POWER OPERATIONS 1. 2/13: Talk by Jacob Lurie This is motivation for what we're talking about ...

24 Pages·2014·0.4 MB·English

POWER OPERATIONS
Abstract. Notesfromthespring2014“Thursdayseminar”atHarvard. TEXedbyAkhilMathew,
whotakesfullresponsibilityforallerrors.
1. 2/13: Talk by Jacob Lurie
This is motivation for what we’re talking about in the Thursday seminar this semester. I’m going
to start by talking about Dijkgraaf-Witten theory.
Let G be a finite group, and fix η ∈ Hn(BG;C×). If Mn is an n-dimensional manifold, you can
associate to it a number Z(Mn). Namely,
(cid:88) (cid:82) P∗η
Z(Mn)= Mn
|Aut(P)|
iso classes of G bundles P
The numerator is as follows. G-bundles on M are given by maps M → BG. Given a G-bundle P,
identified with a map P : M → BG, you get a class by pulling back a class in Hn(M;C×). You can
then integrate this class and then come up with a class in C×.
Example 1. If η =1, then this is counting the number of G-bundles on M with multiplicity.
Ingeneral,itissometwistedversionofthatcount. Thisgivesaninvariantofn-manifolds,andthis
gives the top-level of a topological quantum field theory. If M is a manifold of dimension n−1, then
there is an analogous procedure that you can perform that you can use to perform a complex vector
space. Again, you’re going to think about all G-bundles on M, but instead of thinking of them one
at a time, consider the classifying space of them. That is, consider Map(Mn−1,BG). If I take this,
there is a map
(cid:15):Map(M,BG)×M →BG,
given by evaluation. Moreover, we have a projection map
p:Map(M,BG)×M →Map(M,BG).
IfIstartwithη ∈Hn(BG;C×),pullbackalongtheevaluationmaptogiveaclass(cid:15)∗ηinHn(Map(M,BG)×
M;C×). Then, consider the pushforward along p (cid:82) (cid:15)∗η ∈H1(Map(M,BG);C×).
NowH1 withcoefficientsinC× parametrizeslocalsystemsof1-dimensionalcomplexvectorspaces.
So this class (cid:82) (cid:15)∗η classifies a local system L on Map(M,BG). Earlier, we got a locally constant
M
function — we got something in H0. We were able to sum it over the various connected components.
So, we define
Z(Mn−1)=Γ(Map(M,BG),L ).
M
This is an invariant of any n−1-dimensional manifold; it’s a finite-dimensional complex vector space.
The mapping space Map(M,BG) has finitely many connected components and each connected com-
ponent is the classifying space of a finite group (the automorphism group of a finite G-bundle). So
giving a one-dimensional local system means giving a bunch of characters of a bunch of finite groups
and taking global sections means taking invariants and adding them up.
How are these (n−1)-dimensional invariants related to these n-dimensional invariants? To think
aboutthis,consideran(n−1)-manifoldMn−1,anotherNn−1,andann-manifoldB givingabordism
between the two. In this case, you can look at Map(B,BG), and that projects onto Map(M,BG)×
Map(N,BG). So we get maps
f :Map(B,BG)→Map(M,BG), g :Map(N,BG).
Date:May11,2014.
1
2 POWER OPERATIONS
We get local systems L ,L on the two targets and we can pull back to get f∗L ,g∗L on
M N M N
Map(B,BG).
These are related. These local systems came from pulling back certain cohomology classes on BG
and integrating them. Inside this manifold B, the fundamental cycles of M and N are homologous
– B itself is a boundary between them. So in B, integrating along M and integrating along N are
identified. So we should get f∗L (cid:39) g∗L . If you do this process at the level of cycles, you get a
M N
preferred choice of isomorphism f∗L (cid:39)g∗L .
M N
So, given a section of L , I can pull it back and push it forward to get a section of L . So, I get
M N
a map
Z(Mn−1)Z→(B)Z(Nn−1),
and when M,N are empty, then the two vector spaces are canonically C, so the map is multiplication
by the number we encountered earlier.
More precisely:
Definition 1. Z(Mn−1)=H0(Map(M,BG;L ). (There is no higher cohomology since we are over
M
C.)
