Stable and Unstable Operations in mod p
Cohomology Theories
Andrew Stacey
Sarah Whitehouse
October 17, 2006
Abstract
We consider operations between two multiplicative, complex orientable
cohomology theories. Under suitable hypotheses, we construct a map from
unstable to stable operations, left-inverse to the usual map from stable to
unstable operations. In the main example, where the target theory is one
of the Morava K-theories, this provides a simple and explicit description
of a splitting arising from the Bousfield-Kuhn functor.
1 Introduction
Given two graded cohomology theories, E*(-) and F *(-), we can consider vari-
ous types of operations from one to the other. There are two main types: stable
and unstable; and within the unstable operations are the additive operations.
These are simplest to describe in categorical language. A cohomology theory
is a functor satisfying certain properties. At various levels of forgetfulness *
*we
have the following functors:
1. A functor E*(-) from the (homotopy) category of based topological spaces
to the category of graded abelian groups which intertwines the two sus-
pension functors.
2. A sequence of functors (Ek(-))k2Z from the (homotopy) category of based
topological spaces to the category of abelian groups.
3. A sequence of functors (EkU(-))k2Z from the (homotopy) category of based
topological spaces to the category of sets.
The three types of operation from F *(-), the source theory, to E*(-), the
target theory, are:
1. Stable operations: for l 2 Z, Sl(F, E) is the set of natural transformatio*
*ns
r :F *(-) ! E*(-) of degree l.
2. Additive operations: for k, l 2 Z, Ak+lk(F, E) is the set of natural trans-
formations rk: F k(-) ! Ek+l(-).
3. Unstable operations: for k, l 2 Z, Uk+lk(F, E) is the set of natural trans-
formations rk: FUk(-) ! Ek+lU(-).
1
There is an obvious restriction map Sl(F, E) ! Uk+lk(F, E) for each k, l 2 Z.
In brief, our main theorem shows this map has a left-inverse under certain
conditions on E*(-) and F *(-). The full statement of the theorem is as follows.
Theorem A. Let E*(-) and F *(-) be two multiplicative graded cohomology
theories which are commutative and complex orientable. Let E* := E*(pt) be
the coefficient ring of E*(-). We assume that the following conditions hold.
1. E* has characteristic p.
2. The formal group law of E*(-) has finite height, say n.
3. The coefficient of the first term in the p-series for E*(-) is invertible.
4. The various E*-modules of operations from F *(-) to E*(-) are the duals
over E* to the corresponding E*-modules of co-operations.
Under these conditions, for each k, l 2 Z there is a map
1 : Uk+lk(F, E) ! Sl(F, E)
which is left-inverse to the restriction map.
We postpone to the next section an explanation of what all the conditions
mean. The map 1 has several pleasant properties; to describe most of these
we need to know more about the structure of the spaces of the various types of
operation, knowledge that we also postpone for the next section.
The map itself has a very simple description. To give this in its most topo-
logical form we recall that operations between cohomology theories are closely
related to homotopy classes of maps between certain spaces and between certain
spectra associated to the cohomology theories. In this language, the restriction
map from stable operations to unstable operations is nothing more than the
infinite-loop space functor, 1 . Thus we obtain the following corollary of the-
orem A.
Corollary B. Let E*(-) and F *(-) be cohomology theories as in theorem A.
Let E and F be representing spectra. Let 1 denote the infinite-loop space
functor from spectra to topological spaces. Then there is a map:
1 : [ 1 F, 1 E]+ ! {F, E}0
left-inverse to the map induced by the 1 -functor.
The subscript adorning [X, Y ]+ is to denote homotopy classes of maps which
preserve the basepoint.
Composing 1 with the map coming from the 1 -functor we produce a
projection on [ 1 F, 1 E]+ with the property that a homotopy class lies in the
image of this projection if and only if it is an infinite loop map and, moreove*
*r,
the delooping of this map is unique.
This projection is easy to describe. There are certain maps:
n-1)1
vEn: 1 E ! 2(p E
n-1)1
vFn: 1 F ! 2(p F
2
which come from the p-series of the formal group law associated to each coho-
mology theory. The conditions on E*(-) guarantee that vn is invertible. The
projection is: i j
n-1) F
ae 7! (vEn)-1 2(p ae vn .
The conditions in the theorem are really all about the target theory, E*(-),
even the last one. The main examples to which we wish to apply this theorem
are where E*(-) is one of the Morava K-theories, K(n) *(-), at a prime p. We
discard the case n = 0 as that is just rational cohomology and we take an odd
prime to ensure that the multiplication is commutative. With this choice for the
target theory there is no restriction on choosing F *(-) as the four conditions*
* in
theorem A are automatically satisfied. In this case, corollary B is an elaborat*
*ion
of an application of the Bousfield-Kuhn functor.
This functor, written n, goes from the homotopy category of based p-local
spaces to the homotopy category of p-local spectra. Its key property is that if
G is a p-local spectrum then n 1 (G) is LK(n)G, the K(n) -localisation of G.
In particular, if G is already K(n) -local then n 1 (G) ' G. This is the case
for K(n) itself. Thus the functorial properties of n yield a map:
[ 1 G, 1 K(n) ]+ ! {LK(n)G, K(n)}0.
One of the defining properties of the K(n)-localisation is that there is a natu*
*ral
isomorphism {LK(n)G, K(n)} ~={G, K(n)}. Therefore we have a map
n :[ 1 G, 1 K(n) ]+ ! {G, K(n)}0.
By a similar device we can remove the requirement that G be p-local. Then n
can be compared to the map 1 in corollary B.
Theorem C. Let the source theory be a complex orientable, graded, commu-
tative multiplicative cohomology theory. Then, with target theory K(n) *(-),
1 = n.
This paper is structured as follows. In the next section we describe the
features of cohomology theories that we need. In section 3 we look at the p-
series coming from the complex orientation and use this to define certain key
co-operations. These are the essential ingredients of the proof of theorem A.
In section 4 we prove our main technical result, proposition 4.9, which involves
the relationships between the spaces of co-operations. It is then a short step *
*to
our main result, theorem 5.1 in section 5, which is a more detailed version of
theorem A and of corollary B. We conclude by proving theorem C in section 6.
There is considerable detail in the papers [1] and [2] about operations in
cohomology theories. Most of the background that we need can be found in
those papers. The papers [8] and [10] are the original sources for some of the
structure that we use.
There is some overlap in our main theorem with the work of [5]. The first
splitting of K*(BP ) in that paper is dual to our splitting. We do not go on
to consider further splittings, as is done in [5], because the first splitting *
*has a
good topological description which is missing in the higher ones. The proof of
our theorem and that of [5] run along similar lines.
Related work using the Bousfield-Kuhn functor has appeared in [4], [7], and
[9].
3
Finally, we note some conventions that we shall use throughout this paper.
Firstly, we work throughout in the homotopy categories of spaces and spectra
and shall use the short-hand "map" for a morphism in the appropriate category.
Thus what we mean when we say "map" is really a homotopy class of maps in
the conventional sense. We trust that this will not be overly confusing.
Secondly, following [1] and [2] we grade homology negatively. In order to
get the pairing between homology and cohomology correct one theory has to be
graded negatively. As in [1] and [2], for us the cohomology theory is the object
of study whereas the homology theory is a tool we shall use in the analysis.
Thirdly, and unlike [1] and [2], we shall always be careful to distinguish
between spaces and spectra. The convention of [1] and [2] is to use the same
notation for a space and its suspension spectrum. This is a convenient shorthand
but as our paper is all about the passage from spaces to spectra it is a shorth*
*and
we feel morally obliged to do without.
Fourthly, we shall need to work with both based and unbased spaces. We
shall distinguish between morphisms in the two categories with the notations
[X, Y ] for homotopy classes of all maps and [X, Y ]+ for homotopy classes of
based maps. We recall that when the target, Y , is an H-space and the source,
X, is a based space then there is a natural projection [X, Y ] ! [X, Y ]+ .
