Real and Hyperreal Equivariant and Motivic Computations
This is a modal window.
Das Video konnte nicht geladen werden, da entweder ein Server- oder Netzwerkfehler auftrat oder das Format nicht unterstützt wird.
Formale Metadaten
| Titel |
| |
| Serientitel | ||
| Anzahl der Teile | 29 | |
| Autor | ||
| Mitwirkende | ||
| Lizenz | CC-Namensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. | |
| Identifikatoren | 10.5446/50928 (DOI) | |
| Herausgeber | ||
| Erscheinungsjahr | ||
| Sprache |
Inhaltliche Metadaten
| Fachgebiet | ||
| Genre | ||
| Abstract |
|
7
8
9
24
26
00:00
VerschlingungFormation <Mathematik>Kategorie <Mathematik>Hierarchie <Mathematik>KoeffizientPerspektiveKontrast <Statistik>Komplex <Algebra>Physikalisches SystemKugelEinfach zusammenhängender RaumFunktorKlassische PhysikPunktPunktspektrumModulformGruppenoperationDivergente ReiheMinkowski-MetrikVektorraumbündelChromatisches PolynomBerechenbare FunktionHomotopieÄquivariante AbbildungReelle ZahlAlgebraisches ModellZweiFinitismusElement <Gruppentheorie>Sortierte LogikGradientenverfahrenp-BlockGebäude <Mathematik>Abelsche GruppeHomotopiegruppeThermodynamisches GleichgewichtWärmeübergangFaserbündelGeometrieVektorraumKomplexe EbeneVollständigkeitFreie GruppeProzess <Physik>MultiplikationsoperatorVollständiger VerbandInvarianteEinsLemma <Logik>TopologieEinheitssphäreMathematikMengenlehreDruckspannungFokalpunktCW-KomplexKlasse <Mathematik>ÄquivalenzklasseGefangenendilemmasinc-FunktionComputeranimation
09:47
PrimzahlzwillingeGebäude <Mathematik>Grothendieck-TopologieWitt-AlgebraAbelsche GruppeEinheitssphäreIsomorphieklasseDualitätstheorieFunktorRechter WinkelKategorie <Mathematik>Sortierte LogikMengenlehreWärmeübergangRichtungDifferenteProdukt <Mathematik>KovarianzfunktionFinitismusPhysikalisches SystemObjekt <Kategorie>ÄquivalenzklasseInhalt <Mathematik>UnendlichkeitGeometrieMultifunktionOrbit <Mathematik>VollständigkeitMereologieMultiplikationsoperatorHelmholtz-ZerlegungBurnside-VermutungMorphismusGammafunktionInklusion <Mathematik>EinsGesetz <Physik>Kartesisches ProduktMinkowski-MetrikPunktKonditionszahlPrimidealUntergruppeKugelDiagrammGruppenoperationKoeffizientNichtunterscheidbarkeitHomotopieKontrast <Statistik>BimodulAlgebraische StrukturAdditionÄquivariante AbbildungGibbs-VerteilungComputeranimation
19:16
Machsches PrinzipRuhmasseRangstatistikBurnside-VermutungBerechenbare FunktionInnerer AutomorphismusAxiomWärmeübergangGraphfärbungPunktFunktorGruppenoperationNormalvektorBimodulProdukt <Mathematik>KonditionszahlSchlussregelMengenlehreKlasse <Mathematik>MultiplikationsoperatorHelmholtz-ZerlegungOrbit <Mathematik>Sigma-AlgebraDiagrammKategorie <Mathematik>GammafunktionHomologieIndexberechnungFolge <Mathematik>PunktspektrumFinitismusTermNichtunterscheidbarkeitDifferentialBernštejn-PolynomGeradeArithmetisches MittelSummierbarkeitFunktionalElement <Gruppentheorie>MultifunktionEinfacher RingTeilbarkeitPhysikalische TheorieDreieckVertauschungsrelationZweiMultiplikationSuperstringtheorieNichtlinearer OperatorRechter WinkelMatrizenrechnungSummandLeistung <Physik>Ausdruck <Logik>DifferenteEigentliche AbbildungDelisches ProblemMathematikAuswahlaxiomLie-GruppeRichtungQuotientenraumHomotopieVarianzKovarianzfunktionFourier-TransformierteKonstanteComputeranimation
28:46
Algebraische ZahlMachsches PrinzipEuler-WinkelGruppenkeimMedianwertAlgebraisches ModellSortierte LogikNormalvektorFormale PotenzreiheHierarchie <Mathematik>Kategorie <Mathematik>UntergruppeWärmeübergangFunktorIndexberechnungPunktspektrumAggregatzustandKugelEinfacher RingPhysikalisches SystemFinitismusGeradep-BlockMaßerweiterungHomologieAlgebraisches ModellUnendlichkeitStellenringKartesische KoordinatenPhysikalische TheorieModelltheorieTermNichtlineares GleichungssystemGruppenoperationMereologieQuotientMinimalgradGreen-FunktionHomotopieKoeffizientGruppendarstellungPunktOrdnung <Mathematik>DifferentialBootstrap-AggregationÄquivariante AbbildungAlgebraische StrukturVerband <Mathematik>Objekt <Kategorie>Abelsche GruppeInvarianteTheoremFolge <Mathematik>Abelsche KategorieBimodulKompaktifizierungArithmetisches MittelVektorraumKomplexe EbeneMultiplikationsoperatorKalkülRiemannsche ZahlenkugelHomotopiegruppeKozyklusVerschlingungEndliche GruppeEinheitssphäreMinkowski-MetrikMultifunktionKlasse <Mathematik>Nichtlinearer OperatorInjektivitätProjektive EbeneMengenlehreNichtunterscheidbarkeitBerechenbare FunktionKnotenpunktMultiplikationNegative ZahlKompakter RaumGebäude <Mathematik>KovarianzfunktionPerspektiveCW-KomplexKohomologieOrdnungsreduktionKonstanteNatürliche ZahlProdukt <Mathematik>Computeranimation
38:16
InvarianteMultiplikatorMaß <Mathematik>WärmeübergangKugelMereologieResiduumEinsPunktspektrumÜbergangswahrscheinlichkeitHomologieReelle ZahlFigurierte ZahlBewertungstheoriePunktNegative ZahlTheoremAnalogieschlussMultiplikationsoperatorGradientKoeffizientKozyklusAlgebraische KörpererweiterungFolge <Mathematik>FunktorKompaktifizierungMathematikMinimalgradNichtlinearer OperatorInnerer AutomorphismusKovarianzfunktionMinkowski-MetrikMultiplikationFunktionalDifferentialKomplex <Algebra>Äquivariante AbbildungMotiv <Mathematik>GruppendarstellungFormale PotenzreiheImaginäre ZahlHomologiegruppeSigma-AlgebraPhysikalische TheorieDreiecksfreier GraphKonditionszahlGruppenoperationTermOrbit <Mathematik>KohomologieKonstanteApproximationÄquivalenzklasseBerechenbare FunktionHomotopieFormation <Mathematik>Klasse <Mathematik>Kategorie <Mathematik>Prädikatenlogik erster StufeVorzeichen <Mathematik>Sortierte LogikSinusfunktionDifferenteAssoziativgesetzUrbild <Mathematik>Computeranimation
