18.2.2002
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3
Paul Goerss, Hans-Werner Henn and Mark Mahowald
Abstract. Let V(1) be the Toda-Smith complex for the prime 3. We give a co*
*mplete calcu-
lation of the homotopy groups of the L2-localization of V (1) by making us*
*e of the higher real
K-theory EO2 of Hopkins and Miller and related homotopy fixed point spectr*
*a. In particular
we resolve an ambiguity which was left in an earlier approach of Shimomura*
* whose computation
was almost complete but left an unspecified parameter still to be determin*
*ed.
0. Introduction
The chromatic approach offers at present the most attractive perspective on the*
* stable ho-
motopy category of finite complexes. For any natural prime p there is a tower o*
*f localization
functors Ln with natural transformations Ln ! Ln-1 where Ln is Bousfield locali*
*zation
with respect to a certain multiplicative homology theory E(n)*. For a finite co*
*mplex X the
homotopy inverse limit of these localizations gives the p-localization of X. Th*
*e study of the
localization functors Ln is sometimes referred to as the study of the chromatic*
* primes in
stable homotopy theory. For more details the reader may consult [Ra3].
The solution of the Adams conjecture lead to a good conceptual and calculationa*
*l under-
standing of the localization functor L1 if p is any prime. The case of L2 is re*
*asonably well
understood for primes p > 3 at least from a computational point of view [SY]. T*
*he case
of L2 at the primes p = 3 and p = 2 is harder. The standard approach to underst*
*and the
L2-localization L2S0 (at any prime) is to study L2X for a üs itable" finite com*
*plex X and
to work one's way back to L2S0 through appropriate Bockstein spectral sequences*
* arising
from the skeletal filtration of X. At odd primes the Toda-Smith complexes V (1)*
* (which are
defined as cofibre of a self map A of the mod-p Moore spectrum V (0) such that *
*A induces
multiplication by v1 in Brown-Peterson theory BP*) are suitable in this sense. *
*For primes
p > 3 the homotopy of L2V (1) is relatively easy to understand; the Adams-Novik*
*ov spec-
tral sequence (ANSS for short) converging to ß*L2V (1) collapses at E2 and its *
*E2-term is
known [Ra1]. Starting from this information Shimomura and Yabe were able to com*
*pute
the homotopy of L2S0 for all primes p > 3.
At the prime 3 it is natural to try the same strategy and start with studying ß*
**L2V (1). In
fact, the E2-term of the ANSS converging to ß*L2V (1) has been computed in [H] *
*(see also
[GSS] and [Sh1]) but this time the ANSS for V (1) does not collapse. Using vari*
*ous informa-
tion about homotopy groups of spheres and related complexes in low dimensions S*
*himomura
studied this spectral sequence and arrived at a calculation modulo some ambigui*
*ty; there
____________
The authors would like to thank (in chronological order) the University of Heid*
*elberg, MPI at Bonn, Uni-
versit'e Louis Pasteur at Strasbourg and Northwestern University for providing *
*them with the opportunity
to work together.
Typeset by AM S-T*
*EX
1
2 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
was an unspecified parameter k 2 {0, 1, 2} and several families of homotopy ele*
*ments which
lived in degrees which were only determined up to adding 24k.
The E2-term of the ANSS for L2V (1) can be identified by Morava's change of rin*
*g isomor-
phism with the continuous cohomology of a certain p-adic Lie group G2, (the "ex*
*tended
Morava stabilizer group") with coefficients in F9[u 1]. Devinatz and Hopkins [D*
*H2] have
given a homotopy theoretic interpretation of Morava's change of rings isomorphi*
*sm: the
localization LK(2)S0 of the sphere with respect to the second Morava K-theory K*
*(2) has
the homotopy type of the homotopy fixed point spectrum EhG22which is defined by*
* making
use of the Hopkins-Miller rigidification of the action of G2 on the Lubin-Tate *
*spectrum E2
[Re].
In this paper we analyze the ANSS for L2V (1) by making serious use of group th*
*eoretic
and cohomological properties of G2. In fact, L2V (1) can be identified with LK*
*(2)V (1) '
LK(2)S0 ^ V (1) ' EhG22^ V (1) and we will study L2V (1) by comparing it with E*
*hN2^ V (1)
where EhN2 is the homotopy fixed point spectrum with respect to the normalizer *
*of an
element of order 3 in G2. The use of the spectrum EhN2was suggested by the calc*
*ulation of
the E2-term of the ANSS in [H] which made heavy use of centralizers of elements*
* of order
3 in G2. The other good property of EhN2 on which our method relies is that it*
* can be
analyzed in terms of the Hopkins-Miller higher real K-theory spectrum EO2 at th*
*e prime
3. Both properties together allow us to give an independant calculation of ß*L2*
*V (1) which
is complete and identifies Shimomura's parameter as k = 1.
To state our main result we need some notation. First of all, from now on all *
*spaces or
spectra will be localized at 3.
The homotopy of L2V (1) is annihilated by 3 and a module over the homotopy of L*
*K(2)S0.
Therefore it can be regarded as a module over the algebra F3[fi] (i) where f*
*i is the image
of the generator fi1 2 ß10S0 and i is in ß-1LK(2)S0. The homotopy groups turn o*
*ut to be
periodic of period 144 and on the E2-level this periodicity corresponds to mult*
*iplication by
v92where v2 is the polynomial generator in ß16BP . We do not prove that this pe*
*riodicity
arises geometrically but it is convenient to describe ß*L2V (1) nevertheless as*
* a module over
P := F3[v29, fi] (i). We will use notation like P=(fi5){vl2}l=0,1,5to denote*
* the direct sum
of P -modules each of which is killed by fi5 (more precisely its annihilator id*
*eal is the ideal
generated by fi5) and which have generators named 1 = v02, v2 and v52.
Finally we note that the E2-term has a product structure and it contains elemen*
*ts which
deserve to be named v2 (which is closely related to the generator v2 2 ß16(BP )*
*) , ff (which
detects the image of the generator ff1 2 ß3(S0)), v1=22fi, v1=22ff, fia35, ffa3*
*5, v1=22fia35 and
v1=22fiffa35 and which live in total degree 16, 3, 18, 11, 45, 38, 53 and 56 re*
*spectively. The
reason for these names will become clear once we have discussed the spectrum Eh*
*N2and in
particular EhN2^ V (1).
Theorem. As a module over P = F3[v29, fi] (i) there is an isomorphism
ß*L2V (1) ~=P=(fi5){vl2}l=0,1,5 P=(fi3){vl2ff}l=0,1,2,5,6,7
P=(fi4){vl+1=22fi}l=0,4,5 P=(fi2){vl+1=22ff}l=0,1,2,4,5,6
P=(fi4){vl2fia35}l=0,1,5 P=(fi3){vl2ffa35}l=0,1,2,5,6,7
P=(fi5){vl+1=22fia35}l=0,4,5 P=(fi2){vl+1=22fiffa35}l=0,1,2*
*,4,5,6.
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 3
We note that this result has already been used to carry out the programm to cal*
*culate
ß*(L2S0) that we mentioned above [Sh2], [SW].
The paper is organized as follows. In the first section we discuss the spectru*
*m EO2 and
EO2 ^ V (1). In particular we give a detailed discussion of the ANSS convergin*
*g towards
ß*(EO2). In the second section we introduce the homotopy fixed point spectrum E*
*hN2and
we relate EhN2to EO2, and consequently EhN2^ V (1) to EO2 ^ V (1). In the final*
* section
we compare L2V (1) with EhN2^ V (1) and prove the main theorem.
1. The homotopy of EO2 and EO2 ^ V (1)
1.1. The spectrum EO2.
We begin by recalling the construction of the spectrum EO2 due to Hopkins and M*
*iller. We
refer to [Re] for more details.
