A resolution of the K(2)-local sphere at the prime 3
Paul Goerss* Hans-Werner Henn
Northwestern University Universit'e Louis Pasteur et CNRS
Evanston, IL 60208, U.S.A. 67084 Strasbourg, France
Mark Mahowald Charles Rezk
Northwestern University University of Illinois
Evanston, IL 60208, U.S.A. Urbana, IL 61801, U.S.A.
September 16, 2004
Abstract
We develop a framework for displaying the stable homotopy theory of the *
*sphere, at least after
localization at the second Morava K-theory K(2). At the prime 3, we write t*
*he spectrum LK(2)S0 as the
inverse limit of a tower of fibrations with four layers. The successive fib*
*ers are of the form EhF2where F
is a finite subgroup of the Morava stabilizer group and E2is the second Mor*
*ava or Lubin-Tate homology
theory. We give explicit calculation of the homotopy groups of these fibers*
*. The case n = 2 at p = 3
represents the edge of our current knowledge: n = 1 is classical and at n =*
* 2, the prime 3 is the largest
prime where the Morava stabilizer group has a p-torsion subgroup, so that t*
*he homotopy theory is not
entirely algebraic.
The problem of understanding the homotopy groups of spheres has been central*
* to algebraic topology
ever since the field emerged as a distinct area of mathematics. A period of cal*
*culation beginning with Serre's
computation of the cohomology of Eilenberg-MacLane spaces and the advent of the*
* Adams spectral sequence
culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on *
*periodic phenomena in the
homotopy groups of spheres and Ravenel's nilpotence conjectures. The solutions *
*to most of these conjectures
by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of *
*the "chromatic" point of
view and there followed a period in which the community absorbed these results *
*and extended the qualitative
picture of stable homotopy theory. Computations passed from center stage, to so*
*me extent, although there
has been steady work in the wings - most notably by Shimomura and his coworkers*
*, and Ravenel, and more
lately by Hopkins and his coauthors in their work on topological modular forms.*
* The amount of interest
generated by this last work suggests that we may be entering a period of renewe*
*d focus on computations.
In a nutshell, the chromatic point of view is based on the observation that *
*much of the structure of stable
homotopy theory is controlled by the algebraic geometry of formal groups. The u*
*nderlying geometric object
_________________________________*
The first author and fourth authors were partially supported by the National*
* Science Foundation (USA). The authors would
like to thank (in alphabetical order) MPI at Bonn, Northwestern University, the*
* Research in Pairs Program at Oberwolfach,
the University of Heidelberg and Universit'e Louis Pasteur at Strasbourg, for p*
*roviding them with the opportunity to work
together.
1
is the moduli stack of formal groups. Much of what can be proved and conjecture*
*d about stable homotopy
theory arises from the study of this stack, its stratifications, and the theory*
* of its quasi-coherent sheaves.
See for example, the table in section 2 of [12].
The output we need from this geometry consists of two distinct pieces of dat*
*a. First, the chromatic
convergence theorem of [21], x8.6 says the following. Fix a prime p and let E(n*
*)*, n 0 be the Johnson-
Wilson homology theories and let Ln be localization with respect to E(n)*. Then*
* there are natural maps
LnX ! Ln-1X for all spectra X, and if X is a p-local finite spectrum, then the *
*natural map
X-! holimLnX
is a weak equivalence.
Second, the maps LnX ! Ln-1X fit into a good fiber square. Let K(n)* denote*
* the nth Morava
K-theory. Then there is a natural commutative diagram
(0.1) LnX _______//LK(n)X
| |
| |
fflffl| fflffl|
Ln-1X ____//_Ln-1LK(n)X
which for any spectrum X is a homotopy pull-back square. It is somewhat difficu*
*lt to find this result in the
literature; it is implicit in [14].
Thus, if X is a p-local finite spectrum, the basic building blocks for the h*
*omotopy type of X are the
Morava K-theory localizations LK(n)X.
Both the chromatic convergence theorem and the fiber square of (0.1) can be *
*viewed as analogues of
phenomena familiar in algebraic geometry. For example, the fibre square can be *
*thought of as an analogue
of a Mayer-Vietoris situation for a formal neighborhood of a closed subscheme a*
*nd its open complement (see
[1]). The chromatic convergence theorem can be thought of as a result which det*
*ermines what happens on a
variety X with a nested sequence of closed sub-schemes Xn of codimension n by w*
*hat happens on the open
subvarieties Un = X - Xn (See [6] xIV.3, for example.) This analogy can be made*
* precise using the moduli
stack of p-typical formal group laws for X and, for Xn, the substack which clas*
*sifies formal groups of height
at least n. Again see [12]; also, see [19] for more details.
In this paper, we will write (for p = 3) the K(2)-local stable sphere as a v*
*ery small homotopy inverse
limit of spectra with computable and computed homotopy groups. Specifying a Mor*
*ava K-theory always
means fixing a prime p and a formal group law of height n; we unapologetically *
*focus on the case p = 3 and
n = 2 because this is at the edge of our current knowledge. The homotopy type a*
*nd homotopy groups for
LK(1)S0 are well understood at all primes and are intimately connected with the*
* J-homomorphism - indeed,
this calculation was one of the highlights of the computational period of the 1*
*960s. If n = 2 and p > 3, the
Adams-Novikov spectral sequence (of which more is said below) calculating ss*LK*
*(2)S0 collapses and cannot
have extensions; hence, the problem becomes algebraic, although not easy. Compa*
*re [26].
It should be noticed immediately that for n = 2 and p = 3 there has been a g*
*reat deal of calculations of
the homotopy groups of LK(2)S0 and closely related spectra, most notably by Shi*
*momura and his coauthors.
(See, for example, [23], [24] and [25].) One aim of this paper is to provide a*
* conceptual framework for
organizing those results and produce further advances.
The K(n)-local category of spectra is governed by a homology theory built fr*
*om the Lubin-Tate (or
2
Morava theory) En. This is a commutative ring spectrum with coefficient ring
(En)* = W(Fpn)[[u1, . .,.un-1]][u 1]
with the power series ring over the Witt vectors in degree 0 and the degree of *
*u equal to -2. The ring
(En)0= W(Fpn)[[u1, . .,.un-1]]
is a complete local ring with residue field Fpn. It is one of the rings constru*
*cted by Lubin and Tate in their
study of deformations for formal group laws over fields of characteristic p. Se*
*e [17].
As the notation indicates, En is closely related to the Johnson-Wilson spect*
*rum E(n) mentioned above.
The homology theory (En)* is a complex oriented theory and the formal group *
*law over (En)* is a
universal deformation of the Honda formal group law n of height n over the fie*
*ld Fpn with pn elements.
(Other choices of formal group laws of height n are possible, but all yield ess*
*entially the same results. The
choice of n is only made to be explicit; it is the usual formal group law asso*
*ciated by homotopy theorists to
Morava K-theory.) Lubin-Tate theory implies that the graded ring (En)* supports*
* an action by the group
Gn = Aut( n) o Gal(Fpn=Fp).
The group Aut( n) of automorphisms of the formal group law n is also known as *
*the Morava stabilizer
group and will be denoted by Sn. The Hopkins-Miller theorem (see [22]) says, am*
*ong other things, that we
can lift this action to an action on the spectrum En itself. There is an Adams-*
*Novikov spectral sequence
Es,t2:= Hs(Sn, (En)t)Gal(Fpn=Fp)=) sst-sLK(n)S0.
(See [8] for a basic description.) The group Gn is a profinite group and it ac*
*ts continuously on (En)*.
The cohomology here is continuous cohomology. We note that by [5] LK(n)S0 can b*
*e identified with the
homotopy fixed point spectrum EhGnnand the Adams-Novikov spectral sequence can *
*be interpreted as a
homotopy fixed point spectral sequence.
The qualitative behaviour of this spectral sequence depends very much on qua*
*litative cohomological
properties of the group Sn, in particular on its cohomological dimension. This *
*in turn depends very much
on n and p.
If p - 1 does not divide n (for example, if n < p - 1) then the p-Sylow subg*
*roup of Sn is of cohomological
dimension n2. Furthermore, if n2< 2(p-1)-1 (for example, if n = 2 and p > 3) th*
*en this spectral sequence
is sparse enough so that there can be no differentials or extensions.
However, if p - 1 divides n, then the cohomological dimension of Sn is infin*
*ite and the Adams-Novikov
spectral sequence has a more complicated behaviour. The reason for infinite coh*
*omological dimension is the
existence of elements of order p in Sn. However, in this case at least the virt*
*ual cohomological dimension
remains finite, in other words there are finite index subgroups with finite coh*
*omological dimension. In terms
of resolutions of the trivial module Zp, this means that while there are no pro*
*jective resolutions of the
trivial Sn-module Zp of finite length, one might still hope that there exist "r*
*esolutions" of Zp of finite length
in which the individual modules are direct sums of modules which are permutatio*
*n modules of the form
Zp[[G2=F]] where F is a finite subgroup of Gn. Note that in the case of a discr*
*ete group which acts properly
and cellularly on a finite dimensional contractible space X such a "resolution"*
* is provided by the cellular
complex of X.
This phenomenon is already visible for n = 1 in which case G1= S1 can be ide*
*ntified with Zxp, the units
in the p-adic integers. Thus G1~=Zpx Cp-1if p is odd while G1~=Z2x C2 if p = 2.*
* In both cases there is
a short exact sequence
0 ! Zp[[G1=F]] ! Zp[[G1=F]] ! Zp ! 0
3
of continuous G1-modules (where F is the maximal finite subgroup of G1). If p i*
*s odd this sequence is a
projective resolution of the trivial module while for p = 2 it is only a resolu*
*tion by permutation modules.
These resolutions are the algebraic analogues of the fibrations (see [8])
(0.2) LK(1)S0 ' EhG11! EhF1! EhF1.
We note that p-adic complex K-theory KZp is in fact a model for E1, the homotop*
*y fixed points EhC21can
be identified with 2-adic real K-theory KOZ2 if p = 2 and EhCp-11is the Adams s*
*ummand of KZp if p is
odd, so that the fibration of (0.2) indeed agrees with that of [8].
In this paper we produce a resolution of the trivial module Zp by (direct su*
*mmands of) permutation
modules in the case n = 2 and p = 3 and we use it to build LK(2)S0 as the top o*
*f a finite tower of fibrations
where the fibers are (suspensions of) spectra of the form EhF2where F G2 is a*
* finite subgroup.
In fact, if n = 2 and p = 3, only two subgroups appear. The first is a subgr*
*oup G24 G2; this is a
finite subgroup of order 24 containing a normal cyclic subgroup C3 with quotien*
*t G24=C3 isomorphic to the
quaternion group Q8 of order 8. The other group is the semidihedral group SD16o*
*f order 16. The two
spectra we will see, then, are EhG242and EhSD162.
The discussion of these and related subgroups of G2 occurs in section 1 (see*
* 1.1 and 1.2). The homotopy
groups of these spectra are known. We will review the calculation in section 3.
Our main result can be stated as follows (see Theorem 5.4 and Theorem 5.5).
0.1 Theorem. There is a sequence of maps between spectra
LK(2)S0 ! EhG242! 8EhSD162_ EhG242! 8EhSD162_ 40EhSD162! 40EhSD162_ 48EhG2*
*42! 48EhG242
with the property that the composite of any two successive maps is zero and all*
* possible Toda brackets are
zero modulo indeterminacy.
Because the Toda brackets vanish, this "resolution" can be refined to a towe*
*r of spectra with LK(2)S0
at the top. The precise result is given in Theorem 5.6. There are many curious *
*features of this resolution,
of which we note here only two. First, this is not an Adams resolution for E2, *
*as the spectra EhF2are not
E2-injective, at least if 3 divides the order of F. Second, there is a certain *
*superficial duality to the resolution
which should somehow be explained by the fact that Sn is a virtual Poincar'e du*
*ality group, but we do not
know how to make this thought precise.
As mentioned above, this result can be used to organize the already existing*
* and very complicated
calculations of Shimomura ([24], [25]) and it also suggests an independent appr*
*oach to these calculations.
Other applications would be to the study of Hopkins's Picard group (see [8]) of*
* K(2)-local invertible spectra.