47:46
Pi <Zahl>Ideal <Mathematik>SpektralmaßKeilförmige AnordnungMachsches PrinzipAlgebraische StrukturFunktorIdeal <Mathematik>Folge <Mathematik>KonstantePermanenteWärmeübergangResultanteDreiecksfreier GraphDifferentialBimodulLeistung <Physik>Einfacher RingNichtlinearer OperatorPunktspektrumVertauschungsrelationNormalvektorTensorproduktMonoidNumerische MathematikKlasse <Mathematik>MultiplikationsoperatorFuchs-DifferentialgleichungModulformKonditionszahlKategorie <Mathematik>EinsUrbild <Mathematik>Statistische SchlussweiseTensorNebenbedingungMaßerweiterungMonade <Mathematik>Arithmetisches MittelGruppenoperationPunktZahlensystemHomotopieHomologieSymmetrische MatrixObjekt <Kategorie>KoeffizientMereologieAdditionEinhängung <Mathematik>Ordnung <Mathematik>KohomologieZweiEinfach zusammenhängender RaumComputeranimation
57:15
Machsches PrinzipNormierter RaumMittelwertPrimzahlzwillingeOrdnung <Mathematik>MultiplikationsoperatorSpektralanalyse <Stochastik>NormalvektorEinfacher RingSortierte LogikGruppenoperationKategorie <Mathematik>Algebraische StrukturMonoidTensorGeradeGruppendarstellungResultantePunktspektrumFunktorFolge <Mathematik>MultiplikationElement <Gruppentheorie>Produkt <Mathematik>Nichtlinearer OperatorSummierbarkeitPhysikalische TheorieWärmeübergangTeilbarkeitFilter <Stochastik>Objekt <Kategorie>Orientierung <Mathematik>TheoremHomologieSigma-AlgebraLeistung <Physik>DifferenteDifferentialKartesische KoordinatenQuadratzahlHomotopieÄquivariante AbbildungBerechenbare FunktionSchlussregelKlasse <Mathematik>GammafunktionInnerer AutomorphismusLie-GruppeGraphfärbungComputeranimation
01:06:45
RangstatistikLokales MinimumGewicht <Ausgleichsrechnung>Sortierte LogikMultifunktionMultiplikationMathematikPunktPunktspektrumGruppenoperationDifferentialKonditionszahlPhysikalische TheorieAnalogieschlussNormalvektorAggregatzustandFolge <Mathematik>TermTVD-VerfahrenAlgebraische KörpererweiterungProendliche GruppeHomotopiegruppeMengenlehreModelltheorieBimodulKategorie <Mathematik>Dreiecksfreier GraphEinfacher RingKnotenpunktFinitismusResiduumQuotientMaß <Mathematik>StellenringNichtlinearer OperatorÄquivariante AbbildungTheoremMultiplikationsoperatorHomotopieVektorraumPhysikalisches SystemLie-GruppeMereologieMinimalgradKompakter RaumUntergruppeIndexberechnungGrenzschichtablösungAlgebraisches ModellUnendlichkeitVertauschungsrelationFunktorVerschlingungHierarchie <Mathematik>WärmeübergangAdditionOrdnung <Mathematik>Berechenbare FunktionBootstrap-AggregationComputeranimation
Transkript: Englisch(automatisch erzeugt)
00:17
Thank you, thank you very much for the invitation to speak. I'm really excited to be able to participate
00:25
Albeit remotely. I'm glad that that worked out Less glad that it had to work out kind of wishing that the coronavirus were better under control especially here in the US but Yeah Since I'm given the last talk I get to to also say I think we should thank the organizers
00:48
They've done a wonderful job taking an An in-person conference and very quickly turning it into a great online conference So, please join me, I guess somehow
01:02
Yeah, Thank You Ian and great job This is seriously fantastic. And I know it was a lot of work. So thank you very much Okay. So as I said, I'm I'm gonna talk about Techniques of computation in
01:20
Equivariant and and then some in motiva comitope Trying to hit on as many of the key words from the title of the conference as possible I'm Aiming this to be more introductory than Sort of the research side and please do if you have questions
01:44
Ask away and I'll try to answer them as we go through Most of my focus in the talk is going to be on the equivariant computations a lot of them for for everyone's maybe second favorite group the group with two elements and
02:05
the reason I'm going to be focusing on this one is is Well several fold First the group with two elements, which I'll call c2 from now on is the Galois group of the complex numbers over the reals This ties it to a lot of geometric and algebra geometric
02:25
Concepts namely if I want to talk about say descent for real vector bundles It's the same thing as understanding a complex vector bundle together with this descent data of a c2 action Second a lot of classical and chromatic
02:43
computations can be seen in this c2 equivariant story and actually I know that that you all saw Dan Isaacson's series of talks earlier in the conference where he talked a lot about the connections between Motivic over R. Motivic over C. C2 equivariant homotopy and and
03:04
Classical homotopy, so I'll pick up on some of those themes as well And then finally and this one really I should have led with because in some ways it's the most important from my perspective We can actually do computations here a lot of the literature about equivariant homotopy theory tends to