Nowthereisagoodwayofpullingbackcohomologyclasses,butnotpushingforward. So,crucialto
this discussion was the fact that homology and cohomology were the same — so I could push forward
either. At the level of vector spaces, this comes from the fact that if V is a vector space over C with
an action of a finite group G, then VG (cid:39)V canonically.
G
Two years ago in this seminar, we proved that there is a similar phenomenon of homology and
cohomology giving you the same answer in a different context. Let X be a space which has finitely
many finite homotopy groups. A space like BG is such a space. Moreover, Map(M,BG) is such a
space. Let L be a local system on X of K(n)-local spectra. Then we can talk about C (X;L) and
∗
C∗(X;L) (“chains” and “cochains” of L on X).
Definition 2. C∗(X;L) is the homotopy limit over X of L. (L is a diagram of spectra parametrized
by X.) Similarly, C (X;L) is the homotopy colimit of this diagram. The homotopy colimit is taken
∗
in the world of K(n)-local spectra.
Two years ago, we discussed the fact that if X is as above, there is a canonical equivalence
C (X;L)(cid:39)C∗(X;L).
∗
Thesloganisthatsuchaspace, fromthepointofviewofK(n)-localhomotopytheory, behaveslikea
zero-dimensional space: homology and cohomology are canonically the same. A consequence is that
you can run the same construction for Dijkgraaf-Witten theory.
Let’s restart the discussion. Let E be a Morava E-theory of height n. I’ll review what these are a
little later. This is some kind of cohomology theory, and a theorem of Goerss-Hopkins-Miller implies
that it is a well-structured cohomology theory. E has an essentially unique structure of an E -ring.
∞
There is a multiplication in E-cohomology, but somehow it satisfies all the commutativity that you
would want up to coherent homotopy.
The upshot is that it makes sense to consider GL (E) ⊂ Ω∞E, and this is the subspace of those
1
connected components in Ω∞E which are units at π . That’s a space that you can assign to any kind
0
of multiplication on a spectrum. You just need a way of picking out the units in π . But if you’ve got
0
oneofthesefancymultiplications,thenthisGL (E)isactuallyaninfiniteloopspace. Anyoperadthat
1
acts on E as a spectrum will act on its units as a space. So GL (E) gets an action of an E -operad,
1 ∞
soyoucandeloopitasmanytimesasyouwant. Therefore,youcanwriteGL (E)=Ω∞gl (E)where
1 1
gl (E) is a spectrum.
1
Nowitmakessensetotalkaboutcohomology. Startwithacohomologyclassη ∈Hn(BG;gl (E)).
n 1
By definition, this is
Hn(BG;gl (E))=[BG,Ω−nGL (E)].
1 1
Starting with η, we can mimic all the previous constructions. At various points, we need to invoke
the ambidexterity theorem to avoid distinguishing between homology and cohomology. This means
that we can do the following:
POWER OPERATIONS 3
(1) To every closed n-manifold M, assign Z(M). Before it was a complex number; now it’s a
number in π E.
0
(2) Given an n−1-manifold, assign to it Z(Mn−1), which is going to be an E-module spectrum.
These are related to each other in the previous way – functoriality for bordisms.
Allofthisismotivation; it’snotgoingtobewhattheseminarisgoingtobeabout. ButI’mtelling
you that once you choose such an element η, you can construct a bunch of manifold invariants. If you
thought that this was interesting and wanted to get manifold invariants, you might ask what types of
η to choose.
Now we’re getting to the subject matter of this seminar.
Question: What can we say about H∗(BG;gl (E))?
1
That’ssortoflikeaskingwhatthisspectrumgl (E)lookslike. MaybeIdon’tcareabouteverything
1
here. I just want to map reasonably simple things into it, like the classifying spaces of finite groups.
Let’s start with this question of what gl (E) looks like. How might you answer this question? It’s
1
a spectrum, so it’s got homotopy groups. What are its homotopy groups? That’s pretty easy. Note
that gl (E) is the connective delooping of GL (E), so no homotopy groups in negative degrees, by
1 1
definition. The zeroth space determines the other homotopy groups, so we get

0 if ∗<0

π gl (E)= (π E)× if ∗=0.
∗ 1 0
π E if∗>0
∗
Just knowing the homotopy groups of a spectrum is not necessarily so useful if we want to know, for
instance, how to map into it. Let’s try and say more.