Our method is by brute force. The hard work is really in section 4, where we*
* use the calculations of [13] in
an essential way to produce the short resolution of the trivial G2-module Z3 by*
* (summands of) permutation
modules over the form Z3[[G2=F]] where F is finite (see Theorem 4.1 and Corolla*
*ry 4.2). In section 2, we
calculate the homotopy type of the function spectra F(EhH1, EhH2) if H1is a clo*
*sed and H2a finite subgroup
of Gn; this will allow us to construct the required maps between these spectra *
*and to make the Toda bracket
calculations. Here the work of [5] is crucial. These calculations also explain *
*the role of the suspension by 48
which is really a homotopy theoretic phenomenon while the other suspensions can*
* be explained in terms of
the algebraic resolution constructed in section 4.
4
1 Lubin-Tate Theory and the Morava stabilizer group
The purpose of this section is to give a summary of what we will need about def*
*ormations of formal group
laws over perfect fields. The primary point of this section is to establish not*
*ation and to run through some
of the standard algebra needed to come to terms with the K(n)-local stable homo*
*topy category.
Fix a perfect field k of characteristic p and a formal group law over k. A*
* deformation of to a complete
local ring A (with maximal ideal m) is a pair (G, i) where G is a formal group *
*law over A, i : k ! A=m
is a morphism of fields and one requires i* = ss*G, where ss : A ! A=m is the *
*quotient map. Two such
deformations (G, i) and (H, j) are ?-isomorphic if there is an isomorphism f : *
*G ! H of formal group laws
which reduces to the identity modulo m. Write Def (A) for the set of ?-isomorph*
*ism classes of deformations
of over A.
A common abuse of notation is to write G for the deformation (G, i); i is to*
* be understood from the
context.
Now suppose the height of is finite. Then the theorem of Lubin and Tate [1*
*7] says that the functor
A 7! Def (A) is representable. Indeed let
(1.1) E( , k) = W(k)[[u1, . .,.un-1]]
where W(k) denotes the Witt vectors on k and n is the height of . This is a co*
*mplete local ring with
maximal ideal m = (p, u1, . .,.un-1) and there is a canonical isomorphism q : k*
* ~=E( , k)=m. Then Lubin
and Tate prove there is a deformation (G, q) of over E( , k) so that the natu*
*ral map
(1.2) Homc(E( , k), A) ! Def (A)
sending a continuous map f : E( , k) ! A to (f*G, ~fq) (where ~fis the map on r*
*esidue fields induced by f)
is an isomorphism. Continuous maps here are very simple: they are the local map*
*s; that is, we need only
require that f(m) be contained in the maximal ideal of A. Furthermore, if two d*
*eformations are ?-isomorphic,
then the ?-isomorphism between them is unique.
We'd like to now turn the assignment ( , k) 7! E( , k) into a functor. For t*
*his we introduce the category
FGL nof height n formal group laws over perfect fields. The objects are pairs (*
* , k) where is of height n.
A morphism
(f, j) : ( 1, k1) ! ( 2, k2)
is a homomorphism of fields j : k1! k2 and an isomorphism of formal group laws *
*f : j* 1! 2.
Let (f, j) be such a morphism and let G1 and G2 be the fixed universal defor*
*mations over E( 1, k) and
E( 2, k) respectively. If ef2 E( 2, k2)[[x]] is any lift of f 2 k2[[x]], then w*
*e can define a formal group law
H over E( 2, k2) by requiring that ef: H ! G2 is an isomorphism. Then the pair *
*(H, j) is a deformation of
1, hence we get a homomorphism E( 1, k1) ! E( 2, k2) classifying the ?-isomorp*
*hism class of H - which,
one easily checks, is independent of the lift ef. Thus if Ringscis the category*
* of complete local rings and
local homomorphisms, we get a functor
E(., .) : FGLn -! Ringsc.
In particular, note that any morphism in FGLn from a pair ( , k) to itself is a*
*n isomorphism. The automor-
phism group of ( , k) in FGLn is the "big" Morava stabilizer group of the forma*
*l group law; it contains the
5
subgroup of elements (f, idk). This formal group law and hence also its automor*
*phism group is determined
up to isomorphism by the height of if k is separably closed.
Specifically, let be the Honda formal group law over Fpn; thus the p-serie*
*s of is
n
[p](x) = xp .
From this formula it immediately follows that any automorphism f : ! over a*
*ny finite extension field
of Fpnactually has coefficients in Fpn; thus we obtain no new isomorphisms by m*
*aking such extensions. Let
Sn be the group of automorphisms of this over Fpn; this is the classical Mora*
*va stabilizer group. If we let
Gn be the group of automorphisms of ( , Fpn) in FGL n(the big Morava stabilizer*
* group of ), then one
easily sees that
Gn ~=Sn o Gal(Fpn=Fp)
Of course, Gn acts on E( , Fpn). Also, we note that the Honda formal group law *
*is defined over Fp, although
it won't get its full group of automorphisms until changing base to Fpn.
Next we put in the gradings. This requires a paragraph of introduction. For *
*any commutative ring R,
the morphism R[[x]] ! R of rings sending x to 0 makes R into an R[[x]]-module. *
*Let DerR(R[[x]], R) denote
the R-module of continuous R-derivations; that is, continuous R-module homomorp*
*hisms
@ : R[[x]] -! R
so that
@(f(x)g(x)) = @(f(x))g(0) + f(0)@(g(x)).
P
If @ is any derivation, write @(x) = u; then, if f(x) = aixi,
@(f(x)) = a1@(x) = a1u.
Thus @ is determined by u, and we write @ = @u. We then have that DerR(R[[x]], *
*R) is a free R-module
of rank one, generated by any derivation @u so that u is a unit in R. In the la*
*nguage of schemes, @u is a
generator for the tangent space at 0 of the formal scheme A1Rover Spec(R).
Now consider pairs (F, u) where F is a formal group law over R and u is a un*
*it in R. Thus F defines a
smooth one dimensional commutative formal group scheme over Spec(R) and @u is a*
* chosen generator for
the tangent space at 0. A morphism of pairs
f : (F, u) -! (G, v)
is an isomorphism of formal group laws f : F ! G so that
u = f0(0)v.
Note that if f(x) 2 R[[x]] is a homomorphism of formal group laws from F to G, *
*and @ is a derivation at 0,
then (f*@)(x) = f0(0)@(x). In the context of deformations, we may require that *
*f be a ?-isomorphism.
This suggests the following definition: let be a formal group law of heigh*
*t n over a perfect field k of
characteristic p, and let A be a complete local ring. Define Def (A)* to be equ*
*ivalence classes of pairs (G, u)
where G is a deformation of to A and u is a unit in A. The equivalence relati*
*on is given by ?-isomorphisms
transforming the unit as in the last paragraph. We now have that there is a nat*
*ural isomorphism
Homc(E( , k)[u 1], A) ~=Def (A)*.
6
We impose a grading by giving an action of the multiplicative group scheme G*
*m on the scheme Def (.)*
(on the right) and thus on E( , k)[u 1] (on the left): if v 2 Ax is a unit and *
*(G, u) represents an equivalence
class in Def (A)* define an new element in Def (A)* by (G, v-1u). In the induce*
*d grading on E( , k)[u 1],
one has E( , k) in degree 0 and u in degree -2.
This grading is essentially forced by topological considerations. See the re*
*marks before Theorem 20 of
[28] for an explanation. In particular, it is explained there why u is in degre*
*e -2 rather than 2.
The rest of the section will be devoted to what we need about the Morava sta*
*bilizer group. The group Sn
is the group of units in the endomorphism ring On of the Honda formal group law*
* of height n. The ring On
can be described as follows (See [13] or [20]). One adjoins a non-commuting ele*
*ment S to the Witt vectors
W = W(Fpn) subject to the conditions that
Sa = OE(a)S and Sn = p
where a 2 W and OE : W ! W is the Frobenius. (In terms of power series, S corre*
*sponds to the endomorphism
of the formal group law given by f(x) = xp.) This algebra On is a free W-module*
* of rank n with generators
1, S, . .S.n-1and is equipped with a valuation extending the standard valuati*
*on of W; since we assume
that (p) = 1, we have (S) = 1=n. Define a filtration on Sn by
FkSn = {x 2 Sn | (x - 1) k}.
Note that k is a fraction of the form a=n with a = 0, 1, 2, . ...We have
F0Sn=F1=nSn ~=Fxpn,
Fa=nSn=F(a+1)=nSn ~=Fpn, a 1
and
Sn ~=limaSn=Fa=nSn.
If we define Sn = F1=nSn, then Sn is the p-Sylow subgroup of the profinite grou*
*p Sn. Note that the
Teichm"uller elements Fxpn Wx Oxndefine a splitting of the projection Sn ! F*
*xpnand, hence, Sn is the
semi-direct product of Fxpnand the p-Sylow subgroup.
The action of the Galois group Gal(Fpn=Fp) on On is the obvious one: the Gal*
*ois group is generated by
the Frobenius OE and
OE(a0+ a1S + . .+.an-1Sn-1) = OE(a0) + OE(a1)S + . .+.OE(an-1)Sn-*
*1.
We are, in this paper, concerned mostly with the case n = 2 and p = 3. In th*
*is case, every element of S2
can be written as a sum
a + bS, a, b 2 W(F9) = W
with a 6 0 mod 3. The elements of S2 are of the form a + bS with a 1 mod 3.
The following subgroups of S2 will be of particular interest to us. The firs*
*t two are choices of maximal
finite subgroups. 1 The last one (see 1.3) is a closed subgroup which is, in so*
*me sense, complementary to
the_center.______________________
1The first author would like to thank Haynes Miller for several lengthy and i*
*nformative discussions about finite subgroups
of the Morava stabilizer group.
7
1.1. Choose a pimitive eighth root of unity ! 2 F9. We will write ! for the cor*
*responding element in W
and S2. The element
s = -1_2(1 + !S)
is of order 3; furthermore,
!2s!6 = s2.
Hence the elements s and !2 generate a subgroup of order 12 in S2 which we labe*
*l G12. As a group, it is
abstractly isomorphic to the unique non-trivial semi-direct product of cyclic g*
*roups
C3o C4
Any other subgroup of order 12 in S2 is conjugate to G12. In the sequel, when d*
*iscussing various represen-
tations, we will write the element !2 2 G12as t.
We note that the subgroup G12 S2 is a normal subgroup of a subgroup G24of t*
*he larger group G2.
Indeed, there is a diagram of short exact sequences of groups
1____//_G12__//_G24__//_Gal(F9=F3)_//_1
|| || =||
fflffl| fflffl| fflffl|
1_____//S2____//G2___//_Gal(F9=F3)_//_1.
Since the action of the Galois group on S2 does not preserve any choice of G12,*
* this is not transparent. In
fact, while the lower sequence is split the upper sequence is not. More concret*
*ely we let
_ = !OE 2 S2o Gal(F9=F3) = G2
where ! is our chosen 8th root of unity and OE is the generator of the Galois g*
*roup. Then if s and t are the
elements of order 3 and 4 in G12chosen above, we easily calculate that _s = s_,*
* t_ = _t3 and _2 = t2.
Thus the subgroup of G2 generated by G12and _ has order 24, as required. Note t*
*hat the 2-Sylow subgroup
of G24is the quaternion group Q8 of order 8 generated by t and _ and that indeed
1____//_G12__//_G24__//_Gal(F9=F3)_//_1
is not split.
1.2. The second subgroup is the subgroup SD16generated by ! and OE. This is the*
* semidirect product
Fx9o Z=2 .
and is also known as the semidihedral group of order 16.
1.3. For the third subgroup, note that the evident right action of Sn on On def*
*ines a group homomorphism
Sn ! GLn(W). The determinant homomorphism Sn ! Wx extends to a homomorphism
Gn ! Wx o Gal(Fpn=Fp)
For example, if n = 2, this map sends (a + bS, OEe), e 2 {0, 1}, to
(aOE(a) - pbOE(b), OEe)
8
where OE is the Frobenius. It is simple to check (for all n) that the image of *
*this homomorphism lands in
Zxpx Gal(Fpn=Fp) Wx o Gal(Fpn=Fp) .
If we identify the quotient of Zxpby its subgroup Cp-1of elements of finite ord*
*er with Zp, we get a "reduced
determinant" homomorphism
Gn ! Zp .
Let G1nbe the kernel of this map and S1nresp. S1nbe the kernel of its restricti*
*on to Sn resp. Sn. In particular,
any finite subgroup of Gn is a subgroup of G1n. One also easily checks that the*
* center of Gn is Zxp Wx Sn
and that the composite
Zxp! Gn ! Zxp
sends a to an. Thus, if p doesn't divide n, we have
Gn ~=Zpx G1n.