2 Ingredients
In this section we shall describe the various ingredients needed for our work.
This is not intended to be a detailed reference on cohomology theories, rather
our aim is to establish our notation whilst giving just enough information to
allow the casual reader to follow our argument without constantly referring to
other works. The bulk of this can be found in the expository parts of [1] and
[2] and we largely follow their conventions. The reader familiar with [1] and [*
*2]
may wish to skip to the next section.
2.1 Generalised Cohomology Theories
Let E*(-) be a multiplicative graded generalised cohomology theory that is
commutative and complex orientable. Much of what we are about to say applies
to more general theories but as we shall only use such theories we specialise at
the outset.
As this is a multiplicative theory, the cohomology of a point is a graded ri*
*ng
called the coefficient ring. We write this as E*.
Representing Spaces and Spectrum. Brown's representability theorem,
and its consequences, provide us with a sequence of H-spaces, (E_k)k2Z, which
represent this cohomology theory. That is, we have universal elements 'k 2
Ek(E_k) such that for any space X the map ff 7! ff*'k is an isomorphism of
abelian groups:
[X, E_k] ! Ek(X).
The abelian group structure on the left-hand side comes from the H-space struc-
ture of E_k. The universal class 'k actually lies in the subgroup eEk(E_k) and *
*so
for any based space X the above isomorphism identifies [X, E_k]+ with eEk(X).
4
These spaces are unique up to equivalence. It can be shown that the sus-
pension isomorphism of reduced cohomology, eEk(X) ~=Eek+1( X), defines an
equivalence E_k ! E_k+1. These equivalences allow us to construct an -
spectrum E from the E_k. Using this spectrum we can extend the cohomology
theory to spectra by defining eEk(F ) := {F, E}k and define the associated ho-
mology theory for both based spaces and spectra as eEk(X) := {S, E ^ X}-k.
This extends to unbased spaces by the usual method of adding a disjoint base-
point: Ek(X) := eEk(X+ ). Note that we are following the convention of [1] in
(redundantly) writing the homology and cohomology of spectra as reduced.
In light of the fact that eEk(F ) and {F, E}k are one and the same for spect*
*ra,
we make the same identification for spaces. That is, we consider the isomorphism
[X, E_k] ~=Ek(X) to be so natural as to be worth writing as an equality. We
shall still employ the language of both sides and talk of maps or classes as be*
*st
fits, but shall regard the two dialects as synonymous.
Structure Maps. All of the structure of the cohomology theory E*(-) is
reflected in the spectrum E and the spaces E_k. Essentially, any natural trans-
formation of cohomology theories is represented by maps between the associated
spaces or spectra. The existence of the map can usually be deduced by applying
the natural transformation to the appropriate universal class.
As an example, we have the already-mentioned equivalence E_k ' E_k+1
coming from the natural isomorphism Eek(X) ~= eEk+1( X). To define the
associated map we apply the suspension isomorphism to the space E_k:
Eek(E_k) ~=eEk+1( E_k).
By the representability theorem, the image of the universal class 'k is repre-
sented by a (based) map #k: E_k ! E_k+1. The naturality of the suspension
isomorphism implies that for a general space it is the composition:
eEk(X) = [X, E_k]+ -! [ X, E_k]+ -#k*-![ X, E_k+1]+ = eEk+1( X).
In this fashion we deduce the existence of several maps which we now list.
Suspension. There is a map #k: E_k ! E_k+1 representing the suspension
isomorphism eEk(X) ~=eEk+1( X).
Stabilisation.There is a map of spectra oek: 1 E_k! E of degree k repre-
senting the isomorphism eEk(X) ~=eEk( 1 X).
Multiplication. There is a map of spectra, OE: E ^ E ! E, of degree 0 and
maps of spaces OEk,l:E_k^ E_l! E_k+lrepresenting the multiplication in
the cohomology rings.
Unit. There is a map of spectra, j :S ! E, of degree 0 and maps of spaces
jk: Sk ! E_krepresenting the unit in the cohomology rings.
These maps satisfy various compatibility relations. In particular, the stable
and unstable realms correspond under the stabilisation maps. We record one
particular relation that will be of use later:
#k = OE1,k(j1 ^ 1). (2.1)
5
Using the multiplication we can define the augmentation maps. The stable
augmentation map is:
fflS :Eek(E) = {S, E ^ E}-k -OE*!{S, E}-k = E-k .
The unstable augmentations are:
fflk: El(E_k) ! eEl(E_k) ~=eEl( 1 E_k) oek*--!eEl-k(E) fflS-!Ek-l.
We shall write ffl for fflk where we do not wish to or cannot specify the index.
Duality. The augmentations define a pairing between cohomology and homol-
ogy. An element ff 2 Ek(X) defines a push-forward in homology for X a space
or spectrum; respectively:
ff*:El(X) ! El(E_k),
ff*:El(X) ! El-k(E)
which we compose with the appropriate augmentation to end up in Ek-l.
Under favourable circumstances the induced map E*(X) ! DE*(X) (the
E*-dual of E*(X)) is an isomorphism. To truly understand this statement would
require a lengthy and, for our purposes, unnecessary discussion of the topologi*
*es
involved. The precise circumstances are recorded in [1, theorem 4.14]: if E*(X)
is free as an E*-module then E*(X) is the E*-dual of E*(X). When this occurs
we shall say that X has strong E-duality. If this holds for all spaces and spec*
*tra
then we shall say that E*(-) has strong duality.
2.2 Operations and Co-operations
Another piece of the baggage that comes with a generalised cohomology theory
is the family of operations. As with all the other parts of the structure of
the cohomology theory, these are reflected in maps between the representing
spaces. Also we can consider operations from one cohomology theory to another.
Thus let F *(-) be another generalised cohomology theory (also multiplicative,
commutative, and complex orientable). We shall consider the operations from
F *(-) to E*(-).
Stable and Unstable Operations. As mentioned in the introduction, op-
erations are simplest to describe in the language of category theory. In this
setting, a cohomology theory is a functor on the homotopy category of topologi-
cal spaces and an operation is simply a natural transformation between functors.
With a graded cohomology theory one has two types of operation depending on
whether one considers the cohomology theory as a whole, leading to stable op-
erations, or one takes a single component of it, leading to unstable operations.
We allow degree shifts in both cases.
In the stable case this description needs a little elaboration. Considered as
a whole, a cohomology theory is a functor between two categories each of which
has a suspension functor and the cohomology theory intertwines these functors.
To qualify as a stable operation, a natural transformation has also to respect
the suspension functors. Otherwise, using the restrictions mentioned below, a
6
stable operation would be simply a sequence of unstable operations with no
relations between the components. Respecting the suspension functors imposes
some relations between successive components, as we will see in a moment.
We label the set of stable operations of degree l from F *(-) to E*(-) by
Sl(F, E) and the set of unstable operations from F k(-) to El(-) by Ulk(F, E).
At the most basic level these are abelian groups since operations take values in
abelian groups.
There is an obvious way to define an unstable operation by restricting a
stable operation to a single component. In this way, a stable operation defines
a sequence of unstable operations. This suggests the question as to whether
a sequence of unstable operations patches together to give a stable operation.
This will happen if the unstable operations commute with the suspension iso-
morphisms, modulo a sign. That is, suppose that for each k 2 Z we have
an unstable operation rk: F k(-) ! Ek+l(-) then there is a stable operation
r :F *(-) ! E*+l(-) restricting to rk (modulo the sign issue) if and only if ea*
*ch
rk maps reduced cohomology to reduced cohomology and the following diagram
commutes for each space X up to the indicated sign:
eF k(X)___~=_//_eF(k+1 X)
|rk| (-1)k |rk+1|
fflffl|~ fflffl|
Eek+l(X)___=_//eEk+l+1( X)
The resulting stable operation need not be unique, however. For that one needs
to know that a certain lim1term vanishes. There are technical conditions that
guarantee this which, as we note later, hold in our context.
