 aquavariant maps phi P from X to that Tate construction on X. And you need maps like this for all primes P. Okay. So I'll say more about this in a second. Now we have two definitions of cyclotomic floating around. So let me, to be clear, the previous definition, the one with the geometric fixed points, I'm now going to call genuine cyclotomic. So previous definition I'm going to refer to as genuine cyclotomic. Okay. And we'll call this Niklaus-Schultz cyclotomic or just cyclotomic. Okay. So what's our first observation about this Niklaus-Schultz definition of what it means to be cyclotomic? Well, the first thing that I'd observe is that if I had a genuine cyclotomic spectrum, it gives me a cyclotomic spectrum in the sense of Niklaus and Schultz. So how do I see that? Well, we've actually already seen it. So if X is genuine cyclotomic, we went through all this work above to develop this diagram, the screen lease may, it's also called the isotropy separation diagram. We went through all this work to develop this diagram to help study it. And that, we did it for THH, but really you could do that for any cyclotomic spectrum. And what did that diagram look like? Well, it ended up, once we simplified everything, looking like the following. We had a and so in particular for any genuine cyclotomic, cyclotomic spectrum, then we have this map. We have a map from X to the Tate CP construction for X. That's that map. Now we'll call that Phi CP. That's that map in Niklaus-Schultz's definition of what it means to be cyclotomic. Okay. So a genuine cyclotomic spectrum gives me a cyclotomic spectrum. Oh, and I see there's a question. Yeah. Right. So there's a question about not eventually connective spectrum. Let me come back to that in a minute. So I haven't gotten yet to the point where we need to talk about whether or not our spectra are bounded below. So I'll address that in a moment. Okay. So we've said that a genuine cyclotomic spectra gives you a cyclotomic spectrum, the sense of Niklaus-Schultz. And so in particular, that means that topological Hock-Schultz homology is cyclotomic in this sense. I mean, that's a little unsatisfying to say, well, we knew it was genuine cyclotomic. And so therefore it's Niklaus-Schultz cyclotomic. So Niklaus and Schultz also construct more directly a cyclotomic structure on THH, not going through the genuine structure. So let me just mention that. So Niklaus and Schultz construct a cyclotomic structure directly for THH. And how does that go? Well, I'm not going to go into it in detail, but let me just mention that it uses a map called the Tate diagonal map. So for a spectrum, let's say R, for a spectrum R, there's a map. So in, in spectrum, we don't have in general diagonal maps. We don't have a map from R to R smash R in general. But there is this sort of corrected version of that called the Tate diagonal. So you have a map from R to R smashed with itself P times after taking the Tate CP construction on there. So this is kind of like a replacement for diagonal maps and spectra. And what they do to show to THH has a cyclotomic structure is they use an appropriate subdivision and they show that this Tate diagonal in cooperation with a subdivision induces a map from the topological hawkshield homology of R to the Tate CP of the topological hawkshield homology of R. Tina, there is a question about the definition. Why not to use instead the map from X to X upper TS1 as an alternative to the Nikolaus Schultz definition? We're going to come back to that object, the Tate S1 construction on X momentarily and see how that comes into this Nikolaus Schultz picture as well. So that certainly plays a role here when we get to talk about topological cyclic homology. So we'll get to that momentarily. Okay, so I've said so far that a genuine cyclotomic spectrum is cyclotomic and that topological hawkshield homology is cyclotomic in this new sense. But of course, a big question then is, well, are the notions of cyclotomic the same, right? We would like to know that the Nikolaus Schultz, the argument is that this new definition is capturing what it means to be cyclotomic. So how does it relate to the definition of Bahta Chang and Madsen? And the theorem then from Nikolaus and Schultz, sort of maybe the main theorem there from this work is that, so we have a functor from genuine cyclotomic spectra. So we said any genuine cyclotomic spectra will give us a cyclotomic spectrum in the sense of Nikolaus Schultz. So we have a functor like this. And what they prove is that this is an equivalence of infinity categories as long as your spectra are bounded below. So this is an equivalence of infinity categories when restricted to bounded below spectra. Okay, so in other words, for bounded below spectra, indeed, their definition of cyclotomic recovers the classical definition of cyclotomic that we're so used to using for our trace method approach to algebraic k theory. Okay, so Nikolaus and