2 The K(n)-local category and the Lubin-Tate theories En
The purpose of this section is to collect together the information we need abou*
*t the K(n)-local category and
the role of the functor (En)*(.) in governing this category. But attention! - (*
*En)*X is not the homology of
X defined by the spectrum En, but a completion thereof: see Definition 2.1 belo*
*w.
Most of the information in this section is collected from [3], [4], and [15].
Fix a prime p and let K(n), 1 n < 1, denote the n-th Morava K-theory spect*
*rum. Then K(n)* ~=
Fp[vn] where the degree of vn is 2(pn - 1). This is a complex oriented theory a*
*nd the formal group law over
K(n)* is of height n. As is customary, we specify that the formal group law ove*
*r K(n)* is the graded variant
of the Honda formal group law; thus, the p-series is
n
[p](x) = vnxp .
Following Hovey and Strickland, we will write Kn for the category of K(n)-lo*
*cal spectra. We will write
LK(n)for the localization functor from spectra to Kn.
Next let Kn be the extensionnof K(n) with (Kn)* ~=Fpn[u 1] with the degree o*
*f u = -2. The inclusion
K(n)* (Kn)* sends vn to u-(p -1). There is a natural isomorphism of homology *
*theories
~=
(Kn)* K(n)*K(n)*X -!(Kn)*X
and K(n)* ! (Kn)* is a faithfully flat extension; thus the two theories have th*
*e same local categories and
weakly equivalent localization functors.
If we write F for the graded formal group law over K(n)* we can extend F to *
*a formal group law over
(Kn)* and define a formal group law over Fpn= (Kn)0 by
x + y = (x, y) = u-1F(ux, uy) = u-1(ux +F uy).
Then F is chosen so that is the Honda formal group law.
9
We note that - as in [4] - there is a choice of the universal deformation G *
*of such that the p-series of
the associated graded formal group law G0 over E( , Fpn)[u 1] satisfies
2
[p](x) = v0x +G0v1xp+G0v2xp +G0. . .
with v0= p and 8 k
< u1-p uk 0 < k < n;
vk = : u1-pn k = n;
0 k > n.
This shows that the functor X 7! (En)* BP*BP*X (where (En)* is considered a *
*BP*-module via the
evident ring homomorphism) is a homology theory which is represented by a spect*
*rum En with coefficients
ss*(En) ~=E( , Fpn)[u 1] ~=W[[u1, . .,.un-1]][u 1] .
The inclusion of the subring E(n)* = Z(p)[v1. .,.vn-1, vn1] into (En)* is again*
* faithfully flat; thus, these
two theories have the same local categories. We write Ln for the category of E(*
*n)-local spectra and Ln for
the localization functor from spectra to Ln.
The reader will have noticed that we have avoided using the expression (En)**
*X; we now explain what
we mean by this. The K(n)-local category Kn has internal smash products and (ar*
*bitrary) wedges given by
X ^Kn Y = LK(n)(X ^ Y )
and ` `
Xff= LK(n)( Xff) .
Kn
In making such definitions, we assume we are working in some suitable model *
*category of spectra, and
that we are taking the smash product between cofibrant spectra; that is, we are*
* working with derived smash
product. The issues here are troublesome, but well understood, and we will not *
*dwell on these points. See
[9] or [7]. If we work in our suitable categories of spectra the functor Y 7! X*
* ^Kn Y has a right adjoint
Z 7! F(X, Z).
We define a version of (En)*(.) intrinsic to Kn as follows.
2.1 Definition. Let X be a spectrum. Then we define (En)*X by the equation
(En)*X = ss*LK(n)(En ^ X).
We remark immediately that (En)*(.) is not a homology theory in the usual se*
*nse; for example, it will
not send arbitrary wedges to sums of abelian groups. However, it is tractable, *
*as we now explain. First note
that En itself is K(n)-local; indeed, Lemma 5.2 of [15] demonstrates that En is*
* a finite wedge of spectra of
the form LK(n)E(n). Therefore if X is a finite CW spectrum, then En ^ X is alre*
*ady in Kn, so
(2.1) (En)*X = ss*(En ^ X).
10
Let I = (i0, . .,.in-1) be a sequence of positive integers and let
mI = (pi0, ui11, . .,.uin-1n-1) m (En)*
where m = (p, u1, . .,.un-1) is the maximal ideal in E*. These form a system of*
* ideals in (En)* and produce
a filtered diagram of rings {(En)*=mI}; furthermore
(En)* = limI(En)*=mI.
There is a cofinal diagram {(En)*=mJ} which can be realized as a diagram of spe*
*ctra in the following sense:
using nilpotence technology, one can produce a diagram of finite spectra {MJ} a*
*nd an isomorphism
{(En)*MJ} ~={(En)*=mJ}
as diagrams. See x4 of [15]. Here (En)*MJ = ss*En ^ MJ = ss*LK (n)(En ^ MJ). Th*
*e importance of this
diagram is that (see [15], Proposition 7.10) for each spectrum X
(2.2) LK(n)X ' holimJMJ ^ LnX.
This has the following consequence, immediate from Definition 2.1: there is a s*
*hort exact sequence
0 ! lim1(En)k+1(X ^ MJ) ! (En)kX ! lim(En)k(X ^ MJ) ! 0.
This suggests (En)*X is closely related to some completion of ss*(En ^ X) and t*
*his is nearly the case. The
details are spelled out in x8 of [15], but we won't need the full generality th*
*ere. In fact, all of the spectra we
consider here will satisfy the hypotheses of Proposition 2.2 below.
If M is an (En)*-module, let M^mdenote the completion of M with respect to t*
*he maximal ideal of (En)*.
A module of the form M
( kff(En)*)^m
ff
will be called pro-free.
2.2 Proposition. If X is a spectrum so that K(n)*X is concentrated in even degr*
*ees, then
(En)*X ~=ss*(En ^ X)^m
and (En)*X is pro-free as an (En)*-module.
See Proposition 8.4 of [15].
As with anything like a flat homology theory, the object (En)*X is a comodul*
*e over some sort of Hopf
algebroid of co-operations; it is our next project to describe this structure. *
*In particular, this brings us to
the role of the Morava stabilizer group. We begin by identifying (En)*En.
Let Gn be the (big) Morava stabilizer group of , the Honda formal group law*
* of height n over Fpn. For
the purposes of this paper, a Morava module is a complete (En)*-module M equipp*
*ed with a continuous
Gn-action subject to the following compatibility condition: if g 2 Gn, a 2 (En)*
** and x 2 M, then
(2.3) g(ax) = g(a)g(x) .
11
For example, if X is any spectrum with K(n)*X concentrated in even degrees, the*
*n (En)*X is a complete
(En)*-module (by Proposition 2.2) and the action Gn on En defines a continuous *
*action of Gn on (En)*X.
This is a prototypical Morava module.
Now let M be a Morava module and let
Homc(Gn, M)
be the abelian group of continuous maps from Gn to M where the topology on M is*
* defined via the ideal
m. Then
(2.4) Hom c(Gn, M) ~=limicolimkmap(Gn=Uk, M=miM)
T *
* c
where Uk runs over any system of open subgroups of Gn with kUk = {e}. To give *
*Hom (Gn, M) a structure
of an (En)*-module let OE : Gn ! M be continuous and a 2 (En)*. The we define a*
*OE by the formula
(2.5) (aOE)(x) = aOE(x) .
There also is a continuous action of Gn on Homc(Gn, M): if g 2 Gn and OE : Gn !*
* M is continuous, then
one defines gOE : Gn ! M by the formula
(2.6) (gOE)(x) = gOE(g-1x) .
With this action, and the action of (En)* defined in (2.5), the formula of (2.3*
*) holds. Because M is complete
(2.4) shows that Homc(Gn, M) is complete.
2.3 Remark. With the Morava module structure defined by Equations 2.5 and 2.6, *
*the functor M !
Hom c(Gn, M) has the following universal property. If N and M are Morava module*
*s and f : N ! M is
morphism of continuous (En)* modules, then there is an induced morphism
N -! Homc(Gn, M)
ff7! OEff
with OEff(x) = xf(x-1ff). This yields a natural isomorphism
Hom (En)*(N, M) = Hom(En)*En(N, Homc(Gn, M))
where from continous (En)* module homomorphisms to morphisms of Morava modules.
There is a different, but isomorphic natural Morava module structure on Hom *
*c(Gn, -) so that this
functor becomes a true right adjoint of the forget functor from Morava modules *
*to continuous (En)*-modules.
However, we will not need this module structure at any point and we supress it *
*to avoid confusion.
For example, if X is a spectrum such that (En)*X is (En)*-complete, the Gn-a*
*ction on (En)*X is encoded
by the map
(En)*X ! Homc(Gn, (En)*X)
adjoint (in the sense of the previous remark) to the identity.
12
The next result says that this is essentially all the stucture that (En)*X s*
*upports. For any spectrum X,
Gn acts on
(En)*(En ^ X) = ss*LK(n)(En ^ En ^ X)
by operating in the left factor of En. The multiplication En^En ! En defines a *
*morphism of (En)*-modules
(En)*(En ^ X) ! (En)*X
and by composing we obtain a map
OE : (En)*(En ^ X) ! Homc(Gn, (En)*(En ^ X)) ! Homc(Gn, (En)*X) .
If (En)*X is complete, this is a morphism of Morava modules.
We now record:
2.4 Proposition. For any cellular spectrum X with (Kn)*X concentrated in even d*
*egrees the morphism
OE : (En)*(En ^ X) ! Homc(Gn, (En)*X)
is an isomorphism of Morava modules.
Proof.See [5] and [28] for the case X = S0. The general case follows in the usu*
*al manner. First, it's true
for finite spectra by a five lemma argument. For this one needs to know that th*
*e functor
M 7! Homc(Gn, M)
is exact on finitely generated (En)*-complete modules. This follows from (2.4).*
* Then one argues the general
case, by noting first that by taking colimits over finite cellular subspectra
OE : (En)*(En ^ MJ ^ X) ! Homc(Gn, (En)*(MJ ^ X))
is an isomorphism for any J and any X. Note that En ^ MJ ^ X is K(n)-local for *
*any X; therefore, LK(n)
commutes with the homotopy colimits in question. Finally the hypothesis on X im*
*plies
(En)*(En ^ X) ~=lim(En)*(En ^ MJ ^ X).
and thus we can conclude the result by taking limits with respect to J. *
* |___|
We next turn to the results of Devinatz and Hopkins ([5]) on homotopy fixed *
*point spectra. Let OGn be
the orbit category of Gn. Thus an object in OGn is an orbit Gn=H where H is a c*
*losed subgroup and the
morphisms are continuous Gn-maps. Then Devinatz and Hopkins have defined a func*
*tor
OopGn! K
sending Gn=H to a K(n)-local spectrum EhHn. If H is finite, then EhHnis the usu*
*al homotopy fixed point
spectrum defined by the action of H Gn. By the results of [5], the morphism O*
*E of Proposition 2.4 restricts
to an isomorphism (for any closed H)
~= c
(2.7) (En)*EhHn-! Hom (Gn=H, (En)*).
13
We would now like to write down a result about the function spectra F((En)hH*
*, En). First, some notation.
If E is a spectrum and X = limiXiis an inverse limit of a sequence of finite se*
*ts Xithen define
E[[X]] = holimiE ^ (Xi)+.
2.5 Proposition. Let H be a closed subgroup of Gn. Then there is a natural weak*
* equivalence
En[[Gn=H]]__'_//_F((En)hH, En).
Proof.First let U be an open subgroup of Gn. Functoriality of the homotopy fixe*
*d point spectra construction
of [5] gives us a map EhUn^ Gn=U+ ! En where as usual Gn=U+ denotes Gn=U with a*
* disjoint base point
added. Together with the product on En we obtain a map
(2.8) En ^ EhUn^ Gn=U+ ! En ^ En ! En
whose adjoint induces an equivalence
Y
(2.9) LK(n)(En ^ EhUn) ! En
Gn=U
realizing the isomorphism of (2.7) above. Note that this is a map of En-module *
*spectra. Let FEn(-, En) be
the function spectra in the category of left En-module spectra. (See [9] for de*
*tails.) If we apply FEn(-, En)
to the equivalence of (2.9) we obtain an equivalence of En-module spectra
Y
FEn( E, E) ! FEn(En ^ EhUn, En).