As a warm-up, replace E by complex K-theory K. Morally, this is a special case (not quite; if you
p-adically complete it, then it becomes E ). What can we say about the homotopy groups of gl (K)?
1 1
We have

0 ∗<0
{±1} ∗=0
π gl (E)= .
∗ 1 Z0 ∗∗>>00 aanndd eovdedn
Now we know something about K-theory – it has to do with vector bundles. A point in K-theory is
likeavectorbundle. Soapointingl (K)shouldbelikeavectorbundleinvertiblewithrespecttothe
1
tensor product. What is true is that if you have a vector bundle invertible with respect to the tensor
product, then it gives you a class in gl (K). Now BU(1) = CP∞ has a map to Z×BU = Ω∞K. It
1
maps,infact,to{1}×BU sinceeverylinebundlehasrankone. Thissitsinside{±1}×BU =GL (K).
1
Moreover, this map is an infinite loop map, because the multiplication on BU(1) is precisely the
tensor product of vector bundles, so it maps with the way that you multiply in the classifying space.
So you get an E -map
∞
BU(1)→GL (K),
1
and at the level of homotopy, we have BU(1) = K(Z,2). This map is an isomorphism in degree two
and zero in all other degrees. This is somehow picking out exactly one of the homotopy groups of
gl (K).
1
You can improve this a little bit. You can assign K-theory classes to Z/2-graded vector bundles.
If you have a Z/2-graded vector bundle E =E ⊕E , you can assign to it a K-theory class which is
0 1
justtheformaldifference[E ]−[E ]. Thereasonthatit’snotaformalityisthatthisismultiplicative
0 1
as well. The usual tensor product on graded vector spaces (together with the Koszul sign rule) is
compatible with the way in which you multiply K-theory units. So I could have written
{±1}×BU(1)→GL (K),
1
where the former is the classifying space of Z/2-graded line bundles.
Onhomotopy, Igetamap{±1}×K(Z,2)→GL (K)whichisanisomorphismindegrees≤2and
1
zero everywhere else.
4 POWER OPERATIONS
That’s kind of nice. If you look at the space Z/2×BU(1), this is the “geometric part” GLg(K).
1
The infinite loop structure on Z/2×BU(1) is not the product because of the funny sign rule. But we
can make GLg(K) into an infinite loop space. Note that the composite
1
GLg(K)→GL (K)→τ GL (K),
1 1 ≤2 1
is an equivalence, by considering homotopy groups. The conclusion is:
Proposition1.1. GL (K)splitsasaproductGLg(K)×τ GL (K),andthisisasplittingasinfinite
1 1 ≥4 1
loop spaces.
If you didn’t like this calculation, then you might like the fact that there is a “geometric” part of
gl (K) which is not so big, and some extra stuff τ gl (K) which is “junk.” If you’re interested in
1 ≤4 1
calculating maps from something into gl , you can easily understand maps into the geometric piece,
1
which is pretty close to ordinary cohomology with integer coefficients. Ordinary cohomology gives
you something you can use to twist K-theory by. The other thing is something you can understand
using the logarithm, but maybe is not so interesting from the point of view of producing things like
Dijkgraaf-Witten theory.
We want to replicate this with Morava E-theory in place of K-theory. The homotopy of gl (E) is
1
a big mess, but we imagine that most of it is junk, and we want to isolate the interesting part.
Let’sgobacktoK-theoryforasecond. ConsidertheconnectedcomponentK(Z,2)ofthegeometric
part. It’s an infinite loop space. But it’s better than that. It doesn’t just have a multiplication
commutative up to homotopy and homotopy and homotopy, but you can build it as a topological
abelian group: you can make it commutative on the nose. One way you might try to generalize this
is as follows. Here’s some infinite loop space. It’s big, but can I understand maps from topological
abelian groups into it.
Definition 3. Let R be any E -ring. Consider the space GL (R), a space, and the associated
∞ 1
spectrum gl (R). Now I’m going to define
1
G (R)=Hom (HZ,gl (E)).
m Spectra 1
This is an infinite loop space: in fact, I could consider the spectrum of maps. I prefer to think of it
asaspacebecausewhileI’mgoingtotellyousomethingaboutit,I’mnevergoingtotellyouanything
about the negative homotopy groups.