Schultz now have redefined what it means to be cyclotomic in this new way and checked that at least for bounded below spectra, it agrees with the classical notion. But this new characterization of cyclotomic also allows them to give a new characterization of topological cyclic homology. So in particular, they prove that you can recharacterize topological cyclic homology. So now we've been talking about rings, so let's let A be a ring. You can write the topological cyclic homology then as some equalizer. And it's going to be an equalizer of maps from this from the homotopy S one fixed points of THH. I'm going to take two maps here to the product over primes P of what I get when I take THH of A and take the Tate CP construction and then homotopy S one fixed that. Okay, and there are two. Yes. Another question about gene unification function. Is it describable or is the equivalence proved without having Yeah, right. So that's a great question. So the to go in the inverse direction, it turns out that when you look at these these right, okay, so let me back up for a minute. I haven't really dug into this yet. I'm gonna say something about it in a moment. But when we talked about genuine spectra, what did we need? We needed this genuine notion of a G spectrum. And when you look at Nicolaus Schultz's definition of things, what's happened here? Well, I'm not asking for a genuine G spectrum anymore. I'm asking for a spectrum with an S one action. That's different. That's a much more naive notion of equivariance. And so one key thing is you need to know that you have a map that you need to be able to look at those spectra with G action sitting inside genuine spectra. So that sort of gives you a way of thinking about to go back. And that's that that functor is what's called Burrell completion, which is this this thing that we've been seeing all along, which is like functions from EG plus into your spectrum and then you take right just that. I realized that I wasn't directly addressing your question, which is about the inverse to this functor. Yes, they do need to have some understanding of the inverse to that functor as well. And that's where the bounded below my apologies for that, I feel badly. Okay, so, so, right. Let me continue with where I was before that. So my so niggler central to give a new definition then of topological cyclic homology or or they prove that you can give this new formula for topological cyclic homology. And it looks like the following we can write it as this equalizer, where I have two maps, one of them I'm going to call can for canonical. And the other one I'm going to write p p h s one. So what are those two maps? Well, the, the lower one is sort of easy to see what that is. So I had a map, the cyclotomic structure gives me a map from the topological Hockschild homology to the Tate CP construction. That's the Frobenius map and the definition of what it means to be cyclotomic. And so this map is just the h s one fixed points of that. Homotopy s one fixed points. So that's the lower map that comes directly from that cyclotomic structure. And what is the upper map, this canonical map? Well, the canonical map is then not relying on the cyclotomic structure. To get the canonical map, we say the following. Well, I have these homotopy s one fixed points and I could identify those instead as let's take first the homotopy CP fixed points and then the homotopy s one mod CP fixed points. I'm just breaking that down in that way. And so now s one mod CP as we've discussed is isomorphic to s one. So I could identify this as homotopy CP fixed points followed by homotopy s one fixed points. And then I claim that there's a map from there to the T the CP take construction homotopy s one fixed points. And that map is a canonical map. So what is this map? This last map here. Well, we actually saw it earlier when we developed that diagram, we saw that we always have a map from homotopy fixed points to the take construction. So that's what this canonical map is down here. It's the map from the homotopy fixed points to the take construction. And then we take homotopy s one fixed points of that. Let me just mention that sometimes when you see the Nikolaus-Schultz formula for topological cyclic homology you see it written a bit differently. So let me mention that as well. So you can denote you sometimes see these homotopy s one fixed points denoted as TC minus. So this is notation for homotopy s one fixed points. Somebody was asking about this at the end of the talk yesterday. This is called topological negative cyclic homology. And there's also notation just a moment ago. Somebody was asking about the Tate s one construction. And that is called topological periodic homology. So again, that's notation. So this is topological periodic homology. And Nikolaus and Schultz prove that this right hand side of their equalizer, this thing, this big product that they have over here, that this product is