Gn=U
This equivalence can then be written as
(2.10) En ^ (Gn=U)+ ! F(EhUn, En);
furthermore, an easy calculation shows that this map is adjoint to the map of (*
*2.8).
More generally, letTH be any closed subgroup of Gn. Then there exists a decr*
*easing sequence Uiof open
subgroups Uiwith H = iUiand by [5] we have
EhHn' LK(n)hocolimiEhUin.
Thus, the equivalence of (2.10) and passing to the limit we obtain the desired *
*equivalence. |___|
Now note that if X is a profinite set with continuous H-action and if E is a*
*n H-spectrum then E[[X]] is
an H-spectrum via the diagonal action. It is this action which is used in the f*
*ollowing result.
2.6 Proposition. 1.) Let H1 be a closed subgroup and H2 a finite subgroup of Gn*
*. Then there is a natural
equivalence
En[[Gn=H1]]hH2'__//_F(EhH1n, EhH2n) .
14
2.) If H1 is also an open subgroup then there is a natural decomposition
Y
En[[Gn=H1]]hH2' EhHxn
H2\Gn=H1
where Hx = H2\ xH1x-1 is the isotropy subgroup of the coset xH1 and H2\Gn=H1 is*
* the (finite) set of
double cosets. T
3.) If H1 is a closed subgroup and H1= iUifor a decreasing sequence of open*
* subgroups Uithen
Y
F(EhH1n, EhH2n) ' holimiEn[[Gn=Ui]]hH2' holimi EhHx,in
H2\Gn=Ui
where Hx,i= H2\ xUix-1 is, as before, the isotropy subgroup of the coset xUi. 2
Proof.The first statement follows from Proposition 2.5 by passing to homotopy f*
*ixed point spectra with
respect to H2 and the second statement is then an immediate consequence of the *
*first. For the_third_
statement we write Gn=H1= limiGn=Uiand pass to the homotopy inverse limit. *
* |__|
We will be interested in the En-Hurewicz homomorphism
ss0F(EhH1n, EhH2n) ! Hom(En)*En((En)*EhH1n, (En)*EhH2n)
where Hom(En)*Endenotes morphisms in the category of Morava modules. Let
(En)*[[Gn]] = limi(En)*[Gn=Ui]
denote the completed group ring and give this the structure of a Morava module *
*by letting Gn act diagonally.
2.7 Proposition. Let H1 and H2 be closed subgroups of Gn and suppose that H2 is*
* finite. Then there is an
isomorphism ~
(En)*[[Gn=H1]] H2-=!Hom (En)*En((En)*EhH1n, (En)*EhH2n)
such that the following diagram commutes
H2
ss*En[[Gn=H1]]hH2________//(En)*[[Gn=H1]]
~=|| ~=||
fflffl| fflffl|
ss*F(EhH1n, EhH2n)//_Hom(En)*En((En)*EhH1n, (En)*EhH2n)
where the top horizontal map is the edge homomorphism in the homotopy fixed poi*
*nt spectral sequence, the
left hand vertical map is induced by the equivalence of Proposition 2.6 and the*
* bottom horizontal map is the
En-Hurewicz homomorphism.
_________________________________2
We are grateful to P. Symonds for pointing out that the naive generalization*
* of the second statement does not hold for a
general closed subgroup.
15
Proof.First we assume that H2 is the trivial subgroup and H1 is open, so that G*
*n=H1 is finite. Then there
is an isomorphism
(En)*[[Gn=H1]] ! Hom(En)*(Hom c(Gn=H1, (En)*)), (En)*)
which is the unique linear map which send a coset to evaluation at that coset. *
*Applying Remark 2.3 we
obtain an isomorphism of Morava modules
(En)*[[Gn=H1]] ! Hom(En)*(Hom c(Gn=H1, (En)*)), Homc(Gn, (En)*)).
This isomorphism can be extended to a general closed subgroup H1 by writing H1 *
*as the intersection of a
nested sequence of open subgroups (as in the proof of Proposition 2.5) and taki*
*ng limits. Then we use the
isomorphisms of (2.7) to identify (En)*EhHinwith Homc(Gn=Hi, (En)*). This defin*
*es the isomorphism we
need, and it is straightforward to see that the diagram commutes. To end the pr*
*oof, note that the_case of a
general finite subgroup H2 follows by passing to H2-invariants. *
* |__|
3 The homotopy groups of EhF2 at p = 3
To construct our tower we are going to need some information about ss*EhF2for v*
*arious finite subgroups of
the stabilizer group G2. Much of what we say here can be recovered from various*
* places in the literature (for
example, [11], [18], or [10]) and the point of view and proofs expressed are ce*
*rtainly those of Mike Hopkins.
What we add here to the discussion in [10] is that we pay careful attention to *
*the Galois group. In particular
we treat the case of the finite group G24.
Recall that we are working at the prime 3. We will write E for E2, so that w*
*e may write E* for (E2)*.
In Remark 1.1 we defined a subgroup
G24 G2= S2o Gal(F9=F3)
generated by elements s, t and _ of orders 3, 4 and 4 respectively. The cyclic *
*subgroup C3 generated by s
is normal, and the subgroup Q8 generated by t and _ is the quaternion group of *
*order 8.
The first results are algebraic in nature; they give a nice presentation of *
*E* as a G24-algebra. First we
define an action of G24on W = W(F9) by the formulas:
(3.1) s(a) = a t(a) = !2a _(a) = !OE(a)
where OE is the Frobenius. Note the action factors through G24=C3 ~=Q8. Restr*
*icted to the subgroup
G12= S2\ G24this action is W-linear, but over G24it is simply linear over Z3. L*
*et O denote the resulting
G24-representation and O0its restriction to Q8.
This representation is a module over a twisted version of the group ring W[G*
*24]. The projection
G24-! Gal(F9=F3)
defines an action3 of G24on W and we use this action to twist the multiplicatio*
*n in W[G24]. We should
really write WOE[G24] for this twisted group ring, but we forebear, so as to no*
*t clutter notation. Note that
W[Q8] has a similar twisting, but W[G12] is the ordinary group ring.
_________________________________3
This action is different from that of the representation defined by the form*
*ulas of 3.1.
16
Define a G24-module ae by the short exact sequence
(3.2) 0 ! O ! W[G24] W[Q8]O0! ae ! 0
where the first map takes a generator e of O to
(1 + s + s2)e 2 W[G24] W[Q8]O0.
3.1 Lemma. There is a morphism of G24-modules
ae -! E-2
so that the induced map
F9 W ae ! E0=(3, u21) E0E-2
is an isomorphism. Furthermore, this isomorphism sends the generator e of ae to*
* an invertible element in
E*.
Proof.We need to know a bit about the action of G2 on E*. The relevant formulas*
* have been worked out
by Devinatz and Hopkins. Let m E0 be the maximal ideal and a + bS 2 S2. Then *
*Proposition 3.3 and
Lemma 4.9 of [4] together imply that, modulo m2E-2
(3.3) (a + bS)u au + OE(b)uu1
(3.4) (a + bS)uu1 3bu + OE(a)uu1 .
In some cases we can be more specific. For example, if ff 2 Fx9 W(F9)x G2, t*
*hen the induced map of
rings
ff* : E* ! E*
is the W-algebra map defined by the formulas
(3.5) ff*(u) = ffu and ff*(uu1) = ff3uu1 .
Finally, since the Honda formal group is defined over F3the action of the Frobe*
*nius on E* = W(F9)[[u1]][u 1]
is simply extended from the action on W(F9). Thus we have
(3.6) _(a) = !*OE(a)
for all a 2 E2.
The formulas (3.3) up to (3.6) imply that E0=(3, u21) E0E-2 is isomorphic t*
*o F9 W ae as a G24-module
and, further, that we can choose as a generator the residue class of u. In [10]*
* (following [18], who learned it
from Hopkins) we found a class y 2 E-2 so that
(3.7) y !u mod (3, u1) .
and so that
(1 + s + s2)y = 0
This element might not yet have the correct invariance property with respect to*
* _; to correct this, we average
and set
x = 1_2(y + !-2t*(y) + !-4(t2)*(y) + !-6(t3)*(y) + !-1_*(y) + !-7(_t)*(y) + !-5*
*(_t2)*(y) + !-3(_t3)*(y)).
17
We can now send the generator of ae to x. Note also that the formulas (3.3) up *
*to (3.6) imply that
x 1_2(!u + !3u) modulo(3, u21) .
*
*|___|
We now make a construction. The morphism of G24-modules constructed in this *
*last lemma defines a
morphism of W-algebras
S(ae) = SW (ae) -! E*
sending the generator e of ae to an invertible element in E2. The symmetric alg*
*ebra is over W and the map
is a map of W-algebras. The group G24acts through Z3-algebra maps, and the subg*
*roup G12acts through
W-algebra maps. If a 2 W is a multiple of the unit, then _(a) = OE(a).
Let
Y
(3.8) N = ge 2 S(ae);
g2G12
then N is invariant by G12and _(N) = -N so that the get a morphism of graded G2*
*4-algebras
S(ae)[N-1] -! E*
(where the grading on the source is determined by putting ae in degree -2). Inv*
*erting N inverts e, but in
an invariant manner. This map is not yet an isomorphism, but it is an inclusion*
* onto a dense subring. The
following result is elementary (cf. Proposition 2 of [10]):
3.2 Lemma. Let I = S(ae)[N-1]\m. Then completion at the ideal I defines an isom*
*orphism of G24-algebras
S(ae)[N-1]^I~=E*.
Thus the input for the calculation of the E2-term H*(G24, E*) of the homotop*
*y fixed point spectral
sequence associated to EhG242will be discrete. Indeed, let A = S(ae)[N-1]. Then*
* the essential calculation is
that of H*(G24, A). For this we begin with the following. For any finite group *
*G and any G module M, let
trG= tr: M -! MG = H0(G, M)
P
be the transfer: tr(x) = g2Ggx. In the following result, an element listed as*
* being in bidegree (s, t) is in
Hs(G, At).
If e 2 ae is the generator, define d 2 A to be the multiplicative norm with *
*respect to the cyclic group C3
generated by s: d = s2(e)s(e)e. By construction d is invariant with respect to *
*C3.
3.3 Lemma. Let C3 G12be the normal subgroup of order three. Then there is an e*
*xact sequence
A tr-!H*(C3, A) ! F9[a, b, d 1]=(a2) ! 0
where a has bidegree (1, -2), b has bidegree (2, 0) and d has bidegree (0, -6).*
* Furthermore the action of t
and _ is described by the formulas
t(a) = -!2a t(b) = -b t(d) = !6d
and
_(a) = !a _(b) = b _(d) = !3d .
18
Proof.This is the same argument as in Lemma 3 of [10], although here we keep tr*
*ack of the Frobenius.
Let F be the G24-module W[G24] W[Q8]O0; thus Equation 3.2 gives a short exac*
*t sequence of G24-modules
(3.9) 0 ! S(F) O ! S(F) ! S(ae) ! 0 .
In the first term, we set the degree of O to be -2 in order to make this an exa*
*ct sequence of graded modules.
We use the resulting long exact sequence for computations. We may choose W-gene*
*rators of F labelled x1,
x2, and x3so that if s is the chosen element of order 3 in G24, then s(x1) = x2*
*and s(x2) = x3. Furthermore,
we can choose x1 so that it maps to the generator e of ae and is invariant unde*
*r the action of the Frobenius.
Then we have
S(F) = W[x1, x2, x3]
with the xiin degree -2. Under the action of C3 the orbit of a monomial in W[x1*
*, x2, x3] has three elements
unless that monomial is a power of oe3= x1x2x3 - which, of course, maps to d. T*
*hus, we have a short exact
sequence
S(F) tr-!H*(C3, S(F)) ! F9[b, d] ! 0
where b has bidegree (2, 0) and d has bidegree (0, -6). Here b 2 H2(C3, Z3) H*
*2(C3, W) is a generator and
W S(F) is the submodule generated by the algebra unit. Note that the action o*
*f t is described by
t(d) = !6d and t(b) = -b .
The last is because the element t acts non-trivially on the subgroup C3 G24an*
*d hence on H2(C3, W).
Similary, since the action of the Frobenius on d is trivial and _ acts triviall*
*y on C3, we have
_(d) = !3d and _(b) = b .