03:25
Suggest that computations are essentially impossible Some of which go so far as to say it's impossible to do some of these I I don't find that to be the case and I hope that by the end of my talk You agree that a lot of these computations are much more doable
03:45
Than you may have thought initially Okay, so I'm going to start by just saying that we're going to be working in the following context I'm going to be working for
04:03
I Working in what people sometimes called genuine equivariant homotopy now I
04:23
I Hate the word genuine here. It's for two reasons. It's very value laden especially when you compare it to the Contrast we will talk about naive equivariant or genuine equivariant. There's a Distinct hierarchy established there and it's not actually supported in the math
04:44
So what I would advocate for and I hope I can start to get traction in this is not to call this genuine but rather to call it something instead like complete and the complete here means that we have
05:04
All Transfers and this is a theme that I'm going to spend a little bit of time talking about as we go forward But before I do that, I'm actually going to start with just a little bit of a review
05:22
So, how do we talk about? Computations and where do the invariants live? So, how do we Understand
05:41
How do we understand homotopy groups g-spectra and g-spaces and Dan also
06:01
So sorry, there is a comment by Yuri Sulema He says that one version of Naish g-spectra is often called Borel complete Yeah, that's true. I don't understand the frowny face there Yuri It it is the case that that the homotopically meaningful version of
06:22
Naive g-spectra is the is Borel spectra and here We don't necessarily have transfers. These are things where we sort of free up the action historically And I guess the not having transfers is is probably why you had the frowny face so I'm with you on that one
06:45
So in the equilibrium context the first thing that I run into is I can't get away from thinking about homotopy sheaves instead of homotopy groups and Of course the real way we should be doing algebraic topology and sort of this ideal world where we have complete control over
07:05
Everything is I would be able to just immediately tell you what maps out of any say finite CW complex work I Love to be able to tell you what maps out of any finite CW complex are We approximate this instead by restricting attention to maps out of the building blocks of finite CW
07:27
complexes namely spheres I'm going to do the same thing for for G spaces or G spectra so we'll consider we'll look at
07:41
Functors and these are functors from the category which I'll call fin G op into say abelian groups And this fin G op this is the category of finite
08:01
G sets and equivariant maps And I'm going to do this just via the unita lemma And so the ones I'm going to care about are I'm going to take equivariant homotopy classes of maps
08:21
from I'm describing a functor from t plus so t together with a disjoint base point smashed with the n sphere into some fixed G Spectrum II and my superscript G here is just reminding myself that I'm taking that the collection of equivariant maps
08:46
This is this is a Contravariant functor as written because I'm mapping out of the t plus slot and so in particular it fits into this form and this is called the
09:03
homotopy coefficient system Already, I'm suggesting a way that I should be thinking about my GCW complexes the CW complexes
09:24
I'm going to build not just out of spheres with a trivial action But rather out of spheres Again with a trivial action But I allow myself to to take disjoint unions of these and to permute the copies of the spheres in those stacks
09:40
And that's how the group is going to be acting So I can map out of this and this amounts to picking out maps from spheres into various fixed points so the first thing to notice is Is that if I have a disjoint union of things so T disjoint union T prime and then I
10:03
take the Disjoint base point and smash this with the n sphere map this into E then the inclusions of T and T prime into the disjoint union give me a Pair of maps
10:21
Backwards And so I get a Decomposition oops like this So in other words my functor from finite g-sets up into abelian groups isn't any old functor
10:46
It's one that takes the disjoint union, which is the co-product in finite g-sets Which makes it the product in finite g-sets up to the product in abelian groups So in other words this construction gives me a product preserving functor and I want to stress here that
11:05
I'm in the algebraic context So saying that I'm a product preserving functor is a property of the functor rather than additional structure Well, any g-set has an orbit decomposition. So this functor
11:23
Slot plus smash SN into E is determined by The values on
11:41
Orbits by which I mean transitive g-sets So g mod H as H varies over the subgroups and if you haven't seen this before then I would suggest that you spend a little bit of time thinking about What what's the geometric content?