The point of departure is the Lubin-Tate deformation theory of formal group law*
*s (cf. [LT]),
in particular the universal deformation of the formal group law of height 2 o*
*ver the field
F9 with [p]-series [p](x) = x9. The universal deformation is a lift of to a *
*formal group
law e over WF9[[u1]] (where WF9 are the Witt vectors of F9 and u1 is a formal p*
*ower series
variable). Over the graded ring WF9[[u1]][u 1] (where u is of degree -2 and u1 *
*of degree
0) this formal group law is isomorphic to the one induced from the universal p *
*- typical
formal group law over BP* ~=Z(p)[v1, v2, . .].via the map of algebras which sen*
*ds v1 to
u1u1-p, v2 to u1-p2 and vn to 0 for i > 2. The Landweber exact functor theorem *
*implies
that there is a homology theory E*2represented by a ring spectrum E2 with coeff*
*icients
ß*(E2) = WF9[[u1]][u 1] such that the cohomology theory E2* is complex oriented*
* with
orientation u and such that the associated formal group law is isomorphic to e.*
* To simplify
notation we will abbreviate WF9 by W and E2 by E throughout.
The group S2 of automorphisms of (also known as the Morava stabilizer group) *
*acts on the
ring spectrum E, up to homotopy, and this action extends in a canonical way to *
*an action
of the extended stabilizer group, given as the semidirect product G2 := S2 o C2*
* where the
cyclic group C2 of order 2 acts on S2 via Galois automorphisms of . (Note that*
* is defined
ove F3 and thus we get an action of the Galois group of the extension F3 F9 o*
*n S2.)
Hopkins and Miller have shown how to rigidify this action to a genuine action v*
*ia A1 - maps
[Re] and subsequently Devinatz and Hopkins [DH2] have shown how to construct ho*
*motopy
fixed point spectra EhH with respect to closed subgroups H of G2. Their constru*
*ction has
the following properties: it agrees in the case of finite subgroups with the na*
*ive construction
of the homotopy fixed point spectrum, if H = G2 then EhH ' LK(2)S0, and for any*
* closed
subgroup H and any finite spectrum X there is a spectral sequence
Es,t2(X) = Hscts(H, Et(X)) =) ßt-s(EhH ^ X)
where H*ctsdenotes continous cohomology of the p-adic group H.
The group S2 can also be identified with the group of units of the maximal orde*
*r O2 of the
division algebra D2 over the 3-adic rationals Q3. The maximal order is a free W*
*-module of
rank 2 with basis 1 and S; as a ring it is determined by the relations S2 = 3 a*
*nd Sa = OE(a)S
if OE notes the lift of Frobenius from F9 to W [Ra2, Appendix 2]. From this poi*
*nt of view
4 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
the extended group G2 is the split extension S2 o C2 where the action of C2 is *
*given by
conjugation with S in D2.
Let ! be a fixed 8th root of unity in W. The element s := -1_2(1 + !S) is easil*
*y checked to
be of order 3. Furthermore !2s!-2 = s2 so that s and t := !2 generate a subgrou*
*p G12 of
S2 which is isomorphic to C3o C4 with C4 acting non-trivially on C3. The spectr*
*um EO2 is
defined as the homotopy fixed point spectrum EhG12. We add that G12 is a maxima*
*l finite
subgroup of S2 and every other maximal finite subgroup is conjugate to G12.
1.2. The E2-term of the ANSS converging to ß*(EO2).
We do not claim any originality for the results in this and the following subse*
*ction. The
ANSS for EO2 was first investigated by Hopkins and Miller but unfortunately the*
*ir work
remains unpublished. There is a rather brief discussion of this spectral sequen*
*ce in the still
unpublished paper [N]. A discussion of its E2-term from a different point of vi*
*ew can be
found in [GS]. Neither of these sources suits well our needs and therefore we h*
*ave decided
to give a self-contained treatment here.
In order to describe the E2-term Es,t2~=Hs(G12, Et) we start by analyzing E* as*
* a G12-
algebra. The first step is to locate an appropriate subrepresentation in E-2.
Let Ø be the representation of G12 on W which is trivial on s and on which t ac*
*ts by
multiplication by !2. Define a G12 - module æ by the short exact sequence
0 ! Ø ! W[G12] W[C4]Ø ! æ ! 0
in which in the middle term Ø is considered as a representation of the subgroup*
* C4 generated
by t and where the first map takes a generator e of Ø to
(1 + s + s2)e := (1 + s + s2) e 2 W[G12] W[C4]Ø .
Lemma 1. There is a morphism of G12-modules
æ -! E-2
so that the induced map
æ W F9 ! E-2 E0 E0=(3, u21)
is an isomorphism.
Proof. We need to know something about the action of G2 on E*. Let m = (3, u1) *
* E0
be the maximal ideal. Then Proposition 3.3 and Lemma 4.9 of [DH1] together impl*
*y that,
modulo m2E-2
s*(u) -1_2(u + !3uu1)
s*(uu1) -1_2(3!u + uu1)
t*(u) !2u
t*(uu1) -!2uu1 .
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 5
In particular, we see that E-2 E0 E0=(3, u21) is isomorphic to æ W F9 as a G1*
*2 - module
and that the residue class of u is a G12-module generator. Thus we would like t*
*o find a class
x 2 E-2 with the same reduction as u so that t*(x) = !2x and x+s*(x)+s2*(x) = 0*
*. In fact,
because the action of t on E-2 E0 E0=(3, u21) is diagonalizable with distinct *
*eigenvectors it
suffices to find x such that x u mod (3, u1), up to a scalar in Fx9.
Such an x can be obtained as follows: we start with the element u-2u1 which is *
*the image
of v1 2 BP* with respect to the map BP* ! E* which classifies e (we will denote*
*d this
element simply by v1 in the sequel); v1 is invariant modulo 3 with respect to t*
*he action of
all of S2. More precisely, the structure formulae in BP*BP [Ra2, Appendix 2] yi*
*eld
g*(v1) = v1 + (3 - 33)t1(g) v1 + 3t1(g) mod (32)
for every g 2 S2. Here we use the identification of E*E with the continous func*
*tions from
the profinite group S2 with values in E* equipped with the profinite topology (*
*see [St, Thm.
12] for a convenient reference) and t1 2 E4E is the image of the element with t*
*he same name
in BP*BP .
By definition of t1 we have t1(-1_2(1+!S)) !u-2 mod (3, u1). Hence, if z = 1_*
*3(v1-s*(v1))
then
z !u-2 mod (3, u1) ,
in particular z is non-zero. Clearly we have z + s*(z) + s2*(z) = 0 but z does *
*not yet have
the right degree.
Therefore we consider the class
y = us*(u)s2*(u)z .
Then
y !u mod (3, u1)
and we still have
y + s*(y) + s2*(y) = 0 .
However, y might not yet have the correct invariance property with respect to t*
*he element
t of order 4. Therefore we average and set
x = 1_4(y + !-2t*(y) + !-4t2*(y) + !-6t3*(y)) .
Then we get x !u mod (3, u21) and we are done.
The morphism of G12-modules constructed in the lemma defines a morphism of W [G*
*12]-
algebras
S(æ) -! E*
where S(æ) denotes the symmetric algebra on æ. We note that as an algebra S(æ) *
*is poly-
nomial over W on two generators e and s*(e), and we can choose e such that its *
*image is
the element x of the proof of the lemma above and is therefore invertible in E**
*. Let
Y
N = g*e 2 S(æ) ;
g2G12
then we have a morphism of W[G12]-algebras
S(æ)[N-1 ] -! E* .
Note that inverting N inverts e as well, but in an invariant manner. Let I S*
*(æ)[N-1 ]
be the preimage of the maximal ideal m = (3, u1) E* (now considered as a homo*
*geneous
graded ideal).