A stable operation extends in the obvious way to an operation on the co-
homology of spectra. There is no analogue of an unstable operation in this
case.
Using the same techniques as for the structure maps we can identify opera-
tions with maps between the representing spectra or spaces.
Stable. Stable operations F *(-) ! E*+l(-) correspond to maps of the spec-
trum F to E of degree l, and thus Sl(F, E) ~=eEl(F ).
Unstable. Unstable operations F k(-) ! El(-) correspond to maps F_k! E_l
and thus Ulk(F, E) ~=El(F_k).
Additive Operations. Within the family of unstable operations lie the ad-
ditive operations which we denote by Alk(F, E) Ulk(F, E). A generic unstable
operation need not preserve any of the structure of F k(X), even that of being
an abelian group. An additive operation is one that does preserve the additive
structure. Using the fact that the additive structure of F k(X) comes from the
H-space structure of F_kit is straightforward to show that within El(F_k) the
additive operations are:
* * * l l
ker (~ - p1 - p2 ): E (F_k) ! E (F_kx F_k)
where ~: F_kx F_k! F_kis the H-map and p1, p2 are the projections onto the
two factors. This is the subspace of primitives and is written P El(F_k). Thus
Alk(F, E) ~=P El(F_k).
7
Co-operations. If the spectrum F and the spaces (F_k)k2Z have strong E-
duality then the cohomology rings eE*(F ) and E*(F_k) are the E*-duals of the
corresponding homology groups. Therefore one can analyse the groups of oper-
ations by studying these homology groups. This is often a Good Thing To Do.
Firstly, the topological issues alluded to in the paragraph on duality all occu*
*r on
the cohomology side; homology is discrete. Secondly, it is easier to find expli*
*cit
elements in the homology using push-forwards from key test spaces.
Anything worth studying gets a name, in this case co-operations. As with
operations these come in three flavours: stable, unstable, and additive. The
stable co-operations are eE*(F ). The unstable ones are E*(F_*). The additive
co-operations are the indecomposables of E*(F_*): for each k 2 Z we define
QE*(F_k) := coker (~* - p1*- p2*): E*(F_kx F_k) ! E*(F_k) .
This is a quotient of E*(F_k); let "qkdenote the quotient map. Assuming suffici*
*ent
duality the E*-dual of QE*(F_k) is P E*(F_k), which we know to be isomorphic
to A*k(F, E).
In [2] the authors regrade the additive co-operations by defining Q(E, F )k**
*:=
QE*(F_k), with the total degree of Q(E, F )kibeing k - i. The reason for this is
that the algebraic structure of QE*(F_k) makes more sense with the new grading.
For this paper there is not much difference between the two options as we mainly
deal with all unstable operations and when we do explicitly consider additive
operations then we are concerned with finding identities and these, of course,
hold whatever the grading scheme in use. We choose Q(E, F )**because [2] is the
main background for this paper and so we are trying to use their conventions
whenever possible.
The regraded quotient map of degree k is:
qk: E*(F_k) ! Q(E, F )k*.
The stabilisation map oek: 1 F_k! F induces the stabilisation map of co-
operations:
oek*:E*(F_k) ! eE*(F_k) ~=eE*( 1 F_k) ! eE*-k(F ).
This factors through the quotient to additive co-operations. Thus we can define
the maps:
Qoek*:QEi(F_k) ! eEi-k(F ),
Q(oe):Qki(E, F ) ! eEi-k(F ).
The former is of degree k, the latter of degree 0.
This seems an appropriate place to note that if the spectrum F and the
spaces (F_k)k2Z have strong E*-duality then the potential lim1-problem referred
to above disappears: a stable operation is completely determined by its unstable
components. See [1, x9] for more on this issue.
Operations, Maps, and Functionals. We therefore have three ways of
thinking about operations: as operations, as maps (or classes), and as function-
als on co-operations (assuming sufficient duality). We shall distinguish between
these views using fonts and alphabets: roman [italic] for operations, greek for
8
maps, and gothic for functionals. We shall attempt to make our notation as
transparent as possible: the stable operation r will correspond to the stable
map ae and to the functional r on stable co-operations.
In each of the three cases we have a natural restriction map from the sta-
ble to the unstable operations which factors through the additive ones. These
restriction maps do not correspond exactly: there are signs to insert at the
appropriate junctures. The full diagram (which is an expansion of [2, 6.10]) is:
k+l
Sl(F, E)_____//_Ak+lk(F,_E)__//_Uk (F, E)
|~=| (-1)kl ~=|| ~=||
fflffl|oe* fflffl| fflffl|
eEl(F )____k//_P Ek+l(F_k)___//Ek+l(F_k) (2.2)
|~=| (-1)k ~=|| (-1)k ~=||
fflffl|DQ(oe) fflffl|Dqk fflffl|
DleE*(F )____//DlQk*(E, F_)___//DlE*(F_k).
The reasons for the signs in this diagram are quite subtle so it is worth ta*
*king
some time to explain them carefully. This is an expansion of the scholium on
signs in [2, x6].
Let us start with the (-1)kl in the upper left square. Let r 2 Sl(F, E)
be a stable operation with restriction rk 2 Uk+lk(F, E). Consider the following
diagram.
* ~=
Fe0(F )__oek//_eF(k 1 F_k)____//eF(kF_k)____//_F k(F_k)
|r| (-1)kl r|| r|| |r| (2.3)
fflffl|oe* fflffl|~ fflffl| fflffl|
Eel(F )____/k/eEk+l( 1 F_k)=_//eEk+l(F_k)__//_Ek+1(F_k)
As r is an operation of degree l and oek is a map of spectra of degree k we have
oek*rff = (-1)klroek*ff. This accounts for the sign in this diagram. The other
squares commute since the maps involved have degree 0.
Let us chase the universal class ' 2 eF(0F ) around this diagram. The map
oek was defined so that the image of ' in F k(F_k) is 'k. Therefore the image
of ' in Ek+l(F_k) when taking the upper route in diagram (2.3)is r'k. As rk
is the restriction of r this is also rk'k and so is the class corresponding to *
*the
unstable operation rk. Thus the image of ' in Ek+l(F_k) via the upper route in
diagram (2.3)is the image of r via the upper route in diagram (2.2).
The lower route starts off with the class r' 2 Eel(F ). This is the class
corresponding to r. The rest of the lower route is the map eEl(F ) ! Ek+l(F_k)
from diagram (2.2). Hence the image of ' via the lower route in diagram (2.3)
is the image of r via the lower of the two routes from Sl(F, E) to Ek+l(F_k) in
diagram (2.2).
Therefore as we have the sign (-1)kl in diagram (2.3)we need it also in
diagram (2.2). As additive operations and classes are subsets of the unstable
ones the sign must go between the stable and additive lines.
The signs in the lower squares come from a technical subtlety. Had we put
DQE*(F_k) in the middle of the lower row there would have been no signs.
9
Therefore the signs come from replacing DQE*(F_k) by DQk*(E, F ). Consider
the diagram:
Qoek*
Ek+l(F_k)__"qk//_MQEk+l(F_k)__//eEl(F9)9OO
MMMMqkM -k | Q(oe)sssss
MMM | sss
MM&& | ss
Q(E, F )kk+l
where -k :Q(E, F )k*! QE*(F_k) is the degree shift isomorphism and the total
upper map is oek*which factors as shown. This diagram commutes. When we
dualise, we find that:
D"qk= D( -kqk) = (-1)kDqkD( -k)
DQ(oe)= D(Qoek* -k) = (-1)kD( -k)DQoek*
as in each case both commutants have degree k. Therefore if we define the map
P Ek+l(F_k) ! Dk+lQ(E, F )k*as:
-k
P Ek+l(F_k) ! Dk+lQE*(F_k) D----!DlQ(E, F )k*
we find that we need to add the signs (-1)k to each lower square to make the
resultant diagram commute.
Constant Operations and Based Operations. There is one particular
type of operation that we have to consider, if only so that we know how to
ignore them later. These are constant operations. Each v 2 E* defines an
operation on F *(X) by x 7! v1X , where 1X is the unit in the algebra E*(X).