12:04
Maps out of g mod H equivariant maps out of g mod H into some g space And you can start to see the interplay between fixed points for various subgroups and then she itself Okay, now this is the kind of thing that I didn't need to be working in
12:24
G spectra. Yeah, I could have made sense of this in g spaces Provided N was at least 2 and If I'm in G spectra any kind of G spectra Be it the Borel ones that Yuri brought up be it the complete ones that I'll be working in or anything in between
12:45
I still have these homotopy coefficient systems the key feature of the this complete Equivariant homotopy is that I have not only these Contravariant restriction maps, but also the covariant transfer maps
13:05
so I'm going to define a category the Burnside category of G
13:21
has objects finite g-sets and the morphisms So home in the Burnside category from s to t is going to be
13:43
Well, I'll be a little glib here and just put parenthetically the group completion the set of Correspondences s and t so here I have two
14:06
Equivariant maps f and g and then I'm doing this up to isomorphism So again, I'm doing I'm going to be working in an algebraic context. So I'm working up to isomorphism as
14:22
as you know from Clark's work or Angelica's then I could have instead considered an enrichment of this be it one where I have a two category and Instead of considering this up to isomorphism I remember the isomorphisms as the two categorical part of the data or an infinity category where I build much larger
14:45
Diagrams again recording isomorphisms and various pullback conditions. Oh If I'm going to say that I have a category I need to say what the composition law is and composition is via pullback
15:08
so given to Correspondences then I can pull them back and I get another correspondence Okay, so In this category since the category is the same as that category of
15:23
Excuse me The objects are the same as the category of finite g-sets I can still talk about things like disjoint union and Cartesian product In this category though well the Burnside category it's canonically self dual and
15:41
by canonically, I mean it's the identity on Objects and then I just observe that I have my correspondence which has you know Maps in two different directions and That's just an sort of an artifact of the way I'm writing I'm choosing to read from left to right because that's that's the only way I know how to read English
16:03
but I could have instead swapped it and Gone from right to left then I would be seeing instead from as written right here That would be the same thing as calm from T to S. So a is oh, I should have given this a name
16:21
sorry script a so a is Canonically self dual and the disjoint union is
16:41
now both the product and the co-product and If you haven't spent any time working with this category sort of thinking through what this might look like I would suggest seeing for yourself how the disjoint union could possibly be the product in other words
17:06
See, how do I write down maps in the burn side category from T disjoint union T prime back to T and back to T prime Whereas in general, I'm not going to have those maps just in finite G sets
17:22
So the the players in the equivariant context in this complete one are maki functors these are so a maki functor is a again product preserving
17:48
functor from the burn side category into abelian groups and I'm always going to indicate my maki functors with an underline So that there'll be a little bit of type checking to contrast these with abelian groups
18:07
Again any G set can be decomposed into orbits and if I use that orbit Decomposition I get that a maki functor is determined by a much smaller amount of data
18:22
So let me spell that out For G is CP and I'm going to name a generator CP. So choose a generator to be say gamma Okay a CP maki functor is the following data of first
18:58
an abelian group M of G mod G, which is a point
19:10
Second I have a CP module M
19:21
CP and Third I have maps a restriction Which goes from M of a point to M of CP and a transfer
19:43
M of CP To M of a point and I'll often write this as a little diagram that That people call a Lewis diagram after Gaunt's Lewis
20:02
my restriction goes like this and My transfer goes up and then I have an action of my group CP on the on the CP module and then these satisfy a few axioms
20:24
so first the restriction the restriction lands in The CP fixed points and the transfer factors through
20:48
the coin variance and then second I Have a condition called the maki double coset formula
21:01
Which says that the composite of the restriction with the transfer is The sum over the elements of the group which is sometimes called the trace If you use that Galois theory names
21:23
Okay, so that's it and how am I supposed to connect these to Correspondences. So how am I supposed to see this as something coming from the Burnside category? Well, remember that I have in the Burnside category I have Or excuse me in the category of finite CP sets. I have a map CP to a point
21:46
That's just the crush everything map and I have a map from CP to itself. That's multiplication by gamma and This gives me a little commutative diagram because point is terminal
22:02
Now when I think the category of finite g-sets I can embed that Covariantly as The forward direction map in the Burnside category or I can embed it Contravariantly as the backwards map in the Burnside category and if I embed it
22:23
Contravariantly that gave me my restriction map This one and if I embed it covariantly that gave me the transfer map So both of these two maps here the restriction and the transfer arose from this quotient map from CP to a point and
22:42
Then the first conditions this one about the image of the restriction landing in the fixed points or the transfer factoring through The orbits are exactly summed up in the commutativity of this little triangle So it's it's actually just the funked reality Condition and
23:02
Finally this Mackie double coset condition. This is what you see if you pull back CP over a point with CP over a point and the pullback is CP cross CP and then I want to write that in terms of CP sets so I want to break it up into its orbit decomposition and when you do that you get exactly this condition
23:27
Okay, let me make it a little more concrete Because I will actually do a couple of computations later and I want to be able to to use these so there's the There's the representable functor the Burnside Mackie functor
23:44
the value at a point is given by Z direct sum Z and the value at CP is Z and Then the restriction and transfer maps
24:05
I'll just write them as little as little matrices. This one sends the first thing to one and a second to P and the second is 0 1
24:21
the vile action here is Just by the identity so as I said, this is actually the The functor I get by mapping out of a point in The Burnside category again, this is product preserving because District union was the product and so it's it's that it's literally the universal property of the product to say that hum
24:47
Out of a point is product preserving Okay, if you've also seen the Burnside ring the Burnside ring is the groton-deeke group of
25:01
finite g-sets so I should also be able to connect these two sum ends to finite g-sets and I can this sum end is the g-set point and the district union copies of point this one is the g-set CP as a
25:23
CP set and every CP set breaks up into a district union of points and CP and then my restriction map is just forget the CP action and just remember the set and
25:40