6 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
Proposition 2. The induced map of complete algebras
S(æ)[N-1 ]^I-! E*
is an isomorphism.
Proof. It is enough to show that the induced maps
_Ik_ ! _mk__
Ik+1 mk+1
are isomorphisms for each k.
4
If we identify S(æ) with W[e, s*(e)] then we get N = - es*(e)(e + s*(e) . Fur*
*thermore
it is straightforward to check that the homogeneous graded ideal I is generated*
* by 3 and
e - s*(e) and that the maps in question are isomorphisms.
Our next step is to study H*(G12, S(æ)). We will see in Theorem 6 below how the*
* calculation
of the E2-term can be reduced to that of H*(G12, S(æ)).
If e 2 æ is the generator, let d be the multiplicative norm of e with respect t*
*o the subgroup
generated by s, i.e. d = es*(e)s2*(e). We note that d is of degree -6, it is in*
*variant with
respect to s and furthermore d4 = -N.
For a finite group G and any G module M, let
trG= tr: M -! MG = H0(G, M)
P
be the transfer: tr(x) = g2G gx. The following calculates H*(G12, S(æ)) com*
*pletely if
* > 0 and gives partial information if * = 0; an element listed as being in bid*
*egree (s, t) is
in Hs(G, St(æ)).
Lemma 3. Let C3 G12 be the normal subgroup generated by s. Then there is an e*
*xact
sequence
S(æ) -tr!H*(C3, S(æ)) ! F9[b, d] (a) ! 0
where a has bidegree (1, -2), b has bidegree (2, 0) and d has bidegree (0, -6).*
* Furthermore
the action of t is described by
t*(a) = -!2a t*(b) = -b t*(d) = !6d .
(By abuse of notation we have denoted the image of the invariant class d in the*
* quotient
H0(C3, S-6(æ))=Im trstill by d. )
Proof. Let F be the G12-module W[G12] W[C4]Ø. We can choose a W-basis x1, x2, *
*x3 of
F such that x2 := s*(x1) and x3 := s*(x2), and then we get an identification
S(F ) = W[x1, x2, x3]
with all xi in degree -2. The kernel of the canonical C3-linear algebra map wh*
*ich sends
F to æ is the principal ideal generated by oe1 = x1 + x2 + x3, i.e. we have a *
*short exact
sequence of graded C3-modules
(*) 0 ! S(F ) Ø ! S(F ) ! S(æ) ! 0 .
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 7
(In the tensor product Ø has to be treated as a representation in degree -2 in *
*order to make
the maps degree preserving.)
As a C3-module S(F ) splits into a direct sum of free modules and trivial modul*
*es where the
trivial modules are generated by the powers of the monomial oe3 := x1x2x3. Ther*
*efore we
obtain a short exact sequence
S(F ) -tr!H*(C3, S(F )) ! F9[b, oe3] ! 0
where b has bidegree (0, 2) and oe3 has bidegree (0, -6). Here b is a generator*
* of H2(C3, W) ~=
W=3 and W S(F ) is the submodule generated by the unit of the algebra S(F ). *
*The action
of t is given by the following formula
t*(oe3) = !6oe3 = -!2oe3 and t*(b) = -b.
The short exact sequence (*) and the fact that H1(C3, S(F )) = 0 now imply that*
* there is
an exact sequence
S(æ) -tr!H*(C3, S(æ)) ! F9[a, b, d]=(a2) ! 0.
where d is the image of oe3 and a is the preimage of b 2 H2(C3, W) = H2(C3, S0(*
*F ) Ø)
with respect to the isomorphism
H1(C3, æ) = H1(C3, S1(æ)) ! H2(C3, S0(F ) Ø)
given by the obvious connecting homomorphism. Thus a has bidegree (1, -2) and t*
*he action
of t is twisted by Ø:
t*(a) = -!2a = !6a .
The next step is to compute the invariants S(æ)C3 together with the action of t*
*. For this
we start with the invariants of S(F ) and then we use the exact sequence (*). T*
*he action of
the cyclic group C3 on S(F ) = W[x1, x2, x3] extends in an obvious way to an ac*
*tion 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.
It is clear that the following element
ffl = x21x2 + x22x3 + x23x1 - x22x1 - x21x3 - x23x2
(the ä nti-symmetrizationö f x21x2) is also invariant under the action of C3. *
* We use the
same notation for the images of these elements in S(æ) and we note that oe1 bec*
*omes 0 in
S(æ).
Lemma 4.
a) There is an isomorphism of W-algebras
W[oe1, oe2, oe3, ffl]=(ffl2 - f) ~=S(F )C3
where f is the following polynomial in oe1, oe2, oe3
f = -27oe23- 4oe32- 4oe3oe31+ 18oe1oe2oe3 + oe21oe22.
8 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
Furthermore, the action of t 2 G12 is given by
t*(oe1) = !2oe1 t*(oe2) = -oe2 t*(oe3) = !6oe3 t*(ffl) = !2f*
*fl .
b) There is an isomorphism of W-algebras
W[oe2, oe3, ffl]=(ffl2 - g) ~=S(æ)C3
where g is the following polynomial in oe2, oe3
g = -27oe23- 4oe32
Furthermore, the action of t 2 G12 is given by
t*(oe2) = -oe2 t*(oe3) = !6oe3 t*(ffl) = !2ffl .
Proof. a) It is clear that ffl2 is 3 invariant and therefore it can be express*
*ed as a polynomial
in oe1, oe2 and oe3. To find the precise relation is an elementary exercise. We*
* also leave it
to the reader to verify that the action of t is as claimed. Thus it remains to *
*determine the
algebra structure of S(F )C3.
As a graded C3-module S(F ) decomposes into a direct sum of free modules of ran*
*k 3 and
of trivial modules of rank 1, and each of these summands contributes a summand *
*W to
S(F )C3. From this it is easy to calculate the Poincare series of the invariant*
*s and we find
6
ØS(F)C3(t) = _______1_+_t________(1.- t2)(1 - t4)(1 - t6)
(In this calculation we regrade S(F ) such that F is homogeneous of degree +2.)
On the other hand there is still an action of C2 on S(F )C3, and S(F ) splits a*
*s direct sum
of eigenspaces
S(F ) ~=S(F )+ S(F )- .
Furthermore S(F )+ ~= S(F ) 3 and S(F )- is a module over S(F )- . By the Poin*
*car'e se-
ries calculation it is therefore enough to verify that S(F )- is free as S(F )+*
* -module with
generator ffl.
In fact, it is clear that ffl is in S(F )- and because W[x1, x2, x3] is without*
* zero divisors it
is also clear that ffl generates a free S(F )+ module with the correct Poincar'*
*e series. Now
suppose p 2 S(F )- . We can choose a constant c 2 W of mininimal valuation, say*
* r, such
that cp = fflq for a unique polynomial q 2 W[oe1, oe2, oe3]. Then
ffl(cp) = ffl2q = fq .
If c is divisible by 3 then the formula for f shows that q must be divisible by*
* 3 and then r
was not minimal. Hence r = 0, c 2 Wx and p is in the submodule generated by ffl.
b) This is an immediate consequence of (a) and the vanishing of H1(C3, S(F )).
The next step is to invert the element N. This element is the image of oe43; t*
*hus, we are
effectively inverting the element oe3 2 S(æ)C3. We begin with the observation *
*that if we
divide
ffl2 = -27oe23- 4oe32
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 9
by 4oe63we obtain
(__ffl_2oe3)2 + (_oe2_oe2)3 = -_27_4.
3 4oe3
Thus if we set
c4 = -_oe2_oe2, c6 = _ffl_3, = -__1_4
3 2oe3 4oe3
then we get the following relation
c26- c34= 27
which corresponds to the famous relation from the theory of modular forms [D] (*
*except that
in our case 2 is invertible and hence the usual factor 1728 can be simplified t*
*o 27). The
reader is referred to [GS] for an explanation of this coincidence.