Juxtaposed to constant operations are the based operations. An operation
r : F k(-) ! El(-) is based if it maps zero to zero. This is, of course, automa*
*tic
for an additive operation but not for a general unstable operation.
The reason for mentioning these two types of operation together is that every
(unstable) operation has a decomposition as the sum of a constant operation and
a based operation. For an operation r : F *(-) ! E*(-) let vr 2 E* = E*(pt)
be the image of 0 2 F *= F *(pt) under r, then let "rbe the based operation
given by "r(x) = r(x) - vr1X for x 2 F *(X).
The based operations correspond to the classes in eEl(F_k) and thereby to the
based maps, [F_k, E_l]+ . The based functionals are dual to the reduced homology
groups, eE*(F_k).
In each case, the projection from the unbased to the based version is the ob-
vious one. Where we have a possibly unbased operation r, map ae, or functional
r we shall denote the corresponding based one by "r, "ae, or "r.
Suspension and Looping. There is a method of getting new unstable op-
erations from old. Given an unstable operation rk: F k(-) ! El(-) we can
define another unstable operation rk-1 :F k-1(-) ! El-1(-) via:
rk-1 :F k-1(X) ! eF k-1(X) ~=eF(k X) "rk-!eEl( X) ~=eEl-1(X) El-1(X).
The corresponding idea in the world of maps is to use the equivalences
E_k-1' E_k and so given a map aek: F_k! E_lwe define aek-1 via:
aek-1 :F_k-1' F_k- "aek-! E_l' E_l-1.
10
For functionals the push-forward on co-operations defines the following sus-
pension map:
(-1)k-1#k-1*
: El-1(F_k-1) ! eEl-1(F_k-1) ~=eEl( F_k-1) ---------! eEl(F_k) El(F_k).
The sign here is part of the baggage that comes with dealing with graded and
ungraded objects. Its presence here is a minor nuisance but its absence would
be a minor headache later. We dualise this map to one on functionals.
We shall denote this process of getting one operation, map, or functional
from another by . Thus, for functionals, = D .
The diagram relating these maps is:
~= ~=
Ulk(F, E)_______//El(F_k)________//DlE*(F_k)
|| 1 || (-1)k ||
fflffl|~ fflffl| ~ fflffl|
Ul-1k-1(F,_E)=_//El-1(F_k-1)_=__//Dl-1E*(F_k-1)
It is curious that removing the sign from the definition of the suspension map
on functionals does not make this diagram commute without signs, rather the
sign is -1 regardless of the degree. This is another aspect of the passage from
ungraded to graded objects.
We should emphasise that we have defined looping for unbased operations,
maps, and functionals. However, the construction factors through the projection
to the corresponding based objects.
Colimits. The spectrum F is built from the spaces F_kusing the signed sus-
pension maps (-1)k#k: F_k ! F_k+1. This expresses F as equivalent to the
colimit of the sequence ( 1 F_k) in the category of spectra. Applying E-homol-
ogy leads to:
Ee*(F ) ~=colimkeE*( 1 F_k) ~=colimkeE*(F_k) ~=colimkE*(F_k).
The last isomorphism is because the suspension map factors through the pro-
jection to reduced homology and so this projection defines an isomorphism on
the colimits.
In particular,
eEl(F ) ~=colimkeEl+k(F_k) ~=colimkEl+k(F_k).
As the suspension map also factors through the quotient to additive co-opera-
tions, we can replace E*(F_k) by Q(E, F )k*as appropriate.
Complex Orientation. Our cohomology theories are complex orientable so
they admit universal Chern classes. That is, say for F *(-), there is an element
xF 2 F 2(CP 1 ) which restricts to a generator of eF(*CP1) under the canonical
inclusion CP 1 CP1 . If, identifying once and for all CP 1with S2, xF restricts
to the image of the unit under the natural isomorphisms eF *+2(S2) ~=eF(*S0) ~=
F *then we say that xF is a strict universal Chern class. Any universal Chern
class can be modified to a strict one so there is no loss in assuming that all
universal Chern classes are strict.
11
The existence of a universal Chern class implies that F *(CP 1 ) ~=F *[[xF]].
The F -homology of CP1 is then the free F *-module on generators fiFiof degree
-2i defined so that (xF)i(fiFj) = ffiij.
The H-space structure of CP1 is a map CP1 xCP 1 ! CP1 . In cohomology
this induces a map:
F *[[xF]] ~=F *(CP 1 ) ! F *(CP 1 x CP1 ) ~=F *[[xF1, xF2]].
The image of xF under this map is known as the formal group law of the coho-
mology theory F *(-). We shall write this formal power series as
xF1+FxF2
(the "F " is to indicate the cohomology theory).
In certain circumstances it is possible to substitute elements of an F *-alg*
*ebra
into the formal power series that this represents. (The only difficulty here is
with convergence of the resulting sum; so it works, for example, on nilpotent
elements and it works if the algebra is complete with respect to some filtration
and successive powers of the elements that one is substituting in lie further a*
*nd
further down in the filtration.) The properties of the formal group law imply
that, when this is possible, the resulting operation is associative, commutativ*
*e,
unital, and has inverses - hence the name "formal group law". We shall denote
iterations of this process with the adorned summation notation:
X F
We shall need one more fact about the structure of the formal group law as
a power series. It follows from the basic properties of formal group laws that
there are identities:
xF1+FxF2= xF1+ xF2R1(xF1, xF2) = xF2+ xF1R2(xF1, xF2)(2.4)
for some formal power series R1(xF1, xF2), R2(xF1, xF2).
A particular case where substitution is allowed is the element xF of F *[[xF*
*]].
Substituting this into both variables we define
[2]F(xF) = xF +FxF 2 F *[[xF]].
It is straightforward to see that the resulting formal power series has leading
term 2xF and so can be again substituted in to the formal group law. Iterating
this procedure, we define [n]F(xF) := xF +F[n - 1]F(xF). This formal power
series is called the n-series of F *(-).
There is an alternative derivation of these formal power series. The H-
space structure on CP 1 defines an nth power map CP 1 ! CP 1. Using the
isomorphism F *(CP 1 ) ~=F *[[xF]], the image of xF under the pull-back via this
map is a formal power series in xF and it is not hard to see that it is [n]F(xF*
*).
A particularly important case of this is the p-series for p a prime. This is*
* of
the form: X
[p]F(xF) = pxF + gFj(xF)j+1
j 1
12
for some gFj2 F -2j. The reduction of this modulo p has the form:
X F i
[p]F(xF) vFi(xF)p mod p
i 1
for some vFi2 F -2(pi-1). Note the adorned summation sign.
The Chern class for F *(-) is represented by a map xF :CP 1 ! F_2. Apply-
ing E-homology leads to a push-forward xF*:E*(CP 1 ) ! E*(F_2). As E*(-)
is itself complex orientable the former is the free E*-module on generators fiE*
*i.
Let bi= xF*fiEiand define:
X
b(s) = bisi2 E*(F_*)[[s]].
i 0
We shall use the same notation, i.e. bi, for the images of the biin the addi*
*tive
and stable co-operations.
2.3 Algebraic Structure
The various groups of operations and co-operations have considerable algebraic
structure. The full list is long so we shall only describe what we need. For all
the gory details, see [1] and [2].
The main structures that we shall use are the multiplicative and bimod-
ule structures on the sets of co-operations and the bimodule structure on the
sets of operations. This is further complicated by the fact that there are two
multiplications on the unstable co-operations.
Once we have introduced these algebraic structures we shall consider how
some of the data we have already seen behaves algebraically.
Co-operation Multiplications. The more important - for our purposes -
multiplication is defined using the maps on the spaces F_kand spectrum F which
represent the multiplication in F *(-). That is, the map OEl,k:F_lx F_k! F_l+k
defines a push-forward:
OEl,k*
E*(F_l) x E*(F_k) ! E*(F_lx F_k) ---! E*(F_l+k).