Equivalently, I could have said that G (R) is the collection of maps of infinite loop spaces Z →
m
GL (R). IfIjustwantedmapsfromZ→GL (R), IcouldunderstandthoseasΣ∞Z=S[t±1](thisis
1 1 +
somekindofLaurentpolynomialringoverthesphere; thisobservationissuggestedbythecalculation
of the homotopy groups of this spectrum, as a Laurent polynomial ring over the homotopy groups of
spheres). If I just wanted to study maps Z→Ω∞R of spaces, then tautologically, that’s the same as
maps
Σ∞Z=S0[t±1]→R.
+
IfIwantedtomakethefirstmapintoanE -map,thenIneedtomakeS0[t±1](whichisanE -ring)
∞ ∞
mapping to R via E -ring spectra. So
∞
G (R)=Hom (S[t±1],R).
m E∞
We can summarize this by saying (in quotation marks) that G =SpecS[t±1].
m
Goal: Let’s try to compute G (R) for various R.
m
This is a goal that we’ll do a little bit of in this lecture, but do a lot more of in the seminar. You
could set it as one of the goals for your life.
I want to continue the warm-up where I’m essentially talking about K-theory, but I really want to
p-adically complete, so I’m working with Morava E-theory at height one.
Special case: R=E, Morava E-theory of height one.
What do I mean by that? Let κ be any perfect field of characteristic p. Then there exists an
essentially unique ring spectrum such that π E is the ring of Witt vectors W(κ) and E is even
0
periodic and the formal group associated to E (which is even periodic), i.e., Spfπ ECP∞ is identified
0
with the formal multiplicative group.
POWER OPERATIONS 5
There is a canonical example, where you take κ = Z/p and this Morava E-theory is p-adically
completed K-theory. Another example, though, which you might want to keep in mind, is where
κ = F . In that case, we get an associated Morava E-theory which is the “maximal unramified
p
extension” of p-adic K-theory. I’m going to write this as K .
p∞
It’sacorollaryoftheGoerss-Hopkins-MillerthatthisconstructionistotallyfunctorialinκsoK
p∞
is acted on by the Galois group. In particular, it has a Frobenius automorphism. The Frobenius on
F induces a Frobenius on K and the homotopy fixed points of the Frobenius gives K (p-adically).
p p∞
Here are some examples. We want to compute
Hom (S0[t±1],E).
E∞
Let’s try and do this.
The notation is potentially a little bit confusing. We’re adopting two familiar notations from
algebraic geometry. You can call the units of a ring R either GL (R) or G (R). Those usually have
1 m
the same meaning. I’ve just assigned them two different meanings. For instance, you can see pretty
easily that GL (R) is corepresentable as well. For example
1
Ω∞R=Hom (S0{t},R),
E∞
where S0{t} is the free E -algebra on one generator. The thing that you have to beware of is that
∞
the homotopy groups of that are much more complicated than a simple polynomial ring. This guy
as a spectrum looks like a sum (cid:87) S0 = (cid:87) (BΣ ) . If you’re doing ordinary algebra, if you
n≥0 hΣn n≥0 n +
take the coinvariants of a group acting trivially, you get nothing, but in homotopy theory you get
somethingverycomplicated. Thisisnotsimplyadirectsumofcopiesofthespherespectrumindexed
by Z , which I’ll write by S0[t]. So GL (R) is given by maps from S0{t}[t−1] into R.
≥0 1
Now there is a map
S0(cid:8)t±1(cid:9)→S0[t±1],
and there is a map G (E)→GL (E) which is sort of by definition. The G (E) are those units of E
m 1 m
which are “strictly commutative” units in E.
So, we know of course how to compute GL (E) at the level of homotopy groups. You might try to
1
figure out to what extent this is off from G (E).
m
In the case I’m considering here, the K(1)-localizations of S0(cid:8)t±1(cid:9) and S[t±1] different, but are
not so different from one another. I’m going to just make everything an E-algebra, so the point is
that E(cid:8)t±1(cid:9) and E[t±1]. We would be done if the map
E(cid:8)t±1(cid:9)→E[±1]
is an equivalence, but it’s not. (Let’s work K(1)-locally everywhere.)
Question: What does L E(cid:8)t±1(cid:9) look like?