That takes point to a set with one element and it takes CP to a set with P elements And that was this map Okay, so the other that I want is the constant Mackie functor Z and this one is The value at point is Z the value at the at CP is also Z
26:09
the Restriction map is the identity The vial group. Oops, sorry Vial group action is also the identity and then that forces the transfer to be
26:26
multiplication by P Because since the restriction is injective then I can compute the transfer by Computing the composite of the restriction with the transfer And I see I have no choice here The
26:41
These two Mackie functors are pretty closely connected The target is what sometimes called a co homological Mackie functor Now this one I should pause and connect this Already to what we see in the motivic story Often when we talk about pre-sieves with transfers in motivic homotopy
27:03
we're referring to things like this that are close to co homologic Mackie functors and There I see the same kind of condition that the composite of the transfer and the restriction is multiplication by the index of the group And that's that condition that I'm writing down here
27:23
In equivariant homotopy, we allow these more general kinds of transfers Which you should think of as actually also showing up in the motivic context this is analogous to the transfer along say a Finite at all now, okay
27:44
so Before I continue questions about this so far. I Know a lot of this is review But that doesn't mean that questions won't have come up you said you said a and the line is a
28:03
Bernstein Mackie from yes. Yes. Okay, and it's the usual thing in math where
28:23
proper names become Become adjectives and so you end up with long strings So Sean asks why co homological is this an important distinction These do these do show up a lot and and they're the kinds of
28:42
Mackey functors that you see with group co homology and That's that's one of the reasons why I might Describe them that way So from that perspective, they it is a very natural class of Mackey functors that arises So there's been a lot of work in this for us
29:02
The the constant Mackey functor Z is a fairly easy one to do computations with as well as I'll show you in just a minute and it also arises naturally in the equivariant context
29:22
Yes, group cohomology does take values in these Group cohomology naturally has an extension to a Mackey functor and When I do group cohomology, I consider it in in one of these contexts and they're always coming from
29:41
Modules, it's always something that's a module over the constant Mackey functor Z No, thanks for asking. Okay, so the reason that we talk about Mackey functors in a covariant homotopy is that Mackey functors
30:02
play the role of abelian groups in in Genuine G spectrum in other words all of our usual algebraic invariants
30:35
Actually Mackey functor valued. So for example, normally I might talk about
30:48
homotopy groups of a spectrum and In the equivariant context I have the homotopy Mackey functors an equivariant spectrum I can talk about the
31:04
Generalized cohomology theories value on a space or Spectrum and in the equivariant context. I have a Mackey functor. It's worth of the cohomology of X in some e theory
31:22
So I have a richer structure that I could be working with for those who might Worry or wonder about such things that they're the category of Mackey functors is an abelian category We have enough projectives and injectives so we can do homological algebra the way we normally would and then
31:44
In a little bit. I'll also talk about how the category Mackey functors has a symmetric monoidal product So we were really exactly like with abelian groups as reflecting what we saw in Spectra we build a model in DG abelian groups and we can do the same thing in equivariant spectra. We take a covariance spectra and we compare it to
32:06
to DG Mackey functors or DGAs in Mackey functors So one of the things that I want to be able to do is talk about ordinary Homology and I find it easier when I'm talking about ordinary homology to just show you how to compute this in some examples
32:23
so How do we compute homology Remember, this is supposed to be a Mackey functor, but I'll just tell you the value of this at
32:40
At some point with coefficients in something and now just for simplicity for myself. I'll start at this point switching to the group being c2 so I Had I
33:00
Had I planned I had to use some of the newer technology I would ask via a poll What's your favorite way to compute? ordinary homology You know, it's like we would do in a calculus class do like a quick spot to check But if I were to do that I would guess I would guess you would say
33:24
cellular as opposed to singular Although singular is certainly nice for sort of Formal reasons. Yeah. Thank you. And yes cellular is the way that we actually can compute things easily
33:40
We write down a small chain complex to it. So let's do that here and let's start with an example so We'll just do this via cellular homology and my example is going to be
34:02
Let's look at a representation sphere So I'm going to take s to the C by which I mean the one point compactification of C and then C remember I said earlier my c2 is also the Galois group of C over R and
34:29
So this this naturally has an action of c2 as the as the Galois action So if I were to draw this well, this is the this is the Riemann sphere
34:41
so I have the real line sitting inside the Riemann sphere and then I have the two hemispheres and here was my S to the R sitting as the equator and as they are well, this is just s1 and
35:05
when I think of the two hemispheres in my Riemann sphere Well, I could put them in as Showing up and actually I'm doing a different projection than you're probably thinking of I'm gonna have the positive
35:21
complex part being the upper hemispheres can be the Positive imaginary part being the upper hemisphere and the negative imaginary part being the lower hemisphere so my group acts by swapping the two hemispheres and Leaving the equator fixed So I can build this as an equivariant cell complex. I have
35:45
two copies of The one sphere and they're swapped. So I'm going to have a c2 cross s1 Because that's two copies of the one sphere. I'll draw a cartoon as I go through
36:00
It's my one sphere and my one sphere and they should have been the same And the group acts by swapping them, this is a c2 cross it and I'm going to map this to the one sphere Where I just fold them down So it's via the identity. It's a twisted version of the fold map
36:25
and then I can include these into the corresponding c2 cross discs and that amounts to just putting in a little disc on each of these and
36:40
when I pushed this out So actually let me Let me do it this way When I push this out
37:06
Now I've exactly built my Riemann sphere With the two hemispheres that are swapped so here's my cell structure and
37:20
If I want to take the cellular homology Well, what I need to do is figure out what am I supposed to do when I evaluate what's the homology of one of these? G mod H cross a sphere or G mod H plus smash a sphere So the building block is I'm going to take the homology
37:49
H n of Whoops, I'll say H star of G mod H cross
38:01
An n sphere with coefficients in some Mackie functor M Well, this is going to be I'll do reduced. This is zero if star is not n and it's just evaluate M at G mod H if
38:20
star equals n Now I can start to write down what what my homology is going to look like Notice that this map here This one this is the same thing as C2 to a point
38:42