As OEl,kis a component of an infinite loop map we also get multiplications on
the additive and stable sets of co-operations which all correspond under the
maps from unstable co-operations to additive and to stable. For unstable co-
operations we shall write this multiplication as (a, b) 7! a O b. For the other*
*s we
shall just use the abutment notation. Note that as the quotient from unstable to
additive co-operations has a non-trivial degree, the correct formula on a produ*
*ct
is:
qi+j(a O b) = (-1)j|a|qi(a)qj(b)
for a 2 E*(F_i) and b 2 E*(F_j).
For additive and stable co-operations these multiplications are graded com-
mutative (taking the total degree in the regraded additive realm). For unstable
co-operations this is still true but the issue is somewhat complicated by the f*
*act
that the set of unstable co-operations, E*(F_*), has two indices which are used
in different ways: the first is a genuine grading whereas the second is really
13
only a labelling. However this multiplication does use this second index. To
describe exactly how, we would need to introduce yet more of the structure and
it turns out that, for our purposes, this is unnecessary since any element with
both indices even commutes with everything. On the few occasions where we
need to consider other elements we shall give the explicit commutation formula.
In light of this confusion, we add that when we speak of the degree of an
element in E*(F_*) we shall be using the first index only.
The set of unstable co-operations has another multiplication which comes
from the H-map F_kxF_k ! F_k. This is graded commutative with the "honest"
grading. Note that this product only makes sense for elements which have the
same second index. We shall write this multiplication as (a, b) 7! a * b.
The interaction of the two multiplications is controlled by a coproduct, _,
whichPis induced by the diagonal map F_k ! F_kx F_k. That is, if _(c) =
ic0i c00ithen:
X 0
(a * b) O c = (-1)|b||ci|(a O c0i) * (b O c00i).
i
This is the only place where we use this coproduct.
The reason that the *-product does not appear in the additive or stable
realms is that it is what is being quotiented out when passing to the additive
co-operations. Specifically, the quotient on a *-product is:
qk(a * b) = fflk(a)b + (-1)|a||b|fflk(b)a,
where fflk is the appropriate augmentation.
Bimodule Structure. The various groups of operations from F -cohomology
to E-cohomology have the structure of (E*-F *)-bimodules. The left E*-action
is:
(v . r)(ff) = vr(ff)
whilst the right F *-action is:
(r . v)(ff) = r(vff).
In terms of maps these actions are given by composition with certain maps
of the representing spaces. For v 2 El = El(pt) we define ,v :E_k! E_k+lby:
E_k~=pt x E_kvx1--!E_lx E_kOEl,k--!E_k+l.
In the stable case we use the smash product and view v as an element of eEl(S)
(we could have used the smash product in the unstable case as well since the
multiplication factors through the smash product). Using these maps we define
the left action of E* and right action of F *by appropriate composition:
v . ae = (,v)ae, ae . v = ae(,v).
The left action of E* agrees with the obvious action on E*(F_k).
For co-operations we have an obvious left action of E* as the coefficient ri*
*ng.
The right action of F *is given by push-forwards:
(,v)*: E*(F_k) ! E*(F_k+l).
14
Unpacking the construction of ,v, and using the definition of the O-multiplica-
tion, we see that there is an element [v] 2 E0(F_l) such that the right action *
*of
v on E*(F_k) is: c 7! c O [v]. The element [v] is the image of 1 under the map
v*: E* = E*(pt) ! E*(F_l). There are corresponding actions in the additive
and stable realms since the map ,v is a component of an infinite loop map.
Diagram (2.2)is then a diagram of (E*-F *)-bimodules.
Algebraic Suspension. The suspension map on functionals has a particu-
larly pleasant structure. The suspension isomorphism eE0(S0) ~=eE1(S1) defines
a canonical element u1 2 eE1(S1) as the image of the unit. This element deter-
mines the suspension isomorphism as follows. The E*-module eE*(S1) is free of
rank one generated by u1 so we have the following isomorphisms:
i j
eEk( X) = eEk(S1 ^ X) ~= Ee*(S1) E* eE*(X) ~=eEk-1(X) (2.5)
k
where the final map is u1 c 7! c.
From equation (2.1), the map #k-1 : F_k-1! F_kfactors as OE1,k-1(j1^ 1).
Thus the following diagram commutes:
i ~j ~
eEl-1(F_k-1)~=//_eE*(S1) E* eE*(F_k-1)=//_eEl(S1 ^ F_k-1)=//_eEl( F_k-1)
l |
| |
|j1*|1 |(j1^1)* |
fflffl| fflffl|| |
i e e j__//eE(F ^ F ) |#k-1*|
E* (F_1) E* E*(F_k-1) l l__1 __k-1 |
| |
|OE1,k-1 |
| * |
fflffl| fflffl|
eEl(F_k)__=____//eEl(F_k)
The upper route is, up to sign, the suspension map. Thus from the lower
route, we can see that this map is:
c 7! (-1)k-1e O c
where e = j1*u1 2 eE1(F_1). We shall use the same notation for the image of e
in Q(E, F )11. In the stable realm it maps to the identity (the maps which defi*
*ne
stable co-operations as the colimit of unstable are, up to sign, O-multiplicati*
*on
by e).
The commutation law for, coproduct of, and augmentation of the element e
are:
a O e= (-1)j+ke O a, a 2 Ej(F_k);
_(e) = e 11 + 11 e;
ffl1(e)= 0.
P
Algebraic Chern Class. Returning to the series bisi, the first two terms
are readily identifiable in terms of the algebraic structure. The first, b0, is*
* 12,
15
the *-unit in E0(F_2). The second, as our Chern classes were strict, is -eO2.
These quotient to the (regraded) additives as follows:
q2(b0)= q2(12) = 0
q2(b1)= q2(-e O e) = q1(e)q1(e) = e2.
The bi O-commute with everything as they lie in E2i(F_2). Their coproducts
and augmentations are:
X
_(bk)= bi bj;
i+j=k(
ffl2(bk)= 0 ifk > 0,
1 ifk = 0.
2.4 Morava K-Theory
The Morava K-theories will be our main examples of target theories. These are
a family of multiplicative generalised cohomology theories indexed by primes
and non-negative integers. There are some peculiarities corresponding to prime
2 which we wish to avoid so we fix an odd prime, p. For any prime the the-
ory corresponding to zero is ordinary rational cohomology so any interesting
behaviour peculiar to the Moravian theories would be expected to rear its head
for strictly positive integers, and this is true for the phenomenon we have ob-
served, hence we choose n 1. Thus we have fixed our attention on K(n)*(-),
the nth Morava K-theory at the prime p, for n > 0 and p odd. (The prime is
not explicit in the notation as it is quite unusual to vary it in the course of*
* a
discussion whereas it is sometimes fruitful to consider different values of n.)
The coefficient ring of K(n) *(-) is
K(n)* = Fp[vn, v-1n]
where |vn|= -2(pn - 1). This is a graded field and hence all modules over
this ring are free. Two consequences of this are that K(n) *(-) has a K"unneth
formula and has strong duality.
The p-series for K(n) *(-) is
n
[p]K(n)(s) = vnsp .
3 Analysing the p-Series
In this section we analyse what information can be gleaned from the p-series
of the two cohomology theories under consideration. From now on we assume
that E*(-) and F *(-) satisfy the conditions of theorem A. That is, they are
multiplicative graded cohomology theories which are commutative and complex
orientable and the following conditions hold.
1. The coefficient ring of E*(-) has characteristic p.
2. The formal group law of E*(-) has finite height, say n.
3. The coefficient of the first term in the p-series for E*(-) is invertible.
16
4. The various groups of operations from F *(-) to E*(-) are dual over the
coefficient ring of E*(-) to the corresponding groups of co-operations.
The main tool in our analysis is a result from [8].
Theorem 3.1 (Ravenel-Wilson). The following identity holds in E*(F_*)[[s]]:
b([p]E (s)) = [p]F (b(s)),
where, in expanding out the right-hand side, the coefficients gFjof the p-series
for F *(-) act via the right action of F *on E*(F_*).