K(1)
Let me ask a more basic question first.
Question: What does L E{t} look like?
K(1)
ThisisaK(1)-localE-algebraanditisfreeononegenerator. WhatdoIgetifItakeπ L E{t}?
0 K(1)
This is some ring, and it has a simple interpretation. The elements in here are operations that act on
π of any K(1)-local E-algebra. If you have any K(1)-local E-algebra R and you have some element
0
x∈π R, then you get a map
0
φ:L E{t}→R,
K(1)
sending t → x. Anything else in π of the free thing goes to something else in π R. So, if I have
0 0
an element c ∈ π L E{t}, then c gives an operation O : π R → π R. Namely, you make the
0 K(1) c 0 0
map above φ, and see where φ(c) goes. This is a completely general categorical idea. It’s the same
reason that understanding cohomology operations classically amounts to computing the cohomology
of Eilenberg-MacLane spaces.
Now, there are some operations on the K-theory of spaces that might be familiar to you.
Digression(Adamsoperations). If[e]∈K0(X)isaclass,thenaccordingtothesplittingprinciple,
you’re allowed to pretend that e splits as a direct sum of line bundles e = L ⊕···⊕L . Then you
1 n
6 POWER OPERATIONS
can take the kth powers of those line bundles and get a new class Lk⊕···⊕Lk. That determines a
1 n
class ψk[e], and this defines operations
ψk :K0(X)→K0(X).
This is a map that you can define for any space X.
Fix a prime number p. If you take ψp([(cid:15)]), then ψp((cid:15)]) looks kind of like [(cid:15)]p. If there was one
summand, then these would be the same. In general, of course, [(cid:15)]p has lots of other terms if (cid:15) is a
sum of line bundles — but those summands are permuted by the cyclic group of order p, and the
orbits have two possible shapes. It’s possible to conclude that
ψp((cid:15))≡(cid:15)p mod p.
We can directly argue this when (cid:15) is a sum of line bundles, but we can actually turn this observation
into another observation.
One can introduce a new operation
θ :K0(X)→K0(X),
p
for any space X, and it has the property that
ψp(x)=xp+pθ (x).
p
I’m explaining why you believe that there is such an operation — a priori there might be p-torsion
so that θ is not uniquely defined. Or you could use the splitting principle and write down a formula
p
directly.
This is a story you can tell for K-theory. It turns out that E is any K(1)-local E -ring (fixed at
∞
some prime p), then one has analogous operations ψp,θ .
p
Theorem1.2(McClure). IfE isanyK(1)-localE -ringspectrum(ataprimep), onehasanalogous
∞
operations ψp,θ .
p
In ordinary K-theory, you have lots of these operations ψk. In p-adic K-theory, most of these
operations depend on the fact that K-theory had to do with vector bundles, but ψp is intrinsic to
complex K-theory and has nothing to do with vector bundles. These operations go
θ ,ψp :π E →π E,
p 0 0
and they satisfy the same congruences.
(1) ψp is a ring homomorphism.
(2) θ is not a ring homomorphism: it’s whatever it has to be to make the equation ψp =
p
xp+pθ (x).
p
Theorem 1.3. There are no other operations.
What do I mean by this? Operations that act on the cohomology of E-algebras are in bijection
with π E{t} and the claim is that the two elements I’ve written down explain everything.
0
Theorem 1.4. π L E{t}(cid:39)π E[t,θ t,θ (θ (t)),...].
0 K(1) 0 p p p
Now there is a map
L E{t}→L E[t],
K(1) K(1)
which sends t to t in homotopy. The point is that the θ ’s vanish on t in the target. So in fact, one
p
finds (since pushouts are given by relative tensor products):
Corollary 1.5. There is a pushout square in K(1)-local E-algebras
y(cid:55)→θ(t)(cid:47)(cid:47)
L E{y} L E{t}.
K(1) K(1)
y(cid:55)→0
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
L E L E[t]
K(1) K(1)
POWER OPERATIONS 7
In other words, E[t] is the free K(1)-local E -ring on a generator t with θ(t)=0. There is thus a
∞
cofiber sequence
θ
L E{t}→L E{t}→L E[t].
K(1) K(1) K(1)
So if R is a K(1)-local E-algebra, then
Hom (L E[t],R)→Ω∞R→θ Ω∞R
E∞ K(1)
is a fiber sequence.