Furthermore, c4, c6, and are all invariant under the action of the entire gro*
*up G12. The
elements
ff := ad-1 2 H1(C3, (S(æ)[N-1 ])4)
and
fi := bd-2 2 H2(C3, (S(æ)[N-1 ])12)
are clearly fixed by the action of t and by degree reasons they are acted on tr*
*ivially by c4
and c6. The following result is now straightforward to verify.
Proposition 5.
a) The inclusion
W[c4, c6, 1]=(c26- c34= 27 ) ! S(æ)[N-1 ]G12
is an isomorphism.
b) There is an exact sequence
S(æ)[N-1 ] -tr!H*(G12, S(æ)[N-1 ]) ! F9[ff, fi, 1]=(ff2) ! 0
and c4 and c6 act trivially on ff and fi.
The final step is now to investigate what happens under completion. We continue*
* to use c4,
c6 etc. for the images of these elements in H*(G12, E*) with respect to the map*
* S(æ)[N-1 ] !
E* studied in Proposition 2.
Theorem 6.
a) There is an isomorphism
(E*)G12 ~=W[[c34 -1]][c4, c6, 1]=(c26- c34= 27 )
b) There is an exact sequence
E* -tr!H*(G12, E*) ! F9[ff, fi, 1] ! 0
and c4 and c6 act trivially on ff and fi.
Proof. a) We use Proposition 2 and we use that completion commutes with taking *
*invariants.
We abbreviate S(æ)[N-1 ] by A and we recall that the ideal I A is generated b*
*y 3 and
10 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
e - s*(e). With this it is straightforward to check that both c4 and c6 belong*
* to I. The
relation c26- c34= 27 implies then that I \ AC30is generated by 3 and c34 -1. *
*This implies
(a).
b) It is also straightforward to verify that oe2 -(e - s*(e))2 mod (3) and th*
*is implies that
the ideals I and (3, c34 -1) define the same completion. Abbreviate c34 -1 by z*
*. We have
an isomorphism
E* ~=limkA=(zk) .
Now we consider the short exact sequence
k
0 ! A -z!A ! A=(zk) ! 0 .
Because z acts trivially on Hq(G12, A) for q > 0 (by Proposition 5b) we obtain *
*for each
q > 0 a tower (indexed by k) of short exact sequences
0 ! Hq(G12, A) ! Hq(G12, A=(zk)) ! Hq+1(G12, A) ! 0.
The maps on the right hand side of this tower are also induced by multiplicatio*
*n with z,
hence they are trivial and therefore we obtain an isomorphism
Hq(G12, A) ~=limkHq(G12, A=(zk)) .
On the other hand the graded quotients A=(zk) are of finite type for each k and*
* this implies
that the usual short exact sequences
0 ! lim1kHq-1(G12, A=(zk)) ! Hq(G12, limkA=(zk)) ! limkHq(G12, A=IkA) ! 0
degenerate into isomorphisms
Hq(G12, limkA=(zk)) ~=limkHq(G12, A=IkA)
and the proof is complete.
Remark_ With the same reasoning we can also compute the E2-terms for the homoto*
*py fixed
point spectra EhC32and EhC62where C3 is as before the subgroup generated by s a*
*nd C6
that generated by s and t2. In fact, the G12-invariant = -1=4oe43has a C3-inv*
*ariant 4-th
root 1=4in S(æ)[N-1 ] and we get
(E*)C3 ~=W[[c34 -1]][c4, c6, 1=4]=(c26- c34= 27 ) .
Furthermore there is an exact sequence
E* -tr!H*(C3, E*) ! F9[ff, fi, 1=4] ! 0
and c4 and c6 act trivially on ff and fi.
Similarly,
(E*)C6 ~=W[[c34 -4]][c4, c6, 1=2]=(c26- c34= 27 ) ,
there is an exact sequence
E* -tr!H*(C3, E*) ! F9[ff, fi, 1=2] ! 0
and c4 and c6 act trivially on ff and fi.
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 11
1.3. The homotopy of EO2.
Before we turn to the discussion of the differentials of the spectral sequence *
*we relate the
elements , c4, c6, ff and fi to well known quantities in homotopy theory.
We start by recalling that the elements vk12 Ext0,4BP*BP(BP*, BP*=(3)) define p*
*ermanent
cycles in the classical ANSS of the mod-3 Moore space V (0). Similarly, the ele*
*ment v2 2
Ext0,16BP*BP(BP*, BP*=(3, v1)) defines a permanent cycle in the classical ANSS *
*for the cofibre
V (1) of the Adams self map 4V (0) ! V (0). Furthermore v1 and v2 give rise vi*
*a the Greek
letter construction to generators ff1 2 ß3(S0) ~=Z=3 resp. fi1 2 ß10(S0) ~=Z=3 *
*which are
detected in the classical ANSS by elements with the same name in Ext1,4BP*BP(BP*
**, BP*)
resp. Ext2,12BP*BP(BP*, BP*).
Finally we note that the localization map from a finite spectrum X to LK(2)X to*
*gether
with the Morava change of rings isomorphism and the obvious restriction homomor*
*phism in
group cohomology induce a natural homomorphism
~X : Exts,tBP*BP(BP*, BP*X) ! Hs(G2; EtX) ! Hs(G12; EtX) .
We will denote the images of the elements v1, v2 with respect to ~V (0)resp. ~V*
* (1)still by
v1 resp. v2.
Proposition 7.
a) Reduction modulo (3, u1) sends 2 to the image of v32in H0(G12, E32=(3, u1)).
b) The mod-3 reduction map
H0(G12, Et) ! H0(G12, Et=(3))
sends c4 resp. c6 to the image of v21resp. v31, up to multiplication by a unit *
*in H0(G12, E0=(3))
~= F9[[c34 -1]]. Furthermore there is an element eff2 H1(G12, E12=(3)) and an i*
*somorphism
(of modules over F9[[v61 -1]][v1, 1, fi] (ff))
H*(G12, E*=3) ~=F9[[v61 -1]][v1, 1, fi] (ff){1, ea}=(v1ff, v1ea, ff*
*ea+ v1fi)
c) The map ~S0 sends ff1 to ff and fi1 to fi up to a nontrivial constant in W=3.
Proof. a) The definition of implies immediately that its reduction modulo (3,*
* u1) is equal
to that of u-12. On the other hand the reduction of u-24 is equal to v32.
b) It is clear from our calculation of H*(G12, E*) and the short exact sequence*
* of G12-
modules
(*) 0 ! E* ! E* ! E*=(3) ! 0
that v21and v31are in the image of mod-3 reduction. Furthermore they generate t*
*he invariants
in degree 8 resp. 12 as module over H0(G12, E0=(3)). On the other hand the G12-*
*invariants
in degree 8 resp. 12 of E* are freely generated (as modules over H0(G12, E0)) b*
*y c4 resp.
c6. This proves the statement on c4 and c6 and also gives the result for H0(G12*
*, E*=(3)).
We define effsuch that ffi0(eff) = fi where ffi0 denotes the boundary homomorph*
*ism associated
to the exact sequence (*). Then everything else except perhaps the relation v1*
*fi = ffeffis
straightforward to check. This relation is obtained by calculating
ffi0(v1fi + ffeff) = ffi0(v1)fi - ffffi0(eff) = fffi - fffi =*
* 0
12 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
and by noting that ffi0 is a monomorphism in the relevant bidegree.
c) This is a consequence of the compatibility (with respect to the maps ~X ) of*
* the Greek
letter construction for ExtBP*BP (BP*, -) and an analogous Greek letter constru*
*ction for
H*(G12, -).