crossed with the one sphere and remember my Mackie functors are exactly built so that they know What I'm supposed to do to maps between Orbits so a map C2 to a point. This is something that I can evaluate my Mackie functor on
39:05
Now I can write down the chain complex using that so in degree zero again I'm doing the reduced theory. I have nothing. Here's my degree in degree one Well, I had my one cell. There's only one one cell and it's the one sphere
39:25
so I have M of a point and in degree two, I had a single equivariant to cell it was the one coming from M of C2 and
39:42
The cellular boundary map is just M of C2 going to a point So notice this is the covariant version of this. So this is the transfer from M of C2 to M of a point
40:03
And this tells me how I can write down the this homology for any of these so I get H1 is the co-kernel of the transfer and H2 is
40:21
The kernel of the transfer and it's a little more work You can get the these as instead Mackie value two things It amounts to thinking about G mod H and and putting in another slot where I crossed with some fixed T
40:40
Okay So since the this is a talk in the broader context of a summer school Maybe I'll say as an exercise for you. Whoops as an exercise Figure out
41:04
Homology groups of S K times C with coefficients in any Mackie functor M For all K and M you can use the same idea that I talked about here
41:24
You'll have to think a little bit about what happens in the cohomology version So namely when K is negative, but it's a it's fun to to work through Okay There's one other thing that I want to point out here and that's actually right here. I
41:45
Am going to give a name to this map from S1 Into this C sphere, I'm going to call this a sub Sigma I'm going to call this sometimes the boiler class of
42:06
The sine representation and if I'm being super pedantic, I'd actually call this the suspension of a sigma because a sigma is a map from the zero sphere into
42:23
Now instead it's just the one point compactification of that imaginary axis in C where that's swapped because well, we know how complex conjugation works and Here I'm seeing this co fiber sequence C 2 goes to the zero sphere or C 2 plus goes to the zero sphere
42:45
Then the co fiber is this sine sphere and this whole part that I'm writing down in this case is the suspension of that co fiber sequence So it's something to keep in mind All right, so I I brought this up because just doing these computations
43:05
Understanding the cohomology of these spheres the k times C spheres as k varies Allows you to get a lot of mileage equivariately, so I'll start with a theorem And this is due to lots of people
43:22
Individually, I'm gonna I want to say that the first parts of this is due to Duggar it's due to who creesh and it's due to me Hopkins and Ravenel and that is that there's a filtration on
43:52
the C 2 spectrum of real board ism so em, you are or Again, if you're coming from the motivic context
44:02
You should think of this as MGL and then the theorem is due to different people. So I'm a tip but typically it's due to Hopkins and morale And why and there's a filtration on and you are
44:29
Dang with associated graded
44:42
grr and you are is I'll just write it as the Eilenberg MacLean spectrum Associated to that constant Mackey functor Z and then I'm putting in a bunch of formal indeterminates
45:04
where the degree of each of these indeterminates is Just like in in Dan's talks my Indeterminates are going to be graded by representations or in this case. They're by graded
45:21
Actually have two irreducible representations and this is just I times C or again, I'm using the complex conjugation action on C What this means is if I want to compute the MU our Homology or M you are co homology of some space or spectrum then I have a spectral sequence
45:49
and the e2 term Is given by well say that homology but I'll write it this way I'm going to do the homotopy
46:00
again, homotopy Mackey functors of the function spectrum From say X into HZ I join these indeterminates Well, this is just the homology
46:26
Oops, sorry the negative cohomology X with coefficients in Z and then I had joined a bunch of indeterminates and this spectral sequence converges to
46:47
the MU R Cohomology X So it's like in a tea. Here's a brook spectral sequence, but I'm using this different filtration
47:04
Filtration is the slice filtration named after the motivic slice filtration Wavadsky that was was done by Duggar initially okay, so what I want to focus on and I'm seeing that Time quickly passes. So what I want to focus on is that this is a spectral sequence of
47:26
Mackey functors So this is sort of the the first Order approximation to understanding the way I can do equivariant computations
47:42
Mackey functors form an abelian category, which means I can talk about spectral sequences of Mackey functors And in this case, what does that mean? We have two spectral sequences there's the fixed points
48:00
the value at a point and There's the underlying and then they're connected. I have a map of spectral sequences That's reflecting my restriction map from the fixed was the underlying and I have a map of spectral sequences from the underlying back to the fixed
48:22
points and The underlying was actually a spectral sequence of c2 modules So I have all of this added structure that comes in it's not it's a lot of added structure But it's not an insurmountable I'm out of added structure the biggest thing I can do is I can use the fact that since these are maps of spectral sequences if
48:43
I have a class that's a cycle or maybe a permanent cycle Then the image of that under any map of spectral sequences is a cycle or a permanent cycle If I have a class that's the target of a differential
49:00
Then I know that under a map of spectral sequences. It's still the target of a differential so I get a lot of additional Constraints on this so as a as just an example In the in the spectral sequence computing the homotopy Mach-y functors of
49:28
and you are the ideal generated by two is
49:42
an ideal Of permanent cycles and so why it's just because Two times any class X will remember I'm looking at something where I started with the constant Mach-y functor
50:05
Z and in the constant Mach-y functor Z 2 was Was the transfer of 1 in the underlying and then I have this Frobenius reciprocity condition lets me move the X inside. This is the transfer of 1 times the restriction of X
50:30
So this is the transfer the restriction of X and in the underlying spectral sequence The underlying spectral sequence is just the ordinary Atiyah-Heirsbrook spectral sequence computing the homotopy of MU
50:46
out of the homotopy of MU So it it collapses with no extensions and the restriction here Is a permanent cycle
51:00
Oops, it's not how you spell always So that's giving me a huge amount of information about the structure of the spectral sequence that I don't know how I would have known Otherwise, I needed that this this large number of classes namely twice anything
51:24
Could actually be written via the Mach-y structure as as a permanent cycle Okay, so in the time remaining I need to push into bigger groups and I need to talk a little bit about the
51:41
The added structure I've already started to dance around some of this first You'll notice I used a different wild card here than my asterisk I used a five-pointed star here here Here and then here I just used the ordinary asterisk so this one this wild card was
52:03
following notation of who increased this is the ROC2 grading so I actually have more information That I have at my fingertips and second I talked about an ideal here, which says that I should be thinking about this actually as a spectral sequence of