So, if I wanted to compute G (R), it’s just a subspace of Hom (L E[t],R) given by those
m E∞ K(1)
maps which send t to a unit. So we get a fiber sequence
G (R)→GL (R)→θ Ω∞R.
m 1
Conclusion: we get a long exact sequence in homotopy groups
π G (R)→π GL (R)→θ π R→....
n m n 1 n
Consider the case where R = E, so the homotopy groups of R are concentrated in even degrees.
Using evennness, we get short exact sequences
0→π G (R)→π (GL (E))→θ π E →π G (R)→0.
2m m 2m 1 2m 2m−1 m
When m=0, we should be a little bit careful. When m=0, we get
π G (E)=fibπ GL (E)→π E.
0 m 0 1 0
Let’s calculate π first. Remember, π E = W(κ). We have this Adams operation ψ and we have
0 0
this operation ψ : W(κ) → W(κ) which is actually the Frobenius (i.e., induced by the Frobenius on
κ). There are no choices. This follows from the fact that ψp(x)≡xp mod p.
Sowhatarewelookingfor? Wewanttofindunits intheWittvectorsofκwhichhavetheproperty
thatθ(x)=0. Nowθ(x)= ψp(x)−xp,sotosaythatθ(x)=0istosaythatψp(x)=xp. Whatelements
p
are there in this ring that satisfy this equality?
Exercise: Solutions are just in bijection with the Teichmuller lifts of elements of κ×. For every
element of κ, there’s a canonical way to lift it to an element of W(κ).
It follows that
π G (E)=κ×.
0 m
Now let’s compute θ in positive degrees. It’s going to be a bit tricky. The reason it’s easy to make
a mistake because θ comes to you on π and now I’m thinking of it as an operation on π . I can do
0 n
this by reducing to the case of π by considering ES2m where
0
π ES2m (cid:39)π E⊕(cid:15)π E,
0 0 2m
where (cid:15)2 =0. That’s the ring structure on the function spectrum.
So I want to compute θ(1+x(cid:15)) where x∈π E, but you’re thinking of this as belonging to π by
2m 0
multiplying by β−n. By definition, since everything is torsion-free, I can write this in terms of ψ. So
ψ(1+(cid:15)x)−(1+(cid:15)x)p ??−(1+px(cid:15))
= .
p p
To evaluate the ??, you need to know how ψ acts on something in degree 2m: it acts by multiplying
by pm. So in the end, you get the expression
pm−1(ψ(x)−x))(cid:15).
So we need to understand the map pm−1ψ(x) − x as a map W(κ) → W(κ). Again, ψ is the
Frobenius. There are two cases:
(1) If m≥1, then this map is congruent to the identity mod p and thus an isomorphism. Thus
π G (E)=0 ∗>2 .
∗ m
8 POWER OPERATIONS
(2) We have an exact sequence
0→π G (E)→W(κ)ψ→−1W(κ)→π G (E)→0.
2 m 1 m
So, similarly, we get
π G (E)=Z ,
2 m p
since the fixed points for the Frobenius on W(κ) is W(F ). Finally, we get
p
π G (E)=coker(ψ−1:W(κ)→W(κ)).
1 m
If κ = F ), then the map we’re taking the cokernel of is zero and π G (E) = Z . If we’re taking
p 1 m p
κ algebraically closed, then the map ψ −1 mod p is xp −x and so this is a surjection. So if κ is
algebraically closed, we find that π G (E)=0.
1 m
So for instance

F× ∗=0
 p
π G (K)= Z ∗=1,2 .
∗ m p
0 otherwise
However,
F × ∗=0
 p
π∗Gm(Kp∞)= Zp ∗=2 .
0 otherwise
I like the second calculation better because more things are zero, and because you can get the first
calculation from the second by taking the fixed points of the Frobenius.
This is picking out for us inside the space of units of Morava E-theory a collection (of strictly
commutative pieces) that behaves well.
Theorem 1.6 (Hopkins, Lurie). Let E be any Morava E-theory with an algebraically closed residue
field κ. Then

κ× ∗=0

π G (E)= Z ∗=n+1.
∗ m p
0 otherwise
This is much simpler than the homotopy groups of gl (E). If n is even the Z is happening in a
1 p
place where gl (E) is zero.