really just a completion of topological periodic homology. And so a lot of times when you see the Nikolaus Schultz, sorry now my iPad is mad, this Nikolaus Schultz formula you see it written as follows that it's the equalizer of two maps from negative topological negative cyclic homology to topological periodic homology, the canonical map and a map induced by the Frobenius. Okay. So the so this is the Nikolaus Schultz formula for computing topological cyclic homology. And I am pretty much out of time. But let me just take two minutes to say why this is nice. So what is the advantage of this Nikolaus Schultz approach? So one reason that this is nicer is because in the as I was touching on a moment ago, sometime when my computer crashed, the classical setup, you need to consider topological Hawkshield homology as a genuine s one spectrum. So you need this genuine equivariate theory, it's built out of these spaces coming from representations. Nikolaus Schultz are considering something much much more naive in terms of equivariance. They're considering just THH as an s one equivariate object in spectra. Another way to think about that is if you look at the Nikolaus Schultz definition of topological cyclic homology, what do you need to compute? Well, you need to compute homotopy fixed points, the Tate construction, homotopy fixed points. Those are all those sort of more accessible bottom row kinds of constructions from equivariate stable homotopy theory, not that stuff on the top row that was so hard to get your handle on. So they've moved away from this like really deep notion of equivariance. And it turns out that that's been very sorry to have interrupted you. We received a question, are the negative homotopy groups of TC minus the same as negative homotopy groups of TC or something? No, I mean then the TC the minus part is because TC minus like, you know, yesterday we talked about negative cyclic homology, the algebraic theory, and we wanted some topological analog to that and we said it was topological cyclic homology. And it is, but topological negative cyclic homology is also in some sense the right, yeah, it's hard to explain. The homotopy S1 fixed points here are not the right thing for the Boksa Zhang-Mazen approach to trace methods. So, you know, people, this object, these homotopy S1 fixed points have been around for a long time, but nobody ever studied them because they didn't, it's not the right notion of topological cyclic homology for trace methods, like there's not a nice approximation theorem relating k theory and this TC minus. But the calling this TC minus is because it is more directly, like if you just took negative cyclic homology and algebra and asked what is the topological analog, this is probably the definition that you would write down, homotopy S1 fixed points. So that's how it gets that name, TC minus. Okay, so I want to end with just a concrete explanation of why, of some results showing how this is nice to consider these things in this less deep notion of equivariance. So let me mention a theorem of Hesseholt-Madsen from many, quite a few years ago now. So they proved this amazing theorem, which is that if k is a perfect field of characteristic p, Hesseholt-Madsen computed all the algebraic k theory groups of truncated polynomials over k. And they have some concrete answer in terms of bit vectors. If you were in the first talk this morning, there was some discussion of bit vectors, but if you're not familiar, it doesn't matter. Just that's a concrete calculation of these k groups, the even groups are zero. So they proved this theorem and this is a beautiful theorem, but this was an incredibly hard calculation. So to do this, the gist of what they did is, well, you need to start by understanding fixed points of this truncated polynomial algebra. And it turns out this thing splits as the topological Hawkshield homology of k smash the cyclic bar construction on some space. This is a pointed monoid. It's the pointed monoid zero one x through x to the a minus one, where x to the a is equal to zero. So I'm not going to dwell on that, but let's just say there is this cyclic bar construction that it splits into. Now that seems nice to have this splitting and it is, but the problem is if you want to compute fixed points of this thing on the left, which you need to to do TC via Box to Chang Madsen, what you need on the right is you need the s one equivariant homotopy type of this cyclic bar construction. In other words, you need to know how to build it from representation spheres. And that maybe doesn't sound like a hard question, but that is an incredibly hard question. So Hessehold Madsen have beautiful work where they describe this equivariant homotopy type, and it leads them to some equivariant calculations. So this leads them to calculate, well, you need fixed points of THH, but because you split it, it's just fixed points of THH of k. So that seems good. But because