P
Recall that b(s) = i 0bisi.
To unpack this we use the fact that the maps which represent the addition
and multiplication in F *(-) defined the *- and O-multiplications on E*(F_*).
Therefore, when expanding the right-hand side, we need to translate addition
to *-multiplication and multiplication to O-multiplication. This leads to:
_ !
X
b ps + gEisi+1 = b(s)*p* F b(s)Oi+1O [gFi]. (3.1)
i>0 i>0
3.1 Additive Co-operations
Equation (3.1)looks horrendous but simplifies considerably when we quotient
to the additive co-operations. Throughout this section we shall be working in
the additive realm; that is, with Q(E, F )**and formal power series over this.
As fflb(s) = 1, we find that in Q(E, F )**[[s]]:
_ !
X X
b ps + gEisi+1 = pb(s) + b(s)i+1[gFi].
i>0 i>0
In the additive realm it is a tautology that the left and right Z-actions ag*
*ree.
As E*(F_*) is an E*-module it has characteristic p and thus we may replace both
sides by their reductions modulo p. That is:
_ !
X E i X F i
b vEisp = b(s)p [vFi]. (3.2)
i>0 i>0
From this equation we shall deduce the following result.
n-1
Proposition 3.2. For n 2 N, let ssn = p___p-1. Then in Q(E, F )**:
vEnb1ssn= b1ssn+1-1[vFn]. (3.3)
Proof.Our strategy for proving (3.3)is to equate powers of s in (3.2)and read
off certain identities. To begin, we examine the left-hand side of (3.2)toPfind*
* its
leading term. The left-hand side is of the form b(r(s)) where r(s) = Ei>0vEis*
*pi.
As b(s) has leading term b1s, the leading term of b(r(s)) is the product of b1
and the leading term of r(s).
17
To find this leading term we use the formula in (2.4). Let ri(s), i 2 {1, 2}*
*, be
formal power series in s with leading terms aisliand suppose that 1 l1 < l2.
Then (2.4)shows that
r1(s) +Er2(s)= r1(s) mod sl2
= a1sl1 mod sl1+1.
As r(s) is a summation (using the formal group law of E*(-)) of monomials
of strictly increasing degree, the above shows that its leading term will be the
first non-zero monomial.n Our assumptions on the cohomology theory E*(-)
imply thatnthis is vEnsp . Hence the leading term of the left-hand side of (3.2)
is vEnb1sp .
Now let us consider the right-handiside of (3.2). As b(s) has leading term
b1s, b(s)pi has leading term b1pspi. Therefore the above argument shows that
the leading term of the right-hand side of (3.2)is b1psp[vF1]. If n = 1, equati*
*ng
coefficients of sp yields the identity:
vEnb1 = b1p[vF1].
This is precisely (3.3)with n = 1 since ss1 = 1 and ss2 = p + 1.
If n 6= 1, equating coefficients yields:
0 = b1p[vF1].
This provides the start of a recursion procedure which will lead to our desi*
*red
result. Assume that n > 1 and that for some m 2 N with 1 < m < n we have
0 = b1ssj+1-1[vFj]
for all j 2 N such that 1 j < m. As ss2 = p + 1 we have shown above that
this holds for m = 2.
We multiply (3.2)through by b1ssm -1. The leadingnterm of the left-hand side
of the resulting equation is simply vEnb1ssmsp . Let us examine the right-hand
side. We apply another recursion argument. Suppose that for some 1 j < m
we have X F i X F i
b1ssm -1 b(s)p [vFi] = b1ssm -1 b(s)p [vFi].
i>0 i j
Note that when j = 1 this is a tautology. Using (2.4)we expand this out:
X F i X F i
b1ssm -1 b(s)p [vFi]= b1ssm -1 b(s)p [vFi]
i>0 i j
0 1
j F X F pi F
= b1ssm -1@b(s)p [vj ] +F b(s) [vi ]A
i>j
0 _ ! 1
X F i j
= b1ssm -1@ b(s)p [vFi] + (b(s)p [vFj])R(s)A
i>j
18
for some formal power series R(s) 2 Q(E, F )**[[s]]
_ !
X F i j
= b1ssm -1 b(s)p [vFi] + b(s)p b1ssm -1[vFj]R(s)
i>j
X F i
= b1ssm -1 b(s)p [vFi].
i j+1
The last line follows since j < m and, by assumption, b1ssj+1-1[vFj] = 0.
The last time we can apply our recursion is when j = m - 1. This leaves us
with the equation
X F i X F i
b1ssm -1 b(s)p [vFi] = b1ssm -1 b(s)p [vFi].
i>0 i m
By a now-familiar argument, the leading term of the right-hand side of this is
m F pm ss -1 F pm
b1ssm -1b1p[vm ]s = b1 m+1 [vm ]s .
We now equate coefficients in the modified version of (3.2), i.e. after mult*
*i-
plying both sides by b1ssm -1. If m < n we see that
0 = b1ssm+1-1[vFm]
whence we can continue our recursion.
This recursive argument stops when m = n for then we no longer have
zero on the left-hand side. Equating coefficients at this point yields the desi*
*red
equation __
vEnb1ssn= b1ssn+1-1[vFn]. |__|
Since ssn+1 = pn + ssn we have shown that if h = ssn then the following holds
in Q(E, F )**: n
vEnb1h= b1p +h-1[vFn].
It is entirely possible that this will hold for some smaller value of h and the
minimum such value is an interesting invariant of the cohomology theory F *(-).
In light of the fact that b1 = e2 we get slightly finer control if we consider *
*this
as an identity about e rather than b1.
Definition 3.3. Let E*(-) and F *(-) be complex orientable, graded, commu-
tative, multiplicative cohomology theories. Suppose that the coefficient ring, *
*E*,
has characteristic p andnthat the formalngroup law for E*(-) has finitenheight,
say n. Let vEn2 E-2(p -1)and vFn2 F -2(p -1)be the coefficients of sp that
appear in the formal sum giving the mod p reduction of the p-series of the form*
*al
group laws of E*(-) and F *(-) respectively.
Define the E-additive loop height of F *(-) to be the least positive integer*
* h
for which the identity n
vEneh = e2(p -1)+h[vFn]
holds in Q(E, F )**.
In the case that F *(-) = E*(-) we shall refer to this as the self additive
loop height of E*(-).
19
Examples 3.4. 1.nBy proposition 3.2 the maximum possible E-additive loop
height is 2p_-1_p-1where n is the height of the formal group law of E*(-).
The distinct lack of any relations in K(n) *(BP__*), as demonstrated inn[8*
*],
allows one to conclude that the K(n)-additive loop height for BP is 2p_-1_*
*p-1.
Thus the bound given in proposition 3.2 is the best possible.
2. On the other hand, [10, Proposition 1.1(j)] implies the self additive loop
height of K(n) *(-) is 1.
3.2 Unstable Co-operations
The analysis of the p-series in the unstable realm follows from that in the ad-
ditive context due to a very useful trick: O-multiplication by e factors through
additive co-operations. That is, if we have unstable co-operations a, c such th*
*at
qk(a) = qk(c) then e O a = e O c. Thus we can ignore equation (3.1)and apply
the additive results to the unstable situation.
We have an unstable version of definition 3.3:
Definition 3.5. Let E*(-) and F *(-) be complex orientable, graded, commu-
tative, multiplicative cohomology theories. Suppose that the coefficient ring, *
*E*,
has characteristic p andnthat the formalngroup law for E*(-) has finitenheight,
say n. Let vEn2 E-2(p -1)and vFn2 F -2(p -1)be the coefficients of sp that
appear in the formal sum giving the mod p reduction of the p-series of the form*
*al
group laws of E*(-) and F *(-) respectively.
Define the E-unstable loop height of F *(-) to be the least positive integer
h for which the identity
n-1)+h) F
vEneOh = eO(2(p O [vn ]
holds in E*(F_*).