1
BeingabletomakemapsintothespaceofunitsletsyoumakethingslikeDijkgraaf-Wittentheory.
Here’s another motivation. Let me define µ , the version of the pth roots of unity in a ring spectrum
p
R. By definition, this is going to be maps of E -spaces Z/pZ → GL (R). So µ (R) is the fiber of
∞ 1 p
multiplication by p on G (R). So if you know a theorem like the above, you immediately know what
m
the pth roots of unity look like in Morava E-theory.
So we get
µ (E)=K(Z/p,n),
p
in this case. This is related to an observation, as follows. µ (R) is representable by a group scheme.
p
It’s maps from E[Z/p] to R (the “group algebra”). This ring spectrum has another incarnation.
It arises from a duality phenomenon that we discussed two years ago. It is the function spectrum
EK(Z/p,n).
So
K(Z/p,n)(cid:39)µ (E)=Hom (EK(Z/p,n),E),
p E∞
and the composite map which is a homotopy equivalence is the tautological “evaluation” map.
Corollary 1.7. If X be any space such that π X is a finite p-group for all values of ∗ and 0 for
∗
∗>n. (I.e., a finite Postnikov system where the homotopy groups stop above n.) Then the canonical
map
X →Hom (EX,E),
E∞
is an equivalence.
POWER OPERATIONS 9
This is a version of rational homotopy theory for a very restricted class of spaces.
2. 2/20 Mike Hopkins
2.1. Introduction. Let me remind you a little bit about where we are. So last time Jacob discussed
the following situation. Let R be an E -ring. Today I’m going to focus on the case where R = K).
∞
That means that R is a ring up to homotopy, so that R0(X) for X a space is a ring and has a group
of units R0(X)×. That’s given by
R0(X)× =[X,GL (R)],
1
where GL (R) is a space sitting inside Ω∞R. This has a natural infinite loop space structure
1
GL (R)=Ω∞gl (R),
1 1
whichisaspectrumofunits inR. I.e.,GL (R)hasacoherentlyhomotopycommutativemultiplication.
1
Lasttime,JacobalsointroducedG (R),whichisthemaximal“abeliangroup”mappingtoGL (R)
m 1
(which is only an abelian group up to coherent homotopy). So we can write
G (R)=hom (HZ,gl (R)),
m Sp 1
or infinite loop space maps
G (R)=Hom (Z,GL (R)).
m InfLoop 1
Our goal is to describe a theorem describing the structure of this when R is Morava E-theory.
We might try to calculate this directly, but gl (R) is somewhat mysterious as a spectrum. For
1
example,gl (K)(cid:39)E×B asaspectrum,whereE isa2-stagePostnikovsystem(thepartofK-theory
1
that comes from plus or minus a line) and B has the property that for all p, B(cid:98)p (cid:39) b(cid:100)sup. B has the
homotopy type of b(cid:100)sup at all p-adic completions, but not integrally.
Question: How do we approach the space or spectrum of maps HZ→gl (R)?
1
We can use the fact that spectrum maps HZ→gl (R) are the same thing as E -maps
1 ∞
Σ∞Z→R,
+
i.e.
Hom (HZ,gl (R))(cid:39)Hom (Σ∞Z,R),
Sp 1 E∞ +
andyoushouldthinkofthisasanalogoustotheuniversalpropertyofthegroup ring ofagroup. This
has the advantage that we don’t have to deal with this mysterious gl , but we do have to compute
1
mapping spaces between E -rings.
∞
2.2. Resolutions. How do we compute mapping spaces between E -rings such as these? The only
∞
approach is to find a resolution of Σ∞Z. Say, a simplicial resolution P → Σ∞Z where each P is a
+ • + n
free E -ring spectrum.
∞
I need a temporary name for the free E -ring spectrum. First of all, what is a free E -ring
∞ ∞
spectrum free on? It’s free on a spectrum, and it’s analogous to the symmetric algebra of an abelian
group. Let’s use that notation.
Definition 4. Let X be a spectrum. Then Sym (X) is the wedge (cid:87) (X⊗n)) , i.e. the sum of
∗ n≥0 hΣn
the homotopy-theoretic symmetric powers of X. This is the free E -ring on X.