The short exact sequence (3.9) and the fact that H1(C3, S(F)) = 0 now imply tha*
*t there is an exact sequence
S(ae) tr-!H*(C3, S(ae)) ! F9[a, b, d]=(a2) ! 0 .
The element a maps to
b 2 H2(C3, S0(F) O) = H2(C3, O)
under the boundary map (which is an isomorphism)
H1(C3, ae) = H1(C3, S1(ae)) ! H2(C3, O);
thus a has bidegree (1, -2) and the actions of t and _ are twisted by O:
t(a) = -!2a = !6a and _(a) = !a .
*
*|___|
We next write down the invariants EhF*for the various finite subgroups F of *
*G24. To do this, we work
up from the symmetric algebra S(ae), and we use the presentation of the symmetr*
*ic algebra as given in the
exact sequence (3.8). Recall that we have written S(F) = W[x1, x2, x3] where th*
*e normal subgroup of order
three in G24cyclically permutes the xi. This action by the cyclic group extends*
* in an obvious way to an
action of the symmetric group 3 on three letters; thus we have an inclusion of*
* algebras
W[oe1, oe2, oe3] = W[x1, x2, x3] 3 S(F)C3.
19
There is at least one other obvious element invariant under the action of C3: s*
*et
(3.10) ffl = x21x2+ x22x3+ x23x1- x22x1- x21x3- x23x2 .
This might be called the "anti-symmetrization" (with respect to 3) of x21x2.
3.4 Lemma. There is an isomorphism
W[oe1, oe2, oe3, ffl]=(ffl2- f) ~=S(F)C3
where f is determined by the relation
ffl2= -27oe23- 4oe32- 4oe3oe31+ 18oe1oe2oe3+ oe21oe22.
Furthermore, the actions of t and _ are given by
t(oe1) = !2oe1 t(oe2) = -oe2 t(oe3) = !6oe3 t(ffl) = *
*!2ffl
and
_(oe1) = !oe1 _(oe2) = !2oe2 _(oe3) = !3oe3 _(ffl) = !3*
*ffl .
Proof.Except for the action of _, this is Lemma 4 of [10]. The action of _ is s*
*traightforward |___|
This immediately leads to the following result.
3.5 Proposition. There is an isomorphism
W[oe2, oe3, ffl]=(ffl2- g) ~=S(ae)C3
where g is determined by the relation
ffl2= -27oe23- 4oe32
with the actions of t and _ as given above in Lemma 3.4. Under this isomorphism*
* oe3 maps to d.
Proof.This follows immediately from Lemma 3.4, the short exact sequence (3.9), *
*and the fact (see the proof
of Lemma 3.3) that H1(C3, S(F)) = 0. Together these imply that
S(ae)C3~=S(F)C3=(oe1) .
*
*|___|
The next step is to invert the element N of (3.8). This element is the image*
* of oe43; thus, we are effectively
inverting the element d = oe32 S(ae)C3. We begin with the observation that if w*
*e divide
ffl2= -27oe23- 4oe32
by oe63we obtain the relation
(_ffl_oe3)2+ 4(oe2_2)3= -_27_4.
3 oe3 oe3
Thus if we set
2oe2 !3ffl !6 !2
(3.11) c4= -!___oe2, c6= ___3, = -___4= ___4
3 2oe3 4oe3 4oe3
20
then we get the expected relation 4
c26- c34= 27 .
Furthermore, c4, c6, and are all invariant under the action of the entire gro*
*up G24. (Indeed, the powers
of ! are introduced so that this happens.)
To describe the group cohomology, we define elements
ff = !a_d2 H1(C3, (S(ae)[N-1])4)
and 3
fi = !_b_d22 H2(C3, (S(ae)[N-1])12)
These elements are fixed by t and _ and, for degree reasons, acted on trivially*
* by c4 and c6. The following
is now easy.
3.6 Proposition. 1.) The inclusion
Z3[c4, c6, 1]=(c26- c34= 27 ) ! S(ae)[N-1]G24
is an isomorphism of algebras over the twisted group ring W[G24].
2.) There is an exact sequence
S(ae)[N-1] tr-!H*(G24, S(ae)[N-1]) ! F3[ff, fi, 1]=(ff2) ! 0
and c4 and c6 act trivially on ff and fi.
Then a completion argument, as in Theorem 6 of [10] or [18] implies the next*
* result.
3.7 Theorem. 1.) There is an isomorphism of algebras
(E*)G24~=Z3[[c34 -1]][c4, c6, 1]=(c26- c34= 27 ) .
2.) There is an exact sequence
E*-tr!H*(G24, E*) ! F3[ff, fi, 1]=(ff2) ! 0
and c4 and c6 act trivially on ff and fi.
3.8 Remark. The same kind of reasoning can be used to obtain the group cohomolo*
*gies H*(F, E*) for
other finite subgroups of G2. First define an element
ffi = oe-132 S(ae)[N-1] .
Then = (!2=4)ffi4; thus - has a square root:
3
(- )1=2= !_2ffi2 .
_________________________________4
This is the relation appearing in theory of modular forms [2], except here 2*
* is invertible so we can replace 1728 by 27. There
is some discussion of the connection in [11]. The relation could be arrived at *
*more naturally by choosing, as our basic formal
group law, one arising from the theory of elliptic curves, rather than the Hond*
*a formal group law.
21
The elements t and _ of G24act on ffi by the formulas
t(ffi) = !2ffi and _(ffi) = !5ffi .
The element (- )1=2is invariant under the action of t2 and _ (whereas the evide*
*nt square root of is not
fixed by _).
Let C12be the cyclic subgroup of order 12 in G24generated by s and _. This s*
*ubgroup has a cyclic
subgroup C6 of order 6 generated by s and t2= _2. We have
(E*)C3~=W[[c34 -1]][c4, c6, ffi 1]=(c26- c34= 27 )
(E*)C12~=Z3[[c34 -1]][c4, c6, (- ) 1=2]=(c26- c34= 27 )
(E*)C6~=W Z3(E*)C12
(E*)G12~=W[[c34 -1]][c4, c6, 1]=(c26- c34= 27 ) ~=W Z3(E*)G24.
Furthermore, for all these groups, the analogue of Theorem 3.7.2 holds. For ex*
*ample, there are exact
sequences
E*-tr!H*(C3, E*) ! F9[ff, fi, ffi 1]=(ff2) ! 0
E*-tr!H*(C12, E*) ! F3[ff, fi, (- ) 1=2]=(ff2) ! 0
E*-tr!H*(C6, E*) ! F9[ff, fi, (- ) 1=2]=(ff2) ! 0
E*-tr!H*(G12, E*) ! F9[ff, fi, (- ) 1=2]=(ff2) ! 0
and c4 and c6 act trivially on ff and fi.
These results allow one to competely write down the various homotopy fixed p*
*oint spectral sequences for
computing ss*EhF for the various finite groups in question. The differentials i*
*n the spectral sequence follow
from Toda's classical results and the following easy observation: every element*
* in the image of the transfer
is a permanent cycle. We record:
3.9 Lemma. In the spectral sequence
H*(G24, E*) =) ss*EhG24
the only non-trivial differentials are d5 and d9. They are determined by
d5( ) = a1fffi2 and d9(ff 2) = a2fi5
where a1 and a2 are units in F3.
Proof.These are a consequence of Toda's famous differential (see [29]) and nilp*
*otence. See Proposition 7 of
[10] or, again, [18]. There it is done for G12rather than G24, but because G12i*
*s of index 2 in_G24and_we
are working at the prime 3, this is sufficient. *
* |__|
22
The lemma immediately calculates the differentials in the other spectral seq*
*uences; for example, if one
wants homotopy fixed points with respect to the C3-action, we have, up to units,
d5(ffi) = ffi-3fffi2 and d9(ffffi2) = ffi-6fi5.
It is also worth pointing out that the d5-differential in Lemma 3.9 and some*
* standard Toda bracket ma-
nipulation (see the proof of Theorem 8 in [10]) implies the relation ( ff)ff = *
* fi3 which holds in ss27(EhG24).
The above discussion is summarized in the following main homotopy theoretic *
*result of this section.
3.10 Theorem. In the spectral sequence
H*(C3, E*) =) ss*EhC3
we have an inclusion of subrings
E0,*1~=W[[c34 -1]][c4, c6, c4ffi 1, c6ffi 1, 3ffi 1, ffi 3]=(c34- *
*c26= 27 ) E0,*2.
In positive filtration E1 is additively generated by the elements ff, ffiff, ff*
*fi, ffifffi, fij, 1 j 4 and all
multiples of these elements by ffi 3. These elements are of order 3 and are ann*
*ihilated by c4, c6, c4ffi 1, c6ffi 1
and 3ffi 1. Furthermore the following multiplicative relation holds in ss30(EhC*
*3): ffi3(ffiff)ff = a1!6fi3.
For the case of the cyclic group C6 of order 6 generated by s and t2, one no*
*te that t2(ffi) = -ffi and
the spectral sequence can now be read off Theorem 3.10. This also determines th*
*e case of C12, the group
generated by s and _. We leave the details to the reader but state the result i*
*n the case of G24.
3.11 Theorem. In the spectral sequence
H*(G24, E*) =) ss*EhG24
we have an inclusion of subrings
E0,*1~=Z3[[c34 -1]][c4, c6, c4 1, c6 1, 3 1, 3]=(c34- c26= 27 )*
* E0,*2.
In positive filtration E1 is additively generated by the elements ff, ff, fffi*
*, fffi, fij, 1 j 4 and all
multiples of these elements by 3. These elements are of order 3 and are annih*
*ilated by c4, c6, c4 1,
c6 1 and 3 1. Furthermore ( ff)ff = fi3 in ss30(EhG24).
3.12 Remark. 1.) Note that EhC3 is periodic of period 18 and that ffi3 detects *
*a periodicity class. The
spectra EhC6and EhC12are periodic with period 36 and (- )3=2detects the periodi*
*city generator. Finally
EhG12and EhG24are periodic with period 72 and 3 detects the periodicity genera*
*tor.
2.) By contrast, the Morava module E*EhG24is of period 24. To see this, note*
* that the isomorphism of
(2.7) supplies an isomorphism of Morava modules
E*EhG24~=Homc(G2=G24, E*)
with G2 acting diagonally on the right hand side. Then G24-invariance of impl*
*ies that there is a well
defined automorphism of Morava modules given by ' 7! (g 7! '(g)g*( )).
23
3.) If F has order prime to 3, then ss*(EhF) = (E*)F is easy to calculate. F*
*or example, using (3.5) and
(3.6) we obtain
ss*(EhSD16) = Z3[v1][v21]b E*
with v1= u1u-2 and v2= u-8 and completion is with respect to the ideal generate*
*d by (v41v-12). Similarly
ss*(EhQ8) = Z3[v1][!2u 4]b E*
where completion is with respect to the ideal (v21!2u4). Note that EhSD16is per*
*iodic of order 16 and EhQ8
is periodic of order 8.
We finish this section by listing exactly the computational results we will *
*use in building the tower in
section 5.
3.13 Corollary. 1.) Let F G2 be any finite subgroup. Then the edge homomorphi*
*sm
ss0EhF -! (E0)F
is an isomorphism of algebras.
2.) Let F G2 be any finite subgroup. Then
ss24EhF -! (E24)F
is an injection.
Proof.If F has order prime to 3, both of these statements are clear. If 3 divid*
*es the order of F, then the
3-Sylow subgoup of F is conjugate to C3; hence ss*EhF is a retract of ss*EhC3 a*
*nd the_result follows from
Theorem 3.10. *
* |__|
3.14 Corollary. Let F G2 be a finite subgroup. If the order of F is prime to *
*3, then
ss1EhF = 0 .
For all finite subgroups F,
ss25EhF = 0 .
Proof.Again apply Theorem 3.10. *
* |___|
3.15 Corollary. Let F G2 be any finite subgroup containing the central elemen*
*t !4 = -1. Then
ss26EhF = 0 .
Proof.Equation 3.5 implies that (E*)F will be concentrated in degrees congruent*
* to 0 mod 4. Combining_
this observation with Theorem 3.10 proves the result. *
* |__|
24
4 The algebraic resolution
Let G be a profinite group. Then its p-adic group ring Zp[[G]] is defined as li*
*mU,nZp=(pn)[G=U] where U
runs through all open subgroups of G. Then Zp[[G]] is a complete ring and we wi*
*ll only consider continuous
modules over such rings.