of this whole bit about the equivariant homotopy type, the kinds of homotopy groups you need to compute now are equivariant. This is homotopy graded by representations. So all you need to know about that if you don't know anything about ROG graded equivariant homotopy theory is that it's hard, right? Ordinary homotopy theory is hard. You throw representations in the mix, it gets even worse. So they were able to do those equivariant calculations, understand this cyclic bar construction and produce this beautiful calculation. But this is really difficult mathematics that went into that. In 2020, so that happened in the 90s. In 2020, Martin Spears reproved the Hessehold Madsen result. So he recovers this exact result that I have written up here above of Hessehold Madsen. But now with this new Nikolaus-Schultz approach, you only need to know the only thing you need about this cyclic bar construction. You still split it like this. But the only thing you need to know about that cyclic bar construction is its homology. Well, a little bit more than that, you need to know its homology and the action of what's called the Kahn's operator. But that makes this calculation way, way more approachable. So as a last closing note, let me just note that similarly, in 2014, Hessehold made a conjecture. Actually, I think the conjecture was earlier, but it was published in 2014 where he was interested in the algebraic k-theory of planar hospital curves. And he had a really explicit conjecture. So this is the k-theory of k adjoin x and y, modulo y to the a minus x to the b, where a and b are relatively prime, and k is a perfect Fp algebra. Okay, so he wanted to compute these groups. And he made in 2014, he published a really concrete conjecture of what are the k-theory groups of these planar hospital curves. The missing piece in his conjecture, the thing that he couldn't figure out how to do, nobody could figure out how to do, is that it depended on the equivariant homotopy type of some cyclic bar construction for some other monoid. And as I said, that question, although it seems simple enough, is a really difficult question, understanding these equivariant homotopy types. So that conjecture stood open until very recently. And then in 2019, Hesseholtz and Nikolaus proved his conjecture. And the nice thing about it is that they were able to avoid that equivariant homotopy type altogether by using the approach of Nikolaus Schultz. Okay, I'm overtime, so I'm going to stop. But let me just mention that on Thursday, when I'm back, what we're going to do is, so today we talked about topological cyclic homology. On Thursday, we're going to talk a bit more about topological Hawkshield homology. And in particular, talk about a relationship between topological Hawkshield homology and equivariant norms of Hill Hopkins and Rabinel, and how that leads to some equivariant generalizations of topological Hawkshield homology and Hawkshield homology. But I'm sorry for going over time and for the disruption, but I'll stop there. It's all right. Let's thank the speaker. And so this is the last talk for today. Any questions at all? Can I ask a silly question? Yeah. Is there a classical way of viewing these constructions? In other words, is there any way to recover this in cyclic, usual cyclic homology? Is there an analog of topological cyclic homology from this point of view of Nikolaus Schultz? Or does this really only live in homotopy land? Some of it only lives in homotopy land. I mean, so like ordinary Hawkshield homology is not cyclotomic. So it doesn't really make sense to try to do the exact analog of this topological theory in the algebra because you don't have that. But there certainly is. You can bring this topological perspective to some of those algebraic constructions and you can view, for instance, negative cyclic homology, like you can think of that as homotopy S1 fixed points of a Hawkshield homology spectrum. Ordinary algebraic periodic homology, you can view as like a tape construction on an algebraic on a Hawkshield homology spectrum. So there are pieces of it, which do translate into the algebra setting, but in some general sense, no, because Hawkshield homology is not cyclotomic. Just one last question, the tape construction. If you just do this in group co-homology, can you write down the same formula as you write for spectra? Does that work? I mean, you have nice relationships between the spectral version and the algebraic version, like if you do the tape construction in spectra for Eilenberg McLean things, you get back algebraic things like you would want. So, I mean, in that sense, yes. Okay, great. Thanks. Okay, Tina, we received a couple of questions, further questions. The first one is, I thought I remember that TCOA was something like THH of A lower HSO1. How are these related? Yeah, topological cyclic homology. So it's not the homotopy S1 orbits of THH. It's closer to the homotopy S1 fixed points of