∞
Once I have a resolution P →Σ∞Z, then we get a spectral sequence
• +
π Hom (P ,R) =⇒ π (Σ∞Z,R).
∗ E∞ n ∗ +
Since each P =Sym (V ) for V a spectrum (we’re assuming freeness), then
n ∗ n n
Hom (Sym (V ),R)(cid:39)Hom (V ,R),
E∞ ∗ n Sp n
and that’s something we can calculate in terms of R-cohomology. So, I can do two things to compute
mapping spaces:
• Study the R-cohomology of E -rings and try to understand the E -term of this spectral
∞ 2
sequence.
• Wecouldjustlookaroundandtrytofind“innature”workableexplicitresolutionforP with
•
which we can compute.
10 POWER OPERATIONS
Both of these are viable options, and they give something interesting. I’ll talk about these both in
the case of K-theory.
I want to start with Atiyah’s old paper “Power operations in K-theory.” When we finally get
through it, it won’t give the right information to feed in here, but it informs this picture quite a bit.
Take the free E -ring on S0, which I’ve been calling Sym (S0) = (cid:87) BΣ . To understand this,
∞ ∗ n≥0 n+
we might want to understand:
Question: What is the K-theory of BΣ ?
n
Atiyah, when he introduced the theory of power operations in K-theory, knew that
K0(Σn)(cid:39)Rep(Σn)(cid:98),
was that the completion of the representation ring of Σ at the augmentation ideal. We might try to
n
work with all this before completion.
2.3. Representations of Σ . Let me start by reminding you of the relation between Rep(Σ ) and
n n
thetheoryofsymmetricfunctions. Here’sthisbasicapproach,whichgoesbacktoWeyl’sconstruction
oftherepresentationsofthegenerallineargroupandtoSchur’sthesis. There’sasophisticatedversion
of this in Schur’s thesis in terms of polynomial functors.
Let V be an n-dimensional vector space over C, so V (cid:39) Cn. Choosing such an isomorphism, we
get an action of GL on V. Inside GL , I have the diagonal matrix (t ,...,t ) where the t are
n n 1 n i
algebraically independent. Weyl’s idea is that you take V⊗k which has a commuting action of GL
n
and Σ . So it’s a representation of the product of the groups. That means that I can decompose it
k
into irreducible representations. We can write this
(cid:77)
V⊗k (cid:39) π(cid:2)V ,
π
π
where π is running through the irreducible representations of Σ . V is a representation of GL , and
k π n
Weyl’s theorem is that you can get all of them in this way (plus duals).
Example 2. If π is the trivial representation, then V is Symk(V).
π
Example 3. If π is the sign representation, then V is the exterior algebra (cid:86)kV.
π
The other ones in between give you lots more representations of GL .
n
This gives me some element
∆∈Rep(Σ )⊗Rep(GL ).
k n
Now I want to try to understand Rep(GL ) via their characters, and I have this special diagonal
n
element (t ,...,t ), and I can take the trace of this operator on any GL -representation. This gives
1 n n
a map
Rep(Σ )⊗Rep(GL )→Rep(Σ )⊗Sym(t ,...,t ),
k n k 1 n
taking the trace everywhere. It’s not hard to see that the polynomials you get from applying the
diagonal element are symmetric.
I’mnotgoingtodistinguishthiselement∆fromitsimageinRep(Σ )⊗Sym(t ,...,t ). Notethat
k 1 n
∆ is homogeneous of degree k. That’s easy to see from tensoring all these together.
There is a useful formula for this element ∆. Fix an equivalence relation α on {1,2,...,k}. Inside
Σ is the subgroup Σ ⊂ Σ consisting of permutations which preserve the equivalence classes (i.e.,
k α k
takes each equivalence class to itself). That’s a product of copies of the symmetric groups on the
equivalence classes. Associated to every such α relation is a partition λ = (λ ,...,λ ) which I’m
1 m
goingtowriteinincreasingorder. Theconjugacyclass ofthissubgroupdependsonlyonthepartition.
Let ρ be the permutation subgroup corresponding to this: that is,
λ
ρ =IndSk1.
λ Sα
(Here α is an equivalence relation that yields λ; the choice doesn’t matter.)
Another way to say this is that you’re considering the vector space with basis on equivalence
relations inducing the partition λ.