In the case that F *(-) = E*(-) we shall refer to this as the self unstable
loop height of E*(-).
The argument above produces:
Lemma 3.6. The E-unstable loop height of F *(-) is atnleast the E-additive
loop height and at most one more. In particular, 2p_-1_p-1+ 1 is an upper_bound.
|__|
Careful examination of [10, Proposition 1.1(j)] reveals that the self unstab*
*le
loop height of K(n) *(-) is 1.
4 Splitting Co-operations
In this section we use the results of the previous one to define how to constru*
*ct a
stable operation from an unstable one. Our strategy will be to use the formula
from proposition 3.2, and its unstable version, to define idempotents in the
co-operation algebras which will split the co-operations.
20
4.1 Idempotents
Definition 4.1. Let s 2 E0(F_0) denote the unstable co-operation:
n-1) F
s := (vEn)-1eO2(p O [vn ].
Recall that one of the conditions on the cohomology theory E*(-) is that
the element vEn2 E* is invertible, hence s is well-defined.
Proposition 4.2. Let h be the E-unstable loop height of F *(-). The co-
operation s has the following properties:
1. s O s = s; that is, s is an idempotent for the O-multiplication.
2. e O s = s O e.
3. eOh O s = eOh = s O eOh.
4. There is some s0 such that eOh O s0= s.
Proof. 1.As h is the E-unstable loop height of F *(-) we have the identity:
n-1)+h) F
vEneOh = eO(2(p O [vn ]
which rearranges to:
n-1)+h) F
eOh = (vEn)-1eO(2(p O [vn ].
n-1
Now h 2p___p-1+ 1. As p is odd, it is at least 3 and so h is strictly le*
*ss
than 2(pn - 1). Hence:
n-1) E -1 O2(pn-1) O2(pn-1) F
eO2(p = (vn ) e O e O [vn ],
which leads to:
n-1) F E -1 O2(pn-1) F E -1 O2(pn-1) F
(vEn)-1eO2(p O [vn ] = (vn ) e O [vn ] O (vn ) e O [vn*
* ].
This is another way of saying that s O s = s.
2. As s has both indices zero it O-commutes with everything.
3. From n
eOh = (vEn)-1eO(2(p -1)+h)O [vFn]
we deduce that
eOh = eOh O s.
Then s O eOh = eOh as s O-commutes with everything.
4. As h < 2(pn-1) the element s0= (vEn)-1eO(2(pn-1)-h)O[vFn] is well-defined._
It clearly has the desired property. |__|
Corollary 4.3. There is a split short exact sequence of graded algebras (using
the O-multiplication):
0 ! sE*(F_*) ! E*(F_*) ! E*(F_*)=sE*(F_*) ! 0.
The first splitting map is O-multiplication by s. The second identifies the quo*
*tient
algebra with the ideal generated by (1 - s).
Let h be the E-unstable loop height of F *(-). The map h is an isomorphism
on the ideal generated by s and is null on the ideal generated by (1 - s).
21
Proof.This is essentially a rephrasing of proposition 4.2 in terms of maps rath*
*er
than elements. Define a map:
S :E*(F_*) ! E*(F_*), S(c) = s O c.
As s is an idempotent, S is a projection and an algebra map. The splitting
follows by basic algebra.
Up to sign, the map h is multiplication by eOh. Let S0be the operation of O-
multiplication by s0. The two latter properties of s show that: hS = h = S h
and S0 h = S = hS0. From these we readily see that im h = im S and
ker h = kerS. Thus h restricts to an isomorphism on the image of S and is_
null on the kernel. |__|
Corollary 4.4. All of the above quotients to the additive realm. |___|
4.2 Colimits
We label the various maps in the split short exact sequence as follows:
ss^S
0 ! sE*(F_*) 'S-!E*(F_*) --! E*(F_*)=sE*(F_*) ! 0
with splitting maps ssS and 'S^respectively. Recall that we have the suspen-
sion map : El-1(F_k-1) ! El(F_k). Let S = ssS 'S and S^= ssS^ 'S^.
Using these maps, we can consider the colimits of the families (sE*(F_k)) and
(E*(F_k)=sE*(F_k)).
Proposition 4.5. The maps 'S etc. induce maps on the colimits.
Proof.To do this we need to show that they satisfy identities such as 'S( S)l=
l'S for some l. We shall see that this works for the E-unstable loop height of
F *(-), h. Since 'SssS = S,
'S( S)h= 'S(ssS 'S)h
= 'SssS 'SssS 'S . .s.sS 'S
= S S . .S. 's
= (S )h'S.
Now S is O-multiplication by s and is (up to sign) O-multiplication by e. As
s O e = e O s these two operations commute. Furthermore, as s O eOh = eOh these
operations satisfy S h = h. Putting this together yields the desired identity:
'S( S)h = Sh h'S = h'S.
The other cases are similar; some use the fact that ssS'S = 1. |__*
*_|
Corollary 4.6. There is a split short exact sequence of algebras:
0 ! colimksE*(F_k) ! colimkE*(F_k) ! colimkE*(F_k)=sE*(F_k) ! 0._
|__|
The colimits on the left and right are easily identified.
22
Proposition 4.7. The colimit on the right is null whereas the colimit of the
left is isomorphic to any of its components; that is, the natural map:
sEi(F_j) ! colimksEi+k(F_j+k)
is an isomorphism.
Proof.This follows from the fact that h is an isomorphism on sE*(F_*) and __
null on E*(F_*)=sE*(F_*), where h is the E-unstable loop height of F *(-). |_*
*_|
The "null" part implies that there is an isomorphism of algebras:
colimsE*(F_*) ~=colimE*(F_*);
whilst the other part implies that we can map back from the first colimit to any
of its components.
Definition 4.8. For k, l 2 Z let ffi :Eel(F ) ! El+k(F_k) be the map:
Eel(F ) ~=colimkEl+k(F_k) ~=colimksEl+k(F_k) ~=sEl+k(F_k) ! El+k(F_k),
where the isomorphisms are as above. We refer to ffi as the destabilisation map.
Proposition 4.9. The destabilisation map ffi is right-inverse to the stabilisat*
*ion
map oek*:El+k(F_k) ! Eel(F ). The image of ffi is the image of the iterated
suspension map h :El+k-h(F_k-h) ! El+k(F_k) where h is the E-unstable
loop height of F *(-). In the particular case k = l = 0, ffi is a homomorphism_*
*of
algebras. |__|
The whole of the above can also be done in the additive realm and the two
correspond under the quotient map.
5 Operations and Maps
The results of the previous section readily dualise to operations due to our
assumption that operations from F *(-) to E*(-) are dual to co-operations. In
this section we interpret our results in the languages of operations and maps. *
*It
will be obvious from this formulation that the dual of the destabilisation map
respects composition of operations and maps. We now state our main theorem.
Theorem 5.1. Let E*(-) and F *(-) be two graded multiplicative cohomology
theories that are commutative and complex orientable. Suppose in addition that
the following conditions hold.
1. The coefficient ring, E*, of E*(-) has characteristic p.
2. The formal group law of E*(-) has finite height, say n.
3. The coefficient of the first term in the p-series for E*(-) is invertible.
4. The various E*-modules of operations from F *(-) to E*(-) are the E*-
duals to the corresponding E*-modules of co-operations.
23
Let h be the E-unstable loop height of F *(-).
Then there is a delooping map:
1 : Uk+lk(F, E) ! Sl(F, E)
equivalently: 1: Ek+l(F_k) ! eEl(F )
and: 1 : [F_k, E_k+l] ! {F, E}l
left-inverse to the natural restriction map; thus in the last formulation 1 1
is the identity on {F, E}l.