In fact, the image of v1 2 Ext0BP*BP*(BP*, BP*=(3)) (which is the class u1u-2) *
*defines an
element in H0(G12, E*=(3)). The short exact sequence
0 ! E* ! E* ! E*=(3) ! 0
shows that this class has a nontrivial image ffi0(v1) 2 H1(G12, E4) ~=W=3 and t*
*his latter
group is generated by ff.
Similarly, for fi1 we just need to check that the result of the Greek letter co*
*nstruction on
u-822 H0(G12, E16=(3, u1)) yields a nontrivial element in H2(G12, E12). First w*
*e note that
the boundary map ffi1 associated to the exact sequence
0 ! 4E*=(3) -v1!E*=(3) ! E*=(3, u1) ! 0
maps u-8 nontrivially and hence to eff, up to a nonconstant multiple. In the pr*
*oof of (b) we
have seen that ffi0(eff) = fi and hence we are done.
In the sequel we redefine ff resp. fi such that ff = ~S0(ff1) and fi = ~S0(fi1)*
*. We are now
ready to describe the differentials in our SS.
Theorem 8. In the spectral sequence
Hs(G12, Et) =) ßt-s(EO2)
we have an inclusion of subrings
E0,*1~=W[[c34 -1]][c4, c6, c4 1, c6 1, 3 1, 3]=(c34- c26= 27 ) .
In positive filtration Es,t1is additively generated by the elements ff, fffi, *
*ff, fffi, fij,
j = 1, 2, 3, 4 and their multiples by powers of 3. All these elements are of *
*order 3 and c4
and c6 act trivially on elements in positive filtration.
Proof. First we observe that every element in the image of the transfer is a pe*
*rmanent cycle.
The last proposition implies furthermore that the elements ff and fi are perman*
*ent cycles
detecting homotopy classes with the same name. Next we use Toda's relation ff1f*
*i31= 0 in
ß*(S0). This implies that fffi3 = 0 in ß*(EO2) and this can only happen if d5( *
*) = a1fffi2
for some a1 2 Fx9.
Then we use the Toda bracket relation fi1 2 < ff1, ff1, ff1 > in ß*(S0). Con*
*sequently
we have fi 2 < ff, ff, ff > in ß*(EO2). This and fffi2 = 0 imply that fi3 is*
* in the
indeterminacy of the bracket < fffi2, ff, ff >. This is only possible if ff i*
*s a permanent
cycle and ff(ff ) = fi3, up to a nontrivial constant.
The next possible differential is d9. Up to nontrivial constants we have fi5 =*
* fi2fi3 =
fi2ff(ff ) = 0 in ß*(EO2) and this forces d9( 2ff) = a2fi5 for some a2 2 Fx9. T*
*hen there is
no more room for further differentials and E1 ~= E10 is a stated in the theorem.
Remark_ With the same reasoning we can also compute the homotopy of EhC32and Eh*
*C62. In
the case of C3 we obtain an inclusion of subrings
E0,*1~=W[[c34 -1]][c4, c6, c4 1=4, c6 1=4, 3 1=4, 3=4]=(c34- c26= 2*
*7 )
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 13
and in positive filtration Es,t1is additively generated by the elements ff, fff*
*i, ff, fffi, fij,
j = 1, 2, 3, 4 and their multiples by powers of 3=4. These elements are of or*
*der 3 and c4
and c6 act trivially on elements in positive filtration.
In the case of C6 we obtain an inclusion of subrings
E0,*1~=W[[c34 -1]][c4, c6, c4 1=2, c6 1=2, 3 1=2, 3=2]=(c34- c26= 2*
*7 )
and in positive filtration Es,t1is additively generated by the elements ff, fff*
*i, ff, fffi, fij,
j = 1, 2, 3, 4 and their multiples by powers of 3=2. Again these elements are*
* of order 3
and c4 and c6 act trivially on elements in positive filtration.
In particular we see that EO2 is 72 periodic with periodicity genertor 3, EhC3*
*2is 18 periodic
with periodicity generator 3=4and EhC62is 36 periodic with periodicity generat*
*or 3=2.
1.4. The ANSS converging to ß*(EO2 ^ V (1)).
In this section we calculate ß*(EO2^V (1)). We can do this by using Theorem 8 a*
*nd the long
exact sequences associated to the defining cofibre sequences of V (0) and V (1)*
*. However,
later on we will make use of the structure of the ANSS for L2V (1) and so we ch*
*oose to give
a presentation in terms of the ANSS
Es,t2(V (1)) = Hs(G12, E*V (1)) =) ßt-s(EO2 ^ V (1)) .
First we note that E*(V (1)) is given as F9[u 1]. The element s 2 G12 acts nec*
*essarily
trivially on this ring while t acts via t*(u) = !2u (cf. the proof of Lemma 1).*
* This gives us
the following E2-term
E*,*2~=F9[u 1, y] (x) C4
in which the (s, t)-bidegree of u is (0, -2), that of y is (2, 0) and that of x*
* is (1, 0). The
invariants can then be identified with F9[u 4, fi] (ff) where fi := u-6y is *
*a permanent
cycle detecting fi 2 ß10(V (1)) and ff := u-2x is a permanent cycle detecting f*
*f 2 ß3(V (1))
and where fi and ff are the images of the classical elements fi1 and ff1 in ß*(*
*S0). (This can
easily be checked via the long exact sequences of homotopy groups mentioned abo*
*ve). The
element u-8 is the image of v2 in the E2-term of the ANSS for ß*V (1) with resp*
*ect to the
localization map (cp. the proof of Proposition 7 above).
If k is an integer than we will write from now on vk=22instead of u-4k. If x is*
* an element of
E2 we will denote u-4kx by vk=22x. We note that the periodicity generator 3 of*
* ß72EO2
projects to v9=22.
Theorem 9. There are elements vk=222 ß8k(EO2 ^ V (1)), k = 0, 1, 2, and vk=22f*
*f 2
ß8k+3(EO2 ^ V (1)), k = 0, 1, 2, 3, 4, 5, such that as a module over F9[v29=2, *
*fi] there is
an isomorphism
i *
* j
ß*(EO2 ^ V (1)) ~=F9[v29=2] F9[fi]=(fi5){1, v1=22, v2} F9[fi]=(fi2){ff,*
* . .,.v5=22ff} .
Remark_ We remark that additively ßk(EO2) is nontrivial and of dimension 1 over*
* F9 if
k 10m + 8k mod (72) with 0 m 4, 0 k 2, or if k = 10m + 8k + 3 if 0 *
*m 1
and 0 k 5. For all other k the homotopy group is trivial.
14 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
Proof. Because ff and fi are permanent cycles, the first possible non-trivial d*
*ifferential is d5
and it is determined by its value on the powers of v1=22. By using the long exa*
*ct homotopy
sequences associated to the defining cofibre sequences of V 0) and V (1) togeth*
*er with Theo-
rem 8 and Proposition 7b it is easy to verify that the elements 1, v1=22and v2 *
*are permanent
cycles.
Now E*,*r(V (1)) is a differential graded module over E*,*r(S0). This implies
æ 0 ifk = 0, 1, 2 mod 9
d5(vk=22) = k-3=2
ckv2 fffi2ifk = 3, 4, 5, 6, 7, 8 mod 9
for suitable nontrivial constants ck and therefore
E6 ~=F9[v29=2, fi]{1, v1=2, v2} F9[v29=2, fi]{v6=22ff, v7=22ff, v*
*8=22ff}
F9[v29=2, fi]=(fi2){ff, v1=22ff, v2ff, v3=22ff, v22ff, v5=22*
*ff} .