In this section we will construct our resolution of the trivial Z3[[G2]]-mod*
*ule Z3. In 1.3 we wrote down
a splitting of the group G2 as G12x Z3 and this splitting allows us to focus on*
* constructing a resolution of
Z3 as a Z3[[G12]]-module.
Recall that we have selected a maximal finite subgroup G24 G12; it is gener*
*ated by an element s of
order three, t = !2, and _ = !OE where ! is a primitive eighth root of unity an*
*d OE is the Frobenius. As
before C3 denotes the normal subgroup of order 3 in G24generated by s, Q8 denot*
*es the subgroup of G24of
order 8 generated by t and _ and SD16the subgroup of G12generated by ! and _.
The group Q8 is a subgroup of SD16of index 2. Let O be the sign representati*
*on (over Z3) of SD16=Q8;
we regard O as representation of SD16using the quotient map. (Note that this is*
* not the same O as in
section 3!) In this section, the induced Z3[[G12]]-module
1 def
(4.1) O "G2SD16=Z3[[G12]] Z3[SD16]O
will play an important role. If o is the trivial representation of SD16, there *
*is an isomorphism of SD16=Q8-
modules
(4.2) Z3[SD16=Q8] ~=O o .
1
Thus, we have that O "G2SD16is a direct summand of the induced module Z3[[G12=Q*
*8]] = Z3[[G12]] Z3[Q8]Z3.
The following is the main algebraic result of the paper. It will require the*
* entire section to prove.
4.1 Theorem. There is an exact sequence of Z3[[G12]]-modules
1 G1
0 ! Z3[[G12=G24]] ! O "G2SD16! O "S2D16! Z3[[G12=G24]] ! Z3! 0 .
Some salient features of this "resolution" (we will use this word even thoug*
*h not all of the modules are
projective) are that each module is a summand of a permutation module Z3[[G12=F*
*]] for a finite subgroup F
and that each module is free over K, where K G12is a subgroup so that we can *
*decompose the 3-Sylow
subgroup S12of G2 as K o C3. Important features of K include that it is a torsi*
*on-free 3-adic Poincar'e
duality group of dimension 3. (See before Lemma 4.10 for more on K.)
Since G2~=G12x Z3, we may tensor the resolution of Theorem 4.1 with the stan*
*dard resolution for Z3 to
get the following as an immediate corollary,
4.2 Corollary. There is an exact sequence of Z3[[G2]]-modules
0 ! Z3[[G2=G24]] ! Z3[[G2=G24]] !OO"G2SD16"G2SD16 O "G2SD16
! O "G2SD16 Z3[[G2=G24]] ! Z3[[G2=G24]] ! *
*Z3! 0 .
As input for our calculation we will use H*(S12) := H*(S12, F3) = Ext*Z3[[S1*
*2]](Z3, F3), as calculated by the
second author in [13]. This is an effective starting point because of the follo*
*wing lemma and the fact that
1]] *
(4.3) ExtqZ3[[S12]](M, F3) ~=TorZ3[[S2q(F3, M)
25
for any profinite continuous Z3[[S12]]-module M. Here (-)* means Fp-linear dual.
A profinite group G is called finitely generated if there is a finite set of*
* elements X G so that the
subgroup generated by X is dense. This is true of all the groups in this paper.*
* If G is a p-profinite group
and I Zp[[G]] is the kernel of the augmentation Zp[[G]] ! Fp, then
Zp[[G]] ~=limnZp[[G]]=In.
A Zp[[G]]-module M will be called complete if it is I-adically complete, i.e. i*
*f M ~=limnM=InM.
4.3 Lemma. Let G be a finitely generated p-profinite group and f : M ! N a morp*
*hism of complete
Zp[[G]]-modules. If
Fp f : Fp Zp[[G]]M ! Fp Zp[[G]]N
is surjective, then f is surjective. If
Tor(Fp, f) : TorZp[[G]]q(Fp, M) ! TorZp[[G]]q(Fp, N)
is an isomorphism for q = 0 and onto for q = 1, then f is an isomorphism.
Proof.This is an avatar of Nakayama's Lemma. To see this, suppose K is some com*
*plete Zp[[G]]-module so
that Fp Zp[[G]]K = 0. Then an inductive argument shows
Zp[[G]]=In Zp[[G]]K = 0
for all n; hence K = 0. This is the form of Nakayama's lemma we need.
The result is then proved using the long exact sequence of Torgroups: the we*
*aker hypothesis implies __
that the cokernel of f is trivial; the stronger hypothesis then implies that th*
*e kernel of f is trivial. |__|
We next turn to the details about H*(S12; F3) from [13]. (See Theorem 4.3 of*
* that paper.) We will omit
the coefficients F3 in order to simplify our notation. The key point here is th*
*at the cohomology of the group
S12is detected on the centralizers of the cyclic subgroups of order 3. There ar*
*e two conjugacy classes of such
subgroups of order 3 in S12; namely, C3 and !C3!-1. The element s = s1 is our c*
*hosen generator for C3;
thus we choose as our generator for !C3!-1 the element s2 = !s!-1. The Frobeniu*
*s OE also conjugates C3
to !C3!-1 and a short calculation shows that
(4.4) OEs1OE-1 = OEsOE-1 = s22.
The centralizer C(C3) in S12is isomorphic to C3x Z3, !OE commutes with C(C3) an*
*d conjugation by !2
sends x 2 C(C3) to its inverse x-1 (see [10]). In particular, for every x 2 C(C*
*3) we have
(4.5) !x!-1 = OEx-1OE-1 2 C(!C3!-1) .
Note that C(!C3!-1) = !C(C3)!-1. Write E(X) for the exterior algebra on a set X*
*. Then
H*(C(C3)) ~=F3[y1] E(x1, a1)
and
H*(C(!C3!-1)) ~=F3[y2] E(x2, a2) .
We know that C4 (which is generated by !2) acts on
H*(C(C3)) ~=F3[y1] E(x1, a1)
26
sending all three generators to their negative. This action extends to an actio*
*n of SD16on the product
H*(C(C3)) x H*(C(!C3!-1))
as follows. By (4.4) and (4.5) the action of the generators ! and OE of SD16is *
*given by
(4.6) !*(x1) = x2, !*(y1) = y2, !*(a1) = a2, OE*(x1) = -x2, OE*(y1) = -y2,*
* OE*(a1) = -a2.
(4.7)!*(x2) = -x1, !*(y2) = -y1, !*(a2) = -a1, OE*(x2) = -x1, OE*(y2) = -y1*
*, OE*(a2) = -a1 .
4.4 Theorem. [13]1.) The inclusions !iC(C3)!-i ! S12, i = 0, 1 induce an SD16-e*
*quivariant homomor-
phism
2Y
H*(S12) ! F3[yi] E(xi, ai)
i=1
which is an injection onto the subalgebra generated by x1, x2, y1, y2, x1a1- x2*
*a2, y1a1 and y2a2.
2.) In particular, H*(S12) is free as a module over F3[y1+ y2] on generators*
* 1, x1, x2, y1, x1a1- x2a2,
y1a1, y2a2, and y1x1a1.
We will produce the resolution of Theorem 4.1 from this data and by splicing*
* together the short exact
sequences of Lemma 4.5, 4.6, and 4.7 below. Most of the work will be spent in i*
*dentifying the last module;
this is done in Theorem 4.9.
In the following computations, we will write
Ext(M) = Ext*Z3[[S12]](M, F3) .
This graded vector space is a module over
H*(S12) = Ext*Z3[[S12]](Z3, F3) = Ext(Z3) ,
and, hence is also a module over the sub-polynomial algebra of H*(S12) generate*
*d by y1+y2. If M is actually
a continuous Z3[[G12]]-module, then Ext(M) has an action by SD16~=G12=S12which *
*extends the action by
H*(S12) in the obvious way: if ff 2 SD16, a 2 H*(S12) and x 2 Ext(M), then
ff(ax) = ff(a)ff(x) .
We can write this another way. Let's define Ext(Z3) F3[SD16]to be the algebra*
* constructed by taking
Ext(Z3) F3[SD16]
with twisted product
(a ff)(b fi) = aff(b) fffi .
The above remarks imply that if M is a Z3[[G12]]-module, then Ext(M) is an Ext(*
*Z3) F3[SD16]-module.
This structure behaves well with respect to long exact sequences. If
0 ! M1! M2! M3! 0
is a short exact sequence of Z3[[G12]]-modules we get a long exact sequence in *
*Extwhich is a long exact
sequence of Ext(Z3) F3[SD16]-modules. As a matter of notation, if x 2 Ext(Z3)*
* we will write _x2 Ext(M)
if x is the image of _xunder some unambiguous and injective sequence of boundar*
*y homomorphisms of long
exact sequences.
27
4.5 Lemma. There is a short exact sequence of Z3[[G12]]-modules
0 ! N1! Z3[[G12=G24]] -ffl!Z3! 0
where the map ffl is the augmentation. If we write z for x1a1-x2a22 H2(S21) the*
*n Ext(N1) is a module over
F3[y1+ y2] on generators e, _z, ____y1a1, ____y2a2, and ______y1x1a1of degrees *
*0, 1, 2, 2, and 3 respectively. The last four
generators are free and (y1+ y2)e = 0. The action of SD16is determined by the a*
*ction on Ext(Z3) and the
facts that
!*(e) = -e = OE*(e) .
Proof.As a Z3[[S12]]-module, there is an isomorphism
Z3[[G12=G24]] ~=Z3[[S12=C3]] Z3[[S12=!C3!-1]].
Hence, by the Shapiro Lemma there is an isomorphism
Ext(Z3[[G12=G24]]) ~=H*(C3, F3) x H*(!C3!-1, F3)
and the map Ext(Z3) ! Ext(Z3[[G12=G24]]) corresponds via this isomorphism to th*
*e restriction_map. The
result now follows from Theorem 4.4. *
* |__|
Recall that O is the rank one (over Z3) representation of SD16obtained by pu*
*lling back the sign repre-
sentation along the quotient map " : SD16! SD16=Q8~=Z=2.
4.6 Lemma. There is a short exact sequence of Z3[[G12]]-modules
1
0 ! N2! O "G2SD16! N1! 0 .
The cohomology module Ext(N2) is a freely generated module over F3[y1+ y2] on g*
*enerators _z, ____y1a1, ____y2a2,
and ______y1x1a1of degrees 0, 1, 1, and 2 respectively. The action of ! is dete*
*rmined by the action on Ext(Z3).
Proof.By the previous result the SD16-module F3 Z3[[S12]]N1 is one dimensional *
*over F3 generated by the
dual (with respect to (4.3)) of the class of e and the action is given by the s*
*ign representation along ". Lift
e to an element d 2 N1. Then SD16may not act correctly on d, but we can average*
* d to obtain an element
c on which SD16acts correctly and which reduces to the same element in F3 Z3[[S*
*12]]N1; indeed,
X
c = 1_16 "(ff)-1ff*(d) .
ff2SD16
This defines the morphism 1
O "G2SD16! N1 .
Lemma 4.3 now implies that this map is surjective and we obtain the exact seque*
*nce we need. For the
calculation of Ext(N2) note that we have an isomorphism of S12-modules
1
O "G2SD16~=Z3[[S12]] .
The result now follows from the previous lemma and the long exact sequence. *
* |___|
28
4.7 Lemma. There is a short exact sequence of Z3[[G12]]-modules
1
0 ! N3! O "G2SD16! N2! 0
where Ext(N3) is a free module over F3[y1+ y2] on generators ____y1a1, ____y2a2*
*, ______y1x1a1, and ______y2x2a2of degree 0,
0, 1 and 1 respectively. In fact, the iterated boundary homomorphisms
Ext*(N3) ! Ext*+3(Z3) = H*+3(S12, F3)
define an injection onto an Ext(Z3) F3[SD16]-submodule isomorphic to Ext(Z3[[*
*G12=G24]]).
Proof.The SD16-module F3 Z3[[S12]]N2 is F3 Z3O generated by the class dual to ~*
*z. As in the proof the
last lemma, we can now form a surjective map
1
O "G2SD16! N2 ,
and this map defines N3. The calculation of Ext(N3) follows from the long exact*
* sequence. |___|
To make use of this last result we prove a lemma.