Let rk 2 Uk+lk(F, E) and let aek 2 Ek+l(F_k) be the corresponding class. The
components of 1 rk and 1 aek are:
( 1 rk)m = (-1)lm(vEn)-i( jrk)(vFn)i,
( 1 aek)m = (vEn)-i( jaek)(vFn)i,
where i, j 0 are chosen such that j h and m - k = 2(pn - 1)i - j. In
particular:
n-1) F
( 1 rk)k= (-1)lk(vEn)-1( 2(p rk)vn ,
n-1) F
( 1 aek)k= (vEn)-1( 2(p aek)vn ;
and for m k - h:
( 1 rk)m = (-1)lm k-m rk,
( 1 aek)m= k-m aek.
Moreover, an operation rk: F k(-) ! Ek+l(-) is a component of a stable
operation if and only if it is the h-fold loop of an operation. Similarly, a map
aek: F_k! E_k+lis an infinite loop map if and only if it is an h-fold loop map.
Proof.The delooping map is defined by dualising the destabilisation map, ffi, a*
*nd
using the correspondence between operations, maps, and functionals to translate
it across to the other realms. As the stabilisation map on co-operations is dual
to the restriction map on operations, the map 1 is left-inverse to the natural
restriction map.
To determine the components of 1 rk and 1 aek we first examine the
components of an arbitrary stable operation or map. As h is the E-unstable
loop height of F *(-), we have the identity:
n-1)+h) F
vEneOh = eO(2(p O [vn ].
Under stabilisation the element e maps to the identity co-operation so the above
stabilises to:
vEn= [vFn].
By assumption vEnis invertible. Hence if c is a stable co-operation c = (vEn)-1*
*c[vFn].
Dualising, if r is a stable operation then (vEn)-1r[vFn] = r. Let (rk) be the s*
*e-
quence of unstable operations determined by restricting r to each degree. The
restriction maps are bimodule maps and so we obtain the identity:
rk = (vEn)-1rk-2(pn-1)vFn.
24
Now rk-2(pn-1)= 2(pn-1)rk and hence:
n-1) F
rk = (vEn)-1( 2(p rk)vn .
Thus once we know one component of r, say rk, we can reconstruct the rest
using the following procedure:
1. For m < k simply take the (k - m)-fold loop of rk.
2. For m > k take the j-fold loop of rk where j is such that m - k + j =
2(pn - 1)i for some i > 0. Then the periodicity ensures that:
rm = (vEn)-irm-2(pn-1)i(vFn)i= (vEn)-irk-j(vFn)i= (vEn)-i( jrk)(vFn)i.
Thus the description of components of 1 rk and 1 aek will follow from the
final statement in the theorem: that an h-fold loop map is an infinite loop map.
This is a direct consequence of the fact that the image of the destabilisation
map is the same as the image of the hth iterate of the suspension map. Hence
the image of the delooping map is the image of the hth iterate of the_looping_
map. |__|
As K(n)*(-) has self unstable loop height of 1 we get the following corollar*
*y.
Corollary 5.2. A map ff: K(n)_k! K(n)_lis an infinite loop map if and only_
if it is a loop map. |__|
One further fact to record about the delooping map is that it respects com-
position.
Proposition 5.3. Let E*(-), F *(-), G*(-) be graded multiplicative cohomol-
ogy theories such that the delooping maps 1FE, 1GF, and 1GE are all defined.
Let aej: G_j! F_kand oek: F_k! E_lbe maps. Then:
1GE(oekaej) = 1FE(oek) 1GF(aej).
Proof.Firstly we note that both sides are well-defined. Due to our assump-
tions it is sufficient to show that this equation holds component by component.
Moreover, due to the periodicity and the fact that looping respects composition
it is sufficient to show that it holds for one component. Thus we expand:
1 1 1 1
FE (oek) GF (aej)=j FE (oek) k GF (aej) j
n-1) F F -1 2(pn-1) G
= (vEn)-1( 2(p oek)vn (vn ) ( aej)vn
n-1) G
= (vEn)-1( 2(p (oekaej))vn ;
as required. |___|
6 The Bousfield-Kuhn Functor
In this section we relate our splitting to one that is a direct consequence of *
*the
existence of the Bousfield-Kuhn functor. In [6], Kuhn showed that the K(n) -
localisation of p-local spectra factors through the functor 1 ; this extended
work of Bousfield in [3] for the case n = 1. For each n 1, Kuhn constructed a
25
functor n from p-local based spaces to p-local spectra such that n 1 is the
K(n) -localisation functor, LK(n).
As we now recall, the functorial properties of n define a map
n :Ke(n)k+l(F_k) ! eK(n)l(F )
for any p-local spectrum F . To see this, let F_kbe the zeroth space of kF and
recall that K(n)_k+lis the zeroth space of the p-local spectrum k+lK(n) . As
n is a functor from p-local based spaces to K(n)-local spectra it defines a map
on morphism sets:
[F_k, K(n)_k+l]+ ! {LK(n) kF, LK(n) k+lK(n) }0.
We can simplify the target of this map. The spectrum k+lK(n) is already
K(n) -local allowing us to drop the second LK(n). This, together with sorting o*
*ut
the suspensions, means that the target is naturally isomorphic to {LK(n)F, K(n)*
*}l
which is eK(n)l(LK(n)F ). On the other hand, the source is eK(n)k+1(F_k). Hence
we have a map:
eK(n)k+l(F_k) ! eK(n)l(LK(n)F ) ~=eK(n)l(F ).
In addition, we can remove the condition that F be p-local by noting that the
Morava K-theory is p-local and hence only "sees" the p-localisation of F . We
can also extend this to the unreduced cohomology theory using the canonical
projection of unreduced onto reduced cohomology. Thus for any spectrum F
the Bousfield-Kuhn functor defines a map
n :K(n) k+l(F_k) ! eK(n)l(F ).
Theorem 6.1. Let F *(-) be a graded multiplicative cohomology theory that is
commutative and complex orientable. Then the delooping map
1 : K(n)k+l(F_k) ! eK(n)l(F )
is defined and agrees with n.
Proof.The pair K(n) *(-) and F *(-) satisfy all the conditions for the con-
struction of 1 and so it is at least defined. The map n factors through the
projection to reduced cohomology by construction. The same is true for 1 as
can be seen from the formula in theorem 5.1: recall that our definition of the
loop of an unbased map involved first projecting it to a based map and then
taking the usual loop of the result. Therefore to show that 1 and n agree it
is sufficient to show that they agree on reduced cohomology; equivalently that
they agree on based maps. In this situation the loop of a map is as expected
with no initial projection to based maps.
The first step in showing that 1 and n are the same map is to observe
that, as both are left-inverse to 1 , if a class aek is a component of a stable
class then 1 (aek) = n(aek). By theorem 5.1 any unstable class becomes the
component of a stable class after a finite number of loopings. Therefore it is
enough to show that 1 and n both commute with loops. In both cases this is
immediate from the constructions of the maps. For completeness we review the
definition of the Bousfield-Kuhn functor from [6] and explain how the desired
property follows.
There are three steps in defining n.
26
1. Let Z be a finite CW-complex with a self-map v : dZ ! Z, d > 0.
Composition with v defines a map
v*: Map (Z, X) ! Map ( dZ, X) = d Map(Z, X)
for any based space X. One can therefore define a spectrum with mdth
space Map (Z, X) and structure maps v*. This construction is functorial
in X and so defines a functor 0Zfrom based spaces to spectra.
2. The second step is to compose the functor 0Zwith K(n) -localisation to
produce a functor Z from spaces to K(n) -local spectra.
3. The final step is to define a functor n from spaces to K(n) -local spectra
by taking the direct limit of a sequence of functors, ( Zk), for a suitable
choice of sequence of spaces (Zk).
Both localisation and taking the direct limit of a sequence of spectra com-
mute with the suspension and loop operators acting on the category of spec-
tra. Therefore to show that n, and thus n, commutes with looping it is
sufficient to show that this is true for 0Z. This follows from the fact that
Map (Z, X) = Map (Z, X). Thus the spectrum for 0Z( X) is the spectrum __
0Z(X) and similarly 0Z( ff) = 0Z(ff) for a based map ff: X ! Y . |__|
References
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