The next possible differential is d9 and by using the module structure again we*
* obtain
æ 0 ifk = 0, 1, 2, 3, 4, 5 mod 9
d9(vk=22ff) = k=2-3
c0kv2 fi5ifk = 6, 7, 8 mod 9
for nontrivial constants c0k. The resulting E10-term is isomorphic to the state*
*d result, and
in fact, there is no room for further differentials.
2. The homotopy fixed point spectrum EhN
2.1. The subgroups N and N1.
Next we introduce certain infinite closed subgroups of S2 which are closely rel*
*ated to the
subgroup G12which is used to define EO2. We refer to [H, section 3] for more de*
*tails on the
following discussion. The centralizer C := CS2(C3) of the subgroup C3 G12 gen*
*erated by
s can be identified with the maximal order of the units in the cyclotomic exten*
*sion Q3(i3) of
Q3 generated by a third of unity i3, and is hence isomorphic to C3x C2x Z23. Fu*
*rthermore
C is of index 2 in its normalizer N := NS2(C3). The action of the element n of*
* order 2
in N=C on C is via the Galois automorphism of the cyclotomic extension. The act*
*ion can
be diagonalized, i.e. the splitting of C3 x C2 x Z23. can be chosen to be inv*
*ariant with
respect to the action of n, and n acts trivially on C2 and on one copy of Z3 wh*
*ile it acts by
multiplication by -1 on the other copy and on C3.
Furthermore, there is a homomorphism from S2 ! Zx3! (Zx3)={ 1} given as the com*
*po-
sition of the reduced norm and the canonical projection. Its kernel is denoted *
*by S12, and
S2 decomposes as S12x Z3 where the complimentary factor Z3 comes from the cente*
*r of
the division algebra. There is a corresponding splitting N ~=N1 x Z3 and this s*
*plitting is
preserved by the action of n (where n acts trivially on the complementary facto*
*r Z3). We
observe that the subgroup G12 is contained in N1 and that N1 is a (nonsplit) ex*
*tension of
C2 by C1 := C3 x C2 x Z3 where n preserves the splitting of C1 and acts non-tri*
*vially on
the factors C3 and Z3.
In the sequel we will make use of the homotopy fixed point spectra EhN and EhN1*
* . These
spectra are closer to LK(2)S0 but we will see that they are also closely relate*
*d to EO2.
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 15
2.2. EhN , EhN1 and EO2.
The splitting N ~=N1 x Z3 implies the following result.
Theorem 10. There is a cofibration sequence
1 hN1
EhN ! EhN ! E
where the map EhN1 ! EhN1 is given by id - k if k denotes a topological generat*
*or of the
central Z3.
Proof. There is a canonical map f : EhN1 ! EhN1 induced by the inclusion N1 N.
Furthermore k induces a self map of EhN and (id - k) O f is null. This gives u*
*s a map g
from EhN to the fibre F of id - k. The one shows that g, or equivalently LK(2)*
*(g ^ idE )
is an equivalence. In fact, a slight modification of the argument in [DH2, Prop*
*. 7.1] allows
to identify ß*LK(2)(EhN ^ E) with map cts(G2=N, E*), the continuous maps from t*
*he coset
space G2=N to E*, and likewise ß*LK(2)(EhN1 ^E) with mapcts(G2=N1, E*). Then on*
*e sees
that the map id - k induces a surjection on ß*(LK(2)(- ^ E)) and g induces an i*
*somorphism
between map cts(G2=N, E*) and the kernel.
The spectrum EhN1 itself can be obtained from EO2 in a slightly more sophistica*
*ted fashion.
For this we consider the ring spectrum EhC6 = Eh(C3xC2). The group C6 is normal*
* in G12
and we obtain an induced action of the quotient G12=C6 ~=C2 on the ring spectru*
*m EhC6.
The spectrum EhC6 splits with respect to this action as E+ _ E- where E+ and E-*
* are the
"eigenspectraö f EhC6 with respect to the two non-trivial characters of C2. Fu*
*rthermore
E+ ' (EhC6)hC2 can be identified with EO2 and thus EhC6 and E- become EO2-module
spectra. The following elementary observation introduces a suspension which ma*
*y seem
surprising at first but which becomes very important for the sequel.
Proposition 11. There is an equivalence of EO2-module spectra
E- ' 36EO2 .
Proof. We have seen in section 1.3 above that EhC6 is a periodic ring spectrum *
*with pe-
riodicity generator 3=2of degree 36. Furthermore, the periodicity generator is*
* in the -1
eigenspace of the action of C2 on ß*(EhC6) and represents an element in ß36(E- *
*). Using
the structure of E- as a module spectrum it defines an equivalence between 36E*
*O2 and
E- .
We have other elements of order 2 acting on EhC6, e.g. all elements of order 2*
* in the
group N1=C6 ~=Z3 o C2. In particular, if k1 is a topological generator of Z3 a*
*nd if we
choose the image of t 2 G12 as generator of C2, then k1t is such an element. We*
* refer to
the corresponding eigenspectra of any element ø of order 2 as Eø . In particula*
*r we have
E+t= EO2, E-t= 36EO2.
Theorem 12.
a) There is a cofibration sequence
1 + -
EhN -! Et -! Ek1t
and the map between the eigenspectra is induced by id - k1 (on the level of EhC*
*6).
16 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
b) There is an equivalence
E-k1t' 36EO2 .
Proof. a) This follows the same strategy as the proof of Theorem 10. The map (i*
*d - k1)
induces a self map of EhC6 which we can easily check to induce a map E+t-! E-k1*
*t. Let F
be the fibre of this map. As before the canonical map f : EhN1 ! E+tbecomes nul*
*l after
composing it with id - k1 so that we obtain a map g : EhN1 ! F . This time we *
*get an
identification
ß*LK(2)(EhC6 ^ E) ~= mapcts(G2=C6, E*) ~=Hom cts(Z3[[G2=C6]], E*)
where for a profinite G2-set X = limffXffwith finite G2-sets Xffwe write Z3[[X]*
*] for
limff,nZ3=(3n)[Xff] and where Hom ctsdenotes continuous homomomorphisms. The el*
*ements
t and k1t act on the coset space and after linearization we can pass to the cor*
*responding
eigenspaces which we denote Hom t,ctsetc.
Now id - k1 induces as before a surjective map
Hom t,+cts(Z3[[G2=C6]], E*) ! Hom k1t,-cts(Z3[[G2=C6]], E*)
whose kernel gets identified via g with Hom cts(Z3[[G2=N1]], E*). This finishes*
* the proof of
(a).
b) This is an immediate consequence of Proposition 11 and the fact that the ele*
*ments t
and k1t are conjugate in N1 because 2 is a unit in Z3. Hence they have equivale*
*nt eigen-
spectra.
Remark_ Theorem 10 and Theorem 12 have the following discrete analogues which m*
*ay, in
particular in the case of Theorem 12, help to explain the situation.
It is well known that in the case of an action a discrete infinite cyclic group*
* C1 on a spectrum
(or suspension) X one can obtain XhC1 , up to homotopy as the fibre of the map *
*X ! X,
given by id - k and this fibration may be thought of as being induced from the *
*equivariant
skeletal filtration of the real line R thought of as universal C1 -space EC1 w*
*ith C1 acting
via translations. If we consider C1 o C2 as acting on R via translation and ref*
*lections and
if we ignore 2-primary phenomena then R is still a "reasonable model" for the u*
*niversal
C1 o C2 space E(C1 o C2). This time the 0 - simplices are the integral points *
*on the
real line with isotropy isomorphic to C2 and the isotropy group of a 1 - simple*
*x is C2 with
C2 acting nontrivially on the 1 - simplex. By using the skeletal filtration on*
*ce again the
homotopy fixed points Xh(C1 oC2)can, under suitable assumptions, be obtained as*
* the fibre
of a map as in Theorem 12.