THH, or at least that's like the thing that you might guess it, you would guess it to be is like homotopy S1 fixed points. Because when you think about how we defined TC, it was like you took CP fixed points, then CP squared fixed points, and so on, then you took some limits. So you might think, why didn't I just take S1 fixed points to get TC? The S1 fixed points, you can't do that because they're badly behaved. Like we talked about earlier, sometimes like, in general, those categorical fixed points have bad behavior where you have like weak equivalences that don't induce equivalences on fixed points. The homotopy fixed point S1 fixed points are just not the right thing to be the analog of TC. You don't have good comparison theorems with algebraic A theory. So, you know, you could think of the homotopy S1 fixed points, which we now call TC minus as being some kind of approximation. It's topological cyclic homology, but it's not, it's not the right thing in terms of recovering K theory. Okay, thank you. Another question, can the equivalent homotopy type of encyclopedia recovered from these new calculations now? I mean, in some general sense, no, you're just working around it. I mean, in the specific case of Lars's like conjecture, where it was like he very conjecturally determined this equivariate homotopy type, I'm not sure whether the calculational result forces. Yeah, I'm not sure in some broad sense. No, I mean, you're you the goal is to get around the equivariate homotopy type by only needing to know the homology of that thing. And so you wouldn't expect to recover the homotopy type. In that one case, because the conjecture was so specific, I'm not sure whether you can recover the conjecture from the calculation. Okay, and another question, could you get topological cyclic by taking fixed points with respect to some profinite version of S1? I mean, that's maybe, if I'm understanding the correct question correctly, I mean, you know, maybe that's sort of what you are doing, right? I mean, to get TC with fixed points, you're taking the CP fixed points and CP squared fixed points, CVQ fixed points, et cetera, and taking some limit across all of those. And so in that sense, yes, but maybe I'm not sure what exactly you're asking. Okay. Any other questions for today? We have received three questions. And so. Okay. So, John, you're asking if one has a map from the X to the Tate S1 spectrum where X is naive S1, then is it true that this induces a Nikolaus Schultz? Cyclotomic spectrum. I don't know off the top of my head. I'd have to think about that a little bit. So, if you have a map from X to the topological periodic homology of X, does that guarantee that it's cyclotomic? I'm not sure. I'm sorry. I don't have a good answer for you on that. Immediately clear. Another question, doesn't X equals THH come equipped with such a map from X to XTS1? Yes. Yeah. But again, I don't know if that, like in general, automatically gives you the cyclotomic, but yeah, I'm sorry. It's hard for me to think about that on the fly. I'd have to think about that some more. Okay. I see that there's a question about don't non-homotopy fixed points depend on which model of spectra you're using, which model are we using then? Yeah. So everything that I did is in orthogonal G-spectrum. That was the model that I implicitly was using under everything that I said, but you're right that there are these issues with the non-homotopy fixed points in particular, like if you don't know that in general things aren't fibrant, those aren't going to be well-behaved. I mean, I haven't really said anything about Boxstead's construction of topological Hawkshield homology yet, but the point maybe is that Boxstead's model for topological Hawkshield homology somehow acts like fibrant replacement and ends up giving you this nice construction for which those fixed points are well-behaved, but implicitly everywhere my aqua-variant spectra or orthogonal G-spectrum. Okay. Another question from Andrew Smith. Should not you recover the type of EP wedge and cyclic instead, since we specify the geometric C and fixed points, but not the S1 fixed points? Yeah. I don't know. I mean, I think this is back to the previous question about what would you recover about the aqua-variant homotopy type of the cyclic bar construction? And I don't even know that you would recover that from the K theory calculations themselves. I mean, I would think of it instead as like, you're able to go through some different route where you don't even need that piece as the input data. So I'm not exactly sure what's the best you could get out of that, but I wouldn't expect to fully recover it. Okay, Tina. And the remaining comment, map from John. Yeah. And the dualizing spectrum norm map does it for you? I believe. I think that's just a comment in response to the previous one. Yeah. Two comments. Okay. If there are no other questions, let's thank the speaker again. And so