4.8 Lemma. Let A = Ext(Z3) F3[SD16]and M = Ext(Z3[[G12=G24]]), regarded as an*
* A-module. Then
M is a simple A-module; in fact,
EndA(M) ~=F3 .
Proof.Let e1 and e2 in
Ext(Z3[[G12=G24]]) ~=H*(C3, F3) x H*(!C3!-1, F3)
be the evident two generators in degree 0. If f : M ! M is any A-module endomor*
*phism, we may write
f(e1) = ae1+ be2
where a, b 2 F3. Then (using the notation of Theorem 4.4) we have
0 = f(y2e1) = by2e2.
Since y2e26= 0, we have b = 0. Also, since !*(e1) = e2, we have f(e2) = ae2. Fi*
*nally, since every_homogeneous
element of M is of the from x1yi1e1+ x2yi2e2, we have f = aidM. *
* |__|
This means that in order to prove the following result, we need only produce*
* a map f : N3! Z3[[G12=G24]]
of G12-modules which induces a non-zero map on Extgroups.
4.9 Theorem. There is an isomorphism of Z3[[G12]]-modules
~= 1
N3-! Z3[[G2=G24]] .
This requires a certain amount of preliminaries, and some further lemmas. We*
* are looking for a diagram
(see diagram (4.12) below) which will build and detect the desired map.
The first ingredient of our calculation is a spectral sequence. Let us write
0 ! C3! C2! C1! C0! Z3! 0
29
for the resolution obtained by splicing together the short exact sequences of L*
*emma 4.5, 4.6, and 4.7:
1 G1
0 ! N3! O "G2SD16! O "S2D16! Z3[[G12=G24]] ! Z3! 0 .
By extending the resolution Co ! Z3to a bicomplex of projective Z3[[G12]]-modul*
*es, we get, for any Z3[[G12]]-
module M and any closed subgroup H G12, a first quadrant cohomology spectral *
*sequence
(4.8) Ep,q1= ExtpZ3[[H]](Cq, M) =) Hp+q(H, M) .
In particular, because E0,q1= 0 for q > 3, there is an edge homomorphism
(4.9) Hom Z3[[H]](N3, M) = HomZ3[[H]](C3, M) ! H3(H, M) .
Dually, there are homology spectral sequences
(4.10) E1p,q= TorZ3[[H]]p(M, Cq) =) Hp+q(H, M)
with an edge homomorphism
(4.11) H3(H, M) ! M Z3[[H]]N3 .
That said, we remark that the important ingredient here is that G12contains *
*a subgroup K which is
a Poincar'e duality group of dimension three and which has good cohomological p*
*roperties. The reader is
referred to [27] for a modern discussion of a duality theory in the cohomology *
*of profinite groups.
To define K, we use the filtration on the 3-Sylow subgroup S12= F1=2S12of G1*
*2described in the first
section. There is a projection
S12! F1=2S12=F1S12~=F9 .
We follow this by the map F9 ! F9=F3 ~=C3 to define a group homorphism S12! C3;*
* then, we define
K S12to be the kernel. The chosen subgroup C3 S12of order 3 provides a spli*
*tting of S12! C3; hence
S12can be written as a semi-direct product K o C3. Note that every element of o*
*rder three in S12maps to a
non-zero element in C3 so that K is torsion free.
From [13], we know a good deal about K, some of which is recorded in the fol*
*lowing lemma. Let
j : K ! S12denote the inclusion.
4.10 Lemma. The group K is a 3-adic Poincar'e duality group of dimension 3, and*
* if [K] 2 H3(K, Z3) is
a choice of fundamental class, then
j*[K] 2 H3(S12, Z3)
is a non-zero SD16-invariant generator of infinite order. Furthermore, under th*
*e edge homomorphism (4.11),
the element j*[K] maps to a non-zero SD16-invariant element in F3 Z3[[S12]]N3.
Proof.The fact that K is a Poincar'e duality group is discussed in [13]; this d*
*iscussion is an implementation
of the theory of Lazard [16]. We must now address the statements about j*[K]. F*
*or this, we first compute
with cohomology, and we use the results and notation of Theorem 4.4.
It is known (see Proposition 4.3 and 4.4 of [13]), that the Lyndon-Serre-Hoc*
*hschild spectral sequence
Hp(C3, Hq(K, F3)) =) Hp+q(S12, F3)
30
collapses and that H0(C3, Hq(K, F3)) is one dimensional for 0 q 3; in parti*
*cular, j* : H3(S12, F3) !
H3(K, F3) is onto. Since the composites
H1(C3, F3) ! H1(S12, F3) ! H1(wiC3w-i, F3)
of the inflation with the restriction maps are isomorphisms for i = 1, 2, the i*
*mage of the generator of
H1(C3, F3) is some linear combination ax1+ bx2 with both a 6= 0 and b 6= 0. Thi*
*s implies that j*(xiyi) = 0
for i = 1, 2; for example
aj*(x1y1) = j*(y1(ax1+ bx2)) = 0.
But since j* : H3(S12, F3) ! H3(K, F3) is onto and H3(S12, F3) is generated by *
*xiyiand aiyifor i = 1, 2 it
is impossible that j*(aiyi) is trivial for both i = 1, 2. Because K is a Poinca*
*r'e duality group of dimension 3
we also know that the Bockstein fi : H2(K, F3) ! H3(K, F3) is zero; hence
j*(a1y1- a2y2) = j*(fi(x1a1- x2a2)) = 0
and therefore
j*(a1y1) = j*(a2y2) 6= 0 .
This shows that H3(S12, F3) ! H3(K, F3) is onto and factors through the SD16-co*
*invariants, or dually
H3(K, F3) ! H3(S12, F3) is an injection and lands in the SD16-invariants. Furt*
*hermore, H3(K, F3) even
maps to the kernel of the Bockstein fi : H3(S12, F3) ! H2(S12, F3) and the indu*
*ced map
1, F3) ! H2(S1, F3)
H3(K, F3) ! Kerfi_:_H3(S2____________2Imfi~:=HH3(1S12; Z3) Z31F3
4(S2, F3) ! H3(S2,*
* F3)
is an isomorphism, yielding that j*[K] is a generator of infinite order.
To prove the statement on the edge homomorphism we proceed as follows. Consi*
*der the spectral sequence
of (4.10):
TorpZ3[[K]](F3, Cq) =) Hp+q(K, F3).
*
* 1
We know that C0 = Z3[[G12=G24]] is a free Z3[[K]]-module of rank 2 and C1 = C2 *
*= O "G2SD16are free
Z3[[K]]-modules of rank 3. Since the cohomological dimension of K is 3 we see t*
*hat N3 is projective as a
Z3[[K]]-module and because the Euler characteristic of K is zero, we obtain
ae
TorqZ3[[K]](F3, N3) ~= F30;F3; qq>=00.
Thus we can use Lemma 4.3 to say that N3 is a free Z3[[K]]-module of rank 2. Si*
*nce
F3 Z3[[K]]N3! F3 Z3[[S12]]N3
is a surjective morphism of vector spaces of the same dimension (see Lemma 4.7)*
*, it must be an isomorphism.
Furthermore,
H3(K, F3) -! F3 Z3[[K]]N3
is an injection. The result now follows from the diagram
H3(K, F3)____//_F3 Z3[[K]]N3
j*|| ~=||
fflffl| fflffl|
H3(S12, F3)__//_F3 Z3[[S12]]N3
*
*|___|
31
We will use cap products with the elements [K] and j*[K] to construct a comm*
*utative diagram for
detecting maps N3 ! Z3[[G12=G24]]. In the form we use the cap product, it has *
*a particularly simple
expression. Let G be a profinite group and M a continuous Zp[[G]]-module. If a *
*2 Hn(G, M) and x 2
Hn(G, Zp) we may define a\x 2 H0(G, M) as follows: choose a projective resoluti*
*on Qo ! Zp and represent
a and x by a cocycle ff : Qn ! M and a cycle y 2 Zp Zp[[G]]Qn respectively. The*
*n ff descends to a map
__ff: Z
p Zp[[G]]Qn -! Zp Zp[[G]]M
and a \ x is represented by __ff(y). It is a simple matter to check that this i*
*s well-defined; in particular, if
y = @z is a boundary, then ff(y) = 0 because ff is a cocycle. The usual natural*
*ity statements apply, which
we record in a lemma. Note that part (2) is a special case of part (1) (with K *
*= G).
4.11 Lemma. 1.) If ' : K ! G is a continuous homomorphism of profinite groups, *
*and x 2 Hn(K, Zp)
and a 2 Hn(G, M), then
'*('*a \ x) = a \ ' * (x).
2.) Suppose K G is the inclusion of a normal subgroup and M is a G-module.*
* Then G=K acts on
H*(K, M) and H*(K, Zp) and for g 2 G=K, a 2 Hn(K, M), and x 2 H*(K, Zp)
g*(g*a \ x) = a \ g*(x).
Here is our main diagram. Let i : K ! G12be the inclusion.
(4.12) HomZ3[[G12]](N3, Z3[[G12=G24]])edge//_H3(G12, Z3[[G12=G24]])
| |
| \i*[K]|
fflffl| fflffl|
Hom Z3[SD16](Z3 Z3[[S12]]N3, Z3 Z3[[S12]]Z3[[G12=G24]])ev//_H0(G12, Z3*
*[[G12=G24]])
| |
| |
fflffl| fflffl|
HomZ3[SD16](F3 Z3[[S12]]N3, F3 Z3[[S12]]F3[[G12=G24]])ev//_H0(G12, F3*
*[[G12=G24]])
We now annotate this diagram. The maps labelled evare defined by evaluating *
*a homomorphism at the
image of j*[K] under the edge homomorphism
H3(S12, Z3) ! Z3 Z3[[S12]]N3
resp.
H3(S12, Z3) ! Z3 Z3[[S12]]N3! F3 Z3[[S12]]N3
of (4.11). By Lemma 4.10, this is an SD16-invariant element and hence we get an*
* element in
(Z3 Z3[[S12]]Z3[[G12=G24]])SD16.
We now take the image of that element under the projection map from the invaria*
*nts to the coinvariants
(which in our case is an isomorphism because the order of SD16is prime to 3)
~= 1 1
(Z3 Z3[[S12]]Z3[[G12=G24]])SD16-! H0(G2, Z3[[G2=G24]]) .
32
Similar remarks apply to F3-coefficients.
The diagram commutes, by the definition of cap product. Theorem 4.9 now foll*
*ows from the final two
Lemmas 4.12 and 4.13 below; in fact, once we have proved these lemmas, diagram *
*(4.12) will then show that
we can choose a morphism of continuous G12-modules
f : N3- ! Z3[[G12=G24]]
so that
F3 f : F3 Z3[[S12]]N3- ! F3 Z3[[S12]]Z3[[G12=G24]]
is non-zero. Then Lemmas 4.7, 4.8 and 4.3 imply that f is an isomorphism.
4.12 Lemma. The homomorphism
\i*[K] : H3(G12, Z3[[G12=G24]]) ! H0(G12, Z3[[G12=G24]])
is an isomorphism.
Proof.Recall that we have denoted the inclusion K ! S12by j. We begin by demons*
*trating that
\j*[K] : H3(S12, Z3[[G12=G24]]) ! H0(S12, Z3[[G12=G24]])
is an isomorphism. Since the action of C3 on H3(K, Z3) ~=Z3 is necessarily tri*
*vial we see that [K] is
C3-invariant and Lemma 4.11 supplies a commutative diagram
j*
H3(S12, Z3[[G12=G24]])//_H3(K, Z3[[G12=G24]])C3
\j*[K]|| \[K]||
fflffl| j* fflffl|
H0(S12, Z3[[G12=G24]])H0(K,oZ3[[G12=G24]])C3o_
The morphism \[K] is an isomorphism by Poincar'e duality. As Z3[[S12]]-modules,*
* we have
Z3[[G12=G24]] ~=Z3[[S12=C3]] Z3[[S12=!C3!-1]]
on generators eG24and !G24in G12=G24; hence, as K-modules
Z3[[G12=G24]] ~=Z3[[K]] Z3[[K]]
which shows that j induces an isomorphism on H0(-, Z3[[G12=G24]]). We claim tha*
*t C3 acts trivially on
H0 and thus j* is an isomorphism. In fact, it is clear that the C3-action fixes*
* the coset eG24; furthermore
!C3!-1 is another complement to K in S12and therefore
C3!C3 KC3!C3= S12!C3= K!C3!-1!C3 K!C3 ,
and hence the class of !C3 in H0 is also fixed.