Corollary 13. There is a cofibration sequence
1 36
EhN -! EO2 -! EO2 .
2.3. The homotopy groups of EhN ^ V (1) and of EhN1 ^ V (1).
It is not hard to see that the spectrum EhN11is no longer periodic. Nevertheles*
*s the following
lemma shows that the homotopy groups ßk(EhN ^ V (1)) remain 72-periodic.
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 17
Lemma 14. The map
ß*(EO2 ^ V (1)) ! ß*( 36EO2 ^ V (1))
which is induced by id - k1 is trivial.
Proof. Theorem 9 implies that if ßn(EO2^ V (1)) and ßn( 36EO2^ V (1)) are both *
*nonzero
then
n 2 {0, 10, 20, 36, 46, 56} mod 72 .
Because id-k1 commutes with the action of fi we see that the map is trivial if *
*n 36, 46, 56
and that it suffices to concentrate on the case n 0. Now the periodicity gene*
*rator v9=222
ß72(EO2 ^ V (1)) comes from 3 2 ß72(EO2) and therefore it suffices to show tha*
*t the
composition
ß72(EO2) ! ß72( 36EO2) ! ß72( 36EO2 ^ V (1)))
annihilates 3. In fact, we see from Proposition 7b and Theorem 8 that the ima*
*ge of
ß72( 36EO2) in ß72( 36EO2 ^ V (0)) is divisible by v1 and becomes therefore tri*
*vial in
ß72( 36EO2 ^ V (1))).
The lemma allows us to analyze the ANSS
1
Es,t2~=Hs(N1, E*V (1)) =) ßt-s(EhN ^ V (1)) .
Its E2-term is easily calculated to be
1 0 C4 1=2 -18*
* 0
H*(N1, E*V (1)) ~= F9[u , y] (x, a ) ~=F9[v2 , fi] (ff, u *
*a )
where as before ff = u-2x, fi = u-6y, v1=22= u-4 and the exterior generator a0 *
*is the
contribution of the factor Z3 in the centralizer C1 ~=C3 x C2 x Z3. Its bidegr*
*ee is (1, 0)
and the generator t of C4 acts on it by multiplication by -1. Therefore a35 := *
*u-18a0is a
new invariant class. We note that the elements x and y are a priori not canonic*
*ally defined,
not even up to a nontrivial constant because the corresponding groups are of ra*
*nk 2 over
F9. We1can and will choose them such that ff and fi detect the images of ff1 a*
*nd fi1 in
ß*(EhN2 ^ V (1) for example by defining them via Greek letter constructions in *
*H*(N, -).
Proposition 15.
a) The ANSS
1
Es,t2~=Hs(N1, E*V (1)) ~=F9[v21=2, fi] (ff){1, a35} =) ßt-s(EhN ^ V (*
*1))
splits as the direct sum of the ANSS for EO2^ V (1) and that of 35EO2^ V (1) (*
*where the
summand indexed by 1 corresponds to EO2 ^ V (1) and that by a35 to 35EO2 ^ V (*
*1)).
b) As modules over F9[v29=2, fi] there is an isomorphism
ß*(EhN1 ^ V (1)) ~=ß*(EO2 ^ V (1)){1, a35} .
Remark_ We emphasize that the module structure over v29=2 is (at least at this *
*point) a
purely algebraic accident induced by an algebraic module structure on the level*
* of E2-terms.
18 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
Proof. The fibration sequence
1 36
EhN ^ V (1) ! EO2 ^ V (1) ! EO2 ^ V (1)
induces an exact sequence
1 36
0 ! E*(EhN ^ V (1)) ! E*(EO2 ^ V (1)) ! E*( EO2 ^ V (1)) ! 0
where E*X has to be interpreted as ß*(LK(2)(E ^ X)). In fact, as in the proof o*
*f Theorem
12 this sequence can be identified with the sequence
0 ! Hom cts(Z3[[G2=N1]],E*V (1)) ! Hom t,+cts(Z3[[G2=C6]], E*V (1))
! Hom k1t,-cts(Z3[[G2=C6]], E*V (1)) ! 0 .
In cohomology (i.e. on E2-terms of the relevant ANSS) this sequence induces sho*
*rt exact
sequences for all s 0
0 ! Hs-1(G12, Et( 36V (1))) ! Hs(N1, EtV (1)) ! Hs(G12, Et(V (1)) ! 0
where the monomorphism converges towards the map
1
ßt-s( 35EO2 ^ V (1)) ! ßt-s(EhN ^ V (1))
by the geometric boundary theorem and the epimorphism converges towards the map
1
ßt-s(EhN ^ V (1)) ! ßt-s(EO2 ^ V (1)) .
by naturality. The proposition follows.
Now we turn towards EhN ^ V (1) and consider the ANSS spectral sequence
Es,t2~=Hs(N, EtV (1)) =) ßt-s(EhN ^ V (1)) .
The E2 -term of the SS is easily calculated to be
1 0 C4 1=2
H*(N, E*V (1)) ~= F9[u , y] (x, a , i) ~=F9[v2 , fi] (ff, a*
*35, i)
where the new exterior generator i is the contribution of the central factor Z3*
* in the cen-
tralizer C. Its bidegree is (1, 0) and it is fixed by the action of t.
Proposition 16.
a) The ANSS
Es,t2~=Hs(N, EtV (1)) ~=F9[v21=2, fi] (ff, i){1, a35} =) ßt-s(EhN ^ V *
*(1))
splits as the direct sum of the ANSS of EhN1 ^ V (1) and of -1EhN1 ^ V (1) (wh*
*ere the
summand indexed by 1 corresponds to EhN12^ V (1) and that by i to -1EhN12^ V (*
*1).)
b) As modules over F9[v29=2, fi] there is an isomorphism
1
ß*(EhN2 ^ V (1)) ~=ß*(EO2 ^ V (1)){1, a35, i, ia35} .
Remark_ We emphasize that as before the module structure over v29=2 is (at leas*
*t at this
point) a purely algebraic accident on the level of E1 - terms.
Proof. As before the proof will be an easy consequence of the following result *
*(which is
analogous to Lemma 14) whose proof will, however, make use of the structure of *
*the SS
considered in Proposition 16.
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 19
Lemma 17. The map
1 hN1
(id - k)* : ß*(EhN ^ V (1)) ! ß*(E ^ V (1))
is trivial.
Proof. The action of k on Hs(N1, EtV (1)) is trivial except perhaps on u. The g*
*enerator k
of the central Z3 Dx2is necessarily congruent 1 mod 3 (for example, one can t*
*ake k = 4).
Then u 2 ß-2E gets multiplied with k 1 mod 3. Therefore the action1of id - k *
*is trivial
on the E2 - term of the ANSS. This shows that the action on ßk(EhN ^ V (1)) is*
* trivial
except possibly in degrees
n 0, 3, 8, 11, 16, 19, 35, 38, 43, 45, 46, 48, 53, 56 mod 72
where the total degree n of the E1 - term of the SS is made of two copies of F*
*9. Degrees
35 and 43 resp. 45 and 53 can be excluded from the list because both copies hav*
*e the same
filtration (= 1 resp. 3). Next the action of ff and fi and the compatibility *
*of k with the
fibration sequence of Theorem 12 resp. Corollary 13 imply that degrees 38, 46, *
*48 and 56
can also be excluded. Similarly the action of ff and fi show that it is enough *
*to consider the
cases n 0, 8, 16 mod 72. If id - k acts nontrivially in one of these dimensio*
*ns then there
exists some integer p and there exists q 2 {0, 1, 2} such that
(id - k)*(v9p+q=22) = cv(9(p-1)+q+3)=22fiffa35
for some nontrivial constant c. This implies then that in the ANSS for EhN ^ V*
* (1) the
element v(9(p-1)+q+3)=22fiffia35 (which is a permanent cycle by Proposition 14 *
*and the fact
that i detects a homotopy class which comes from LK(2)S0) does not survive and *
*hence
that it is in the image of a differential. At this point we turn attention towa*
*rds the analysis
of the SS to show that htis cannot happen.