52:25
rings And that's true, but I won't go too much into it so in fact The slice spectral sequence is a spectral sequence of
52:45
Commutative monoids in Mach-y functors And these are called green functors
53:12
Sean asks if there's a that there's a result saying differentials are power operations It yes, yes
53:25
The Maybe the the best way to say what the differentials are in the classical a tea Here's a spectral sequence. Is that they're cohomology operations because their maps connecting there between on the McLean spectra and here in for
53:41
for something like MUR the the Fibers are again suspensions of Allenburg McLean spectra. So the initial differentials are exactly Cohomology operations in this case from cohomology with constant Z coefficients to itself and then all of the higher differentials can be expressed as
54:06
Secondary or or higher order operations just as we would see with the ordinary a tea Here's a book spectral sequence In this case It's a it's a consequence of knowing the form of the spectral sequence knowing that the fibers are all these
54:24
Generalized I'll ever McLean spectra But yeah, I can't think of them in exactly that way Okay, so I'm in that the time remaining and I've said that already I want to do one
54:41
Last added bit of structure so here I've used that the Mackie structure shows up and it gives me a way to produce a bunch of permanent cycles and to transport differentials Then I know that this is a spectral sequence of these ring objects these green functors So I understand that at each page. I have a ring and it for each G mod H
55:05
I have a ring the restriction maps are all ring maps and so I can use all of this to To continue to bind classes to other classes and simplify the problem The last part is to use the norm
55:20
so we have Also multiplicative transfers and these are actually arising from functors Quite generally on on the complete
55:42
Spectra, so I have a norm functor from H spectra 2g spectra and This is a symmetric monoidal functor That's going to take some e and you should think about it as going to I'm going to smash together G mod H copies of e
56:04
ie This is a tensor induction and these norm maps have the property that since the tensor product is the
56:22
co-product on Commutative rings I have canonical maps We have canonical maps for any
56:40
Commutative ring in G spectra I Have a map from the norm of the restriction of R
57:00
back to R this endows the homotopy maki functors of R with these external norm maps and Here I have to use the grading by the representation ring. Oops
57:35
so just as earlier when I talked about the sum over the Vial group or the sum over G being the trace. I'm using the Galois theoretic language there here
57:46
I'm also using the Galois theoretic language. You should think of this as being a as being heuristically the product over over G mod H of
58:03
some element and Since my ring is commutative if the group is acting by permuting now the tensor factors around but the multiplication is actually commutative So it doesn't care what order that they were in This gives me a way to take an element that's fixed by H and produce an element that's fixed now by G
58:25
The this structure was first studied by Tambara who looked at these and called them TNR functors these sort of maki functors together with multiplicative Transfers and the last result is the slice spectral sequence
58:42
is a spectral sequence of Tambara functors This one I don't know how to show this and in the motivic context the analogous
59:01
Operations the analogous norms and normed which it expectra were done by Bachman and hoy wah, they described how you can think about the norm maps and how to build these added norm External norm maps on commutative monoids in this context and I would expect that the slice spectra should have this property
59:24
in the equivariant context The somewhat surprising feature is that the slice filtration is actually the universal filtration That has the property that it takes commutative monoids in spectra to a spectral sequence of
59:41
Tambara functors and if there are questions about that afterwards, I'll answer it So I I saw you pop in Padri so I know that I'm That I am almost out of time. So let me just say a punchline And that is what the hell does it mean?
01:00:00
to be a spectral sequence of timbara functors. So it means that we have... So I was hoping for questions, but you can take time. Oh, okay. I reserve more questions. All right. So we have a twisted version of the Leibniz rule.
01:00:28
And let me just spell that out in one case. So I have the, just move this one. I have the differential on some class that I'll write as the norm of X.
01:00:41
Again, heuristically, so I'll do this in a different color because this part's a lie. This is supposed to be X times the conjugate of X. And then I know from the Leibniz rule how to compute the differential on a product. This is the differential on X times gamma X
01:01:01
plus X times the differential on gamma X. Remember the differentials were maps of Mackey functors. So I can pull the vile action out. So this is DX times gamma X plus X times gamma of DX.
01:01:28
And this is the same thing then as one plus gamma on X times, oops, not the one I wanna do.
01:01:41
DX times gamma X. And now I can make this true statement, which is the transfer of DX times gamma X. So this is the version of the Leibniz rule that shows up in this case.
01:02:02
The differential on the norm, it's just like on a product, but I'm supposed to remember that the norm was a kind of product where the group permuted the factors around. And so the differential is going to take that to a sum where again, the group permutes now the sum ends around
01:02:23
and that's exactly the role of the transfer. You can do better though. This is saying something just about D and of the norm, it's just the usual Leibniz property. And so this is the very last thing I'll say,
01:02:42
and we can do better. If DN of X, so now I'm in my slice spectral sequence is Y, then I actually get a longer differential,
01:03:02
D2N minus one of that same classroom before, a sigma times the norm of X is the norm of Y. For this, I don't know a classical antecedent.
01:03:21
This is saying that instead that once I put in this a sigma, it actually the differential almost behaves like a ring map at the expense of shearing it from that DN to D2N minus one. So it's like saying, if I know the differential on X,
01:03:41
then I can is Y, then there's some kind of differential on X squared that looks like some kind of Y squared. And these collections of properties I've described them are the way that people are doing computations with these spectral sequences. So I think I'll stop there.
01:04:06
Okay, so first, Mike, thanks a lot for a nice talk. And so we are, so let's fire the first question. So can you read it or do I read it?
01:04:23
So Sean asks, couldn't I view a sigma N and N as two different power operations? And then it would look like some of Bruner's work. Yeah, I think that's exactly the way that I wanna do this. I should be able to connect a sigma N to some kind of actually in this case,
01:04:46
Dyer-Lashov operation, because I'm looking at an operation on homology. But I don't quite know how to make those work. I'd love to talk to you more about that.
01:05:05
Yes, Sean also asked, do these norms prolong the category of filtered equivariant spectra? Yes, they do. Yuri asks, is the universal property of the slice filtration written down anywhere?