In addition, since Hq(K, Z3[[G12=G24]]) = 0 if q 6= 3, the Lyndon-Serre-Hoch*
*schild spectral sequence
shows that j* is an isomorphism.
33
To finish the proof, we continue in the same manner. Let r : S12! G12be the*
* inclusion, so that
i = rj : K ! G12. By Lemma 4.10 j*[K] is SD16-invariant and then 4.11 supplies *
*once more a commutative
diagram
*
H3(G12, Z3[[G12=G24]])r//_H3(S12, Z3[[G12=G24]])SD16
\i*[K]|| \j*[K]||
|fflffl r* fflffl|
H0(G12, Z3[[G12=G24]])H0(S12,oZ3[[G12=G24]])SD16.o_
We have just shown that \j*[K] is an isomorphism. The map r* sends invariants *
*to coinvariants and,
since the order of SD16is prime to 3, is an isomorphism. Again, because the ord*
*er of SD16is prime to 3
the spectral sequence of the extension S12! G12! SD16collapses at E2 and theref*
*ore the map_r* is an
isomorphism. This completes the proof. *
* |__|
4.13 Lemma. The edge homomorphism
Hom G12(N3, Z3[[G12=G24]]) ! H3(G12, Z3[[G12=G24]])
is surjective.
Proof.We examine the spectral sequence of (4.7):
Ep,q1~=ExtpZ3[[G12]](Cq, Z3[[G12=G24]]) =) Hp+q(G12, Z3[[G12=G2*
*4]])
We need only show that
ExtpZ3[[G12]](Cq, Z3[[G12=G24]]) = 0
1 G1
for p + q = 3 and q < 3. If q = 1 or 2, then Cq = O "G2SD16. Now O "S2D16is pro*
*jective as a Z3[[G12]]-module
and therefore ExtpZ3[[G12]](Cq, Z3[[G12=G24]]) is trivial for p > 0.
If q = 0, then C0= Z3[[G12=G24]]and by the Shapiro lemma we get an isomorphi*
*sm
Ext3G12(C0, Z3[[G12=G24]]) = H3(G24, Z3[[G12=G24]]) ~=H3(C3, Z3[[G12*
*=G24]])Q8 .
The profinite C3-set G12=G24is an inverse limit of finite C3-sets Xiand thus we*
* get an exact sequence
0 ! limi1H2(C3, Z3[Xi]) ! H3(C3, Z3[[G12=C3]]) ! limiH3(C3, Z3[Xi])*
* ! 0 .
Now each Xiis made of a finite number of C3-orbits. The contribution of each or*
*bit to H3(C3, -) is trivial
and to H2(C3, -) it is either trivial or Z=3. Therefore limiis clearly trivial *
*and lim1iis trivial_because the
Mittag-Leffler condition is satisfied. *
* |__|
34
5 The tower
In this section we write down the five stage tower whose homotopy inverse limit*
* is LK(2)S0 = EhG22and the
1
four stage tower whose homotopy inverse limit is EhG22. As before we will write*
* E = E2 and we recall that
we have fixed the prime 3.
To state our results, we will need a new spectrum. Let O be the representati*
*on of the subgroup SD16 G2
that appeared in (4.1) and let eO be an idempotent in the group ring Z3[SD16] t*
*hat picks up O. The action
of SD16on E gives us a spectrum EO which is the telescope associated to this id*
*empotent: EO := eOE.
Then we have an isomorphism of Morava modules
E*EO ~=Hom Z3[SD16](O, E*E)) ~=HomZ3[SD16](O, Homc(G2, E*))
~=Hom Z3[[G2]](O "G2SD16, Homc(G2, E*)) ~=HomcZ3(O "G2SD16, *
*E*) .
We recall that Homc(G2, E*) is a Morava module via the diagonal G2-action, and *
*a Z3[SD16]-module via the
translation action on G2. The group HomcZ3(-, -) is the group of all homomorphi*
*sms which are continuous
with respect to the obvious p-adic topologies.
It is clear from (4.2) that EO is a direct summand of EhQ82and a module spec*
*trum over EhSD162. In
fact, it is easy to check that the homotopy of EO is free of rank 1 as a ss*(Eh*
*SD16)-module on a generator
!2u42 ss8(EhQ8) ss*(E): this generator detemines a map of module spectra from*
* 8EhSD16to EO which
is an equivalence. From now on we will use this equivalence to replace EO by 8*
*EhSD16. We note that EO
is periodic with period 16.
5.1 Lemma. There is an exact sequence of Morava modules
0 ! E* ! E*EhG24! E* 8EhSD16 E*EhG24!E* 8EhSD16 E* 40EhSD16
! E* 40EhSD16 E* 48EhG24! E* 48EhG24! 0*
* .
Proof.Take the exact sequence of continuous G2-modules of Corollary 4.2 and app*
*ly HomcZ3(o, E*). Then
use the isomorphism E* 8EhSD16= HomcZp(O "G2SD16, E*) above and the isomorphisms
E*EhF ~=Homc(G2=F, E*)
supplied by (2.7) to get an exact sequence of Morava modules
0 ! E* ! E*EhG24! E* 8EhSD16 E*EhG24!E* 8EhSD16 E* 8EhSD16
! E* 8EhSD16 E*EhG24! E*EhG24! 0 .
Finally, we use that 8EhSD16' 40EhSD16because EhSD16is periodic of period 16 *
*and E*EhG24~=_
E* 48EhG24as Morava modules (see Remark 3.12.2). *
* |__|
5.2 Remark. In the previous lemma, replacing 8EhSD16by 40EhSD16is merely aest*
*hetic: it empha-
sizes some sort of duality. However, EhG24and 48EhG24are different spectra, ev*
*en though E*EhG24~=
E* 48EhG24. This substitution is essential to the solution to the Toda bracket*
* problem which arises in
Theorem 5.5.
In the same way, one can immediately prove
35
5.3 Lemma. There is an exact sequence of Morava modules
1 hG 8 hSD 40 hSD 48 hG
0 ! E*EhG2! E*E 24! E* E 16! E* E 16! E* E 24! 0 .
5.4 Theorem. The exact sequence of Morava modules
0 ! E* ! E*EhG24! E* 8EhSD16 E*EhG24!E* 8EhSD16 E* 40EhSD16
! E* 40EhSD16 E* 48EhG24! E* 48EhG24! 0
can be realized in the homotopy category of K(2)-local spectra by a sequence of*
* maps
LK(2)S0 ! EhG24! 8EhSD16_ EhG24! 8EhSD16_ 40EhSD16! 40EhSD16_ 48EhG24! 48*
*EhG24
so that the composite of any two successive maps is null homotopic.
Proof.The map LK(2)S0 ! EhG24is the unit map of the ring spectrum EhG24. To pro*
*duce the other maps
and to show that the successive composites are null homotopic, we use the diagr*
*am of Proposition 2.7. It is
enough to show that the En-Hurewicz homomorphism
ss0F(X, Y ) ! HomE*E(E*X, E*Y )
is onto when X and Y belong to the set { 8EhSD16, EhG24}. (Notice that the othe*
*r suspensions cancel out
nicely, since EhSD16is 16-periodic.) Since 8EhSD16is a retract of EhQ8, it is *
*sufficient to show that
ss0F(EhK1, EhK2) ! HomE*E(E*EhK1, E*EhK2)
is onto for K1 and K2 in the set {Q8, G24}. Using Proposition 2.6 and the short*
* exact lim-lim1sequence for
the homotopy groups of holimwe see that it is enough to note that (E1)K = 0 and
ss0EhK ! (E0)K
is surjective whenever K K2\ xK1x-1. The first part is trivial and for the se*
*cond part we can appeal to
Corollary 3.13.
To show that the successive compositions are zero, we proceed similarly, aga*
*in using Proposition 2.7, but
now we have to show that various Hurewicz maps are injective. In this case, the*
* suspensions do not cancel
out, and we must show
ss0F(EhK1, 48kEhK2) ! HomE*E(E*EhK1, E* 48kEhK2)
is injective for k = 0 and 1, at least for K1 of the form G2, G24, or Q8 and K2*
* of the form G24or Q8. Since
all the spectra involved in Proposition 2.6 are 72-periodic, the lim-lim1sequen*
*ce for the homotopy groups
of holimshows one more that it is sufficient to note that E24k+1is trivial, tha*
*t ss24k+1(EhK) is finite and
(5.1) ss24kEhK ! (E24k)K
is injective for k = 0 and 1 and K K2\ xK1x-1.
Again the first part is trivial while the other parts follow from Theorem 3.*
*10 and Corollary_3.13. Note
that for k = 1 the map in (5.1) need not be an isomorphism. *
* |__|
The following result will let us build the tower.
36
5.5 Theorem. In the sequence of spectra
LK(2)S0 ! EhG24! 8EhSD16_ EhG24! 8EhSD16_ 40EhSD16! 40EhSD16_ 48EhG24! 48*
*EhG24
all the possible Toda brackets are zero modulo their indeterminacy.
Proof.There are three possible three-fold Toda brackets, two possible four-fold*
* Toda bracket and one possible
five-fold Toda bracket. All but the last lie in zero groups.
Because 8EhSD16is a summand of EhQ8, the three possible three-fold Toda bra*
*ckets lie in
ss1F(EhG2, EhQ8_ 32EhQ8), ss1F(EhG24, 32EhQ8_ 48EhG24), ss1F(EhQ8_ EhG2*
*4, 48EhG24)
which are all zero by Proposition 2.6, Corollary 3.14 and Corollary 3.15. The m*
*ost interesting calculation is
the middle of these three and the most interesting part of that calculation is
Y
ss1F(EhG24, 48EhG24) ~=ss25(holimi EhHx) .
G24\Gn=Ui)
This is zero by Corollary 3.14 and Corollary 3.15 (note that the element -1 = !*
*4 2 G24is in the center of
G2 and it is in Hx for every x); however, notice that without the suspension by*
* 48 this group is non-zero.
The two possible four-fold Toda brackets lie in
ss2F(EhG2, 32EhQ8_ 48EhG24) and ss2F(EhG24, 48EhG24)
We claim these are also zero groups. All of the calculations here present some *
*interest. For example, consider
Y
ss2F(EhG24, 48EhG24) ~=ss26(holimi EhHx) .
G24\Gn=Ui)
where Hx = xUix-1\ G24. Since the element -1 = !4 2 G24is in the center of G2, *
*it is in Hx for every x
and the result follows from Corollary 3.15 and the observation that ss27(EhKx) *
*is finite.
Finally, the five-fold Toda bracket lies in
ss3F(EhG2, 48EhG24) ~=ss27EhG24~=Z=3 .
Thus, we do not have the zero group; however, we claim that the map EhG2! EhG24*
*at the beginning of
our sequence supplies a surjective homomorphism
ss*F(EhG24, 48EhG24) ! ss*F(EhG2, 48EhG24) .
This implies that the indeterminancy of the five-fold Toda bracket is the whole*
* group, completing the proof.
To prove this claim, note the EhG2! EhG24is the inclusion of the homotopy fi*
*xed points by a larger
subgroup into a smaller one. Thus Proposition 2.6 yields a diagram
F(EhG24, EhG24)__//_F(EG2, EhG24)
' || |'|
fflffl| fflffl| '
E[[G2=G24]]hG24__//_E[[G2=G2]]hG24//_EhG24
and the contribution of the coset eG24in E[[G2=G24]]hG24shows that the horizont*
*al map_is a split surjection
of spectra. *
* |__|
37
The following result is now an immediate consequence of Theorems 5.4 and 5.5:
5.6 Theorem. There is a tower of fibrations in the K(n)-local category
LK(2)S0__________//X3________________//X2________________//X1________//_EhG2*
*4OOOOOOOO
| | | |
| | | |
| | | |
44EhG24 45EhG24_ 37EhSD16 6EhSD16_ 38EhSD16 7EhSD16_ -1EhG24
Using Lemma 5.3 and the very same program, we may produce the following resu*
*lt. The only difference
will be that the Toda brackets will all lie in zero groups.
5.7 Theorem. There is a tower of fibrations in the K(n)-local category
EhG12_________//_Y2________//_Y1_____//EhG24OOOOOO
| | |
| | |
| | |
45EhG24 38EhSD16 7EhSD16
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