01:05:23
No. Yes, I don't think, maybe, but I don't recall. Oh, wait, maybe in the handbook of homotopy, in the chapter I wrote for the handbook of homotopy, I believe I talked about the universal property of the slice filtration there.
01:05:43
So thank you for making me remember that. And then anonymous attendee asks, could I mention some of the applications to chromatic homotopy as promised in the abstract? Also, yes. So the applications of some of this,
01:06:05
and I'll be quick, the applications are, first, let me recall a theorem of Han-Shi. And this says that the Lubin-Tate spectra,
01:06:32
EN, for any N, are real orientable.
01:06:42
In other words, I have a map of ring objects in the homotopy category, MUR to EN. Knowing this, if N is two to the K, minus one times anything, so times M,
01:07:05
then the Hopkins-Miller theorem says that C two to the K acts on EN, which gives me then via just the norm forget a junction
01:07:26
I described above a map from C two to C two to the K of MUR into EN. That's again, a map of ring objects, but now in the homotopy category
01:07:40
of C two to the K spectra. And then recent work of Wu Tui, me, Dingxi, and Ming Kong Zheng says that you can use this
01:08:04
to build a model for E theory. EN as the KN localization of some quotient of MUR
01:08:39
and to spell out exactly what stuff is
01:08:42
would take me a little far afield. But the important thing is the slice spectral sequence here has a describable, a more understandable,
01:09:07
E two and then et cetera terms than the corresponding Lubin-Tate theory had. We knew the C two action on EN,
01:09:20
but being able to describe the C two to the N action on EN in a way that we could write down the homotopy fixed point spectral sequence, that was sort of the bloody edge of the state of the art. Using these sorts of equivariant methods and stepping through the norms and these sort of quotients of the norms of MU,
01:09:42
the slice E two term is very easy to write down and to describe. And then you can use the techniques that I was describing over the course of the talk to sort of bootstrap differentials inducting up over the order of the group and use this to get a lot of information
01:10:02
about the homotopy groups of the Hopkins-Miller spectrum in ways that we never were able to before. Okay, so I have a question. What's the link between Tambara and Green-Fung-Tors?
01:10:25
So there's a forgetful functor. Every Tambara functor has an underlying Green-Fung-Tor. So a Tambara functor, you can think of as a Green-Fung-Tor together with these additional multiplicative norm maps.
01:10:41
And then there's actually, just as there was a hierarchy that started with coefficient systems and it ended in Mackie functors, where I start to put in more and more transfers, there's a hierarchy between Green-Fung-Tors, which are Tambara functors with no multiplicative transfers,
01:11:00
all the way up to Tambara functors, which have all multiplicative transfers. And this hierarchy, it's exactly analogous to the additive hierarchy for the Mackie functor case. And this is an important feature. So thank you for bringing it up.
01:11:21
Zariski localization does not work well in Tambara functors. So for equivariant commutative rings, Zariski localization doesn't preserve the property of being a commutative ring. It does always preserve the property though, of being sort of the spectral version of a Green-Fung-Tor.
01:11:40
So an algebra over an E infinity operat, but it's a very particular kind of E infinity operat, one in which the group doesn't act. And so that's a subtlety that shows up and makes some of the computations a little trickier. Okay, so I have a kind of maybe vague or broader question.
01:12:06
So you described Mackie-Fung-Tors and they are defined for finite groups, but is there a theory for other groups? So first example would be profinite groups like Galois. Yeah, yeah, yes.
01:12:21
And there are several versions of these. The cases that were most studied in classical homotopy theory were compact Lie groups, where again, we have a good notion and you have Mackie-Fung-Tors for a compact Lie group and they're describing the homotopy groups
01:12:42
of a genuine G spectrum for G compact Lie. The multiplicative version of these only shows up for finite index. So pairs of finite index subgroups. There's no sort of degree shifting part that can show up in the compact Lie.
01:13:03
For profinite, Dress and Siebenacher have a bit vector construction for sort of profinite groups that's generalizing the ordinary bit vectors. And they're describing the profinite version
01:13:23
of the Burnside ring in that case, because the bit vectors of Z is where the truncated bit vectors are exactly giving me the various Burnside rings as I look for like C, N
01:13:40
or whatever my truncation system was. Barwick also in his spectral Mackie-Fung-Tors has a really beautiful approach to understanding the profinite case of Mackie-Fung-Tors as well. Beyond this, nothing that I was writing down really depended on the group being finite.
01:14:04
I could still talk about finite G sets for G not finite. I start to run into pathologies, like if G is divisible, there aren't any interesting finite G sets. And then I'm gonna start to run into trouble.
01:14:21
But aside from those cases, you can talk then about, you can talk about Mackie-Fung-Tors, you can do all the same thing. Last question, I was also thinking about, I don't know if you know these Ross cycle modules. So it looks really like Mackie-Fung-Tors, but there are two operations that are added.
01:14:42
So it's kind of multiplication by unit and the residue maps for variations. We could see that as Mackie-Fung-Tors for the so-called Motivi Galois groups, where you have transcendental extension.
01:15:01
So have you seen something like that in, I guess? Yes, I haven't, but that's an interesting thing. I'll think about that. I think, I mean, one of the reasons that I wanted to, yeah, yeah. One of the reasons I wanted to give this talk
01:15:20
is I think a lot of the techniques that we've been using in the, recently in the equivariant context should port through without change into the motivic one. All the stuff that we've been seeing with the multiplicative transfers, anything showing up in the Bachmann-Hoy-Wah normed motivic spectra, we should have analogs of these two kinds of conditions
01:15:44
on differentials in certain spectral sequences. And this last one, the one that changes degree, it's allowing you to lift differentials multiplicatively in a way that can actually be pretty surprising to get new ones.
01:16:00
So I'd love to see the analog of that motivically. Okay, so it seems we have no more questions. So again, Mike, thanks a lot for a nice talk.