In the ANSS converging to ß*(EhN ^ V (1)) the elements ff, fi and i come from t*
*he sphere
(or at least from the K(2)-local sphere). Furthermore we know from [Ra2, Table *
*A3.4] that
the Greek letter element fi6=32 Ext2,84BP*BP(BP *, BP*) is a permanent cycle in*
* the ANSS
for S0. By [Sh1, Lemma 2.4] and Corollary 19 below this element is detected in *
*E2,842in
our SS and therefore agrees with v9=22fi, up to a nontrivial constant. Because *
*E2 is free over
F9[fi] we deduce that the first differential is linear with respect to v9=22.
So for the first differential we need to study the elements
vr=22, a35vr=22ifr 0, 1, . .,.8 mod 9 .
Degree reasons (i.e. calculating modulo total degree 8) shows that the first po*
*ssibility for a
differential is d5. The possible targets are as follows:
o d5(vr=22) is a linear combination of v(r-3)=22fffi2, v(r-7)=22fi2a35 and v(r-*
*6)=22fffiia35.
o d5(a35vr=22) is a multiple of v(r-3)=22fffi2a35.
Now we use that 1, v2 and v52are permanent cycles coming from V (1) (see [Sh1, *
*Thm 2.6]).
This implies that for r 0, 1, 2 mod 9 we have
d5(vr=22) = 0 ,
in particular v(9(p-1)+l+3)=22fiffia35is not in the image of d5, hence it survi*
*ves and the proof
of the lemma is complete.
20 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
3. The homotopy of L2V (1)
In this section we will calculate the homotopy of L2V (1) ' EhG2 ^ V (1) by com*
*paring it to
that of EhN2^ V (1). The E2-term of the ANSS converging to ß*L2V (1) is isomorp*
*hic to
H*(G2, F9[u 1]) ~=(H*(S2, F9[u 1])C2
where C2 acts via conjugation on S2 and via Frobenius on F9 (hence the action i*
*s free and the
spectral sequence of the extension S2 ! G2 ! C2 degenerates into the stated iso*
*morphism).
Furthermore there is a canonical isomorphism
x
H*(S2, F9[u 1]) ~=(H*(S2, F9) F9F9[u 1])F9
where S2 is the 3-Sylow subgroup of S2 and acts trivially, and the invariants a*
*re taken with
respect to the action of the quotient S2=S2 which can be naturally identified w*
*ith Fx9gen-
erated by !. The generator ! of Fx9acts diagonally on this tensor product, via *
*conjugation
on S2 and via multiplication with ! on u, so that taking invariants amounts to *
*taking the
eigenspace decomposition of H*(S2, F9) with respect to the action of ! (determi*
*ned im-
plicitly by Theorem 18 below) and tensoring the eigenspace E!i (with eigenvalue*
* !i) with
powers u-i+8k to get invariants.
In [H, Prop. 3.4 and Thm. 4.2] H*(S2, F3) was studied via the restriction map*
* to the
centralizers CS2(Ei) ~= C3 x Z23, i = 1, 2, where the Ei are representatives of*
* the two
different conjugacy classes of C3's in S2. We can choose E1 to be the subgroup *
*generated
by s 2 G12 and then E2 can be choosen to be !-1E1! so that the restriction map
Y2 Y2
H*(S2, F3) ! H*(CS2(Ei), F3) ~= F3[yi] (xi, ii, ai0)
i=1 i=1
becomes Fx9-equivariant where the Fx9-action on the right is induced from the c*
*onjugation
action of NS2(Ei)=CS2(Ei) ~=C4 Fx9. We note that t 2 G12 NS2(E1) projects *
*to a
generator in C4. We can choose the cohomology classes such that yi and xi corre*
*spond to
the generators of the cohomology of the cyclic subgroup, iito the cohomology of*
* the central
factor Z3, and ai0to that of the noncentral factor Z3 on which t acts by multip*
*lication by
-1. This notation differs somewhat from that in [H] but is consistent with our *
*notation in
section 2.
Theorem 18 [H].
a) The restriction map
Y2 Y2
H*(S2, F3) ! H*(CS2(Ei), F3) ~= F3[yi] (xi, ii, ai0)
i=1 i=1
is an Fx9-invariant monomorphism whose image is the F3-subalgebra generated by *
*x1, x2,
y1, y2, i1 + i2, x1a10- x2a20, y1a10and y2a20.
b) In particular H*(S2, F3) is a free module over F3[y1+ y2] (i1+ i2) genera*
*ted by 1, x1,
x2, y1, x1a10- x2a20, y1a10, y2a20and y1x1a10.
We note that a priori the elements xi, yiand ai0are not canonical (because they*
* depend on
the chosen decomposition of CS2(E1)). The theorem implies, however, that xi and*
* yi (as
the Bockstein of xi) are distinguished (at least up to nontrivial constants).
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 21
Next we describe the invariants of H*(S2, F9[u 1]) with respect to the action o*
*f Fx9(which
is determined by !*(u) = !u). By using the Fx9-linear monomorphism these invari*
*ants can
be identified with a subring of
Y2 x
( H*(CS2(Ei), F9[u 1])F9 ~=(H*(CS2(E1), F9[u 1])C4 ~=H*(NS2(E1), F9[u 1])
i=1
where C4 is as before generated by t 2 G12. Its action on H*(CS2(E1), F3) is gi*
*ven by
t*(y) = -y, t*(x) = -x, t*(a0) = -a0, t*(i) = i
(where we have omitted the indices for simplicity). The following corollary is *
*now straight-
forward to verify. As in section 1.4. and chapter 2 we write vk=22for u-4k.
Corollary 19.
a) The restriction map
1 0 C4
H*(S2, F9[u 1]) ! H*(N, F9[u 1]) ~= F9[u , y] (x, i, a )
is a monomorphism. Its target is isomorphic to
F9[v21=2, fi] (ff, i, a35)
(with fi = u-6y, ff = u-2x, i and a35 = u-18a0) and its image is the F9-subalge*
*bra of
F9[v21=2, fi] (ff, i, a35) generated by v21, ff, v1=22ff, fi, v1=22fi, i, ffa3*
*5, fia35 and v1=22fia35.
b) In particular H*(S2; F9[u 1]) is the free F9[v21, fi] (i)-submodule of F9*
*[v21=2, fi]
(ff, i, a35) generated by 1, ff, v1=22ff, v1=22fi, ffa35 fia35, v1=22fia35, an*
*d v1=22fiffa35.
The restriction map above is the comparison map at the E2-level between the two*
* ANSS
converging to ßt-s(EhS2 ^ V (1)) resp. ßt-s(EhN ^ V (1)). We still have to deal*
* with the
Galois action of C2 if we want to get at L2V (1).
Now the Galois generator OE 2 C2 acts on S2 Dx2by conjugation by S, hence it *
*is clear
that !OE centralizes s = -1_2(1 + !S) and thus everything in the commutative su*
*bfield of
D2 generated by s. In particular !OE commutes with the units in the maximal ord*
*er of this
subfield, i.e. with CS2(E1). Therefore the group CG2(E1) (which is generated by*
* CS2(E1)
and !OE) is an abelian group and !OE acts trivially on H*(CS2(E1), F3). The act*
*ion on the
coefficient ring F9[u 1] is given by (!OE)*(cuk) = OE(c)!kuk if c 2 F9.
The monomorphism of Theorem 18 (with coefficients extended to F9[u 1]) is actua*
*lly linear
even with respect to Fx9o C2 where the Galois generator OE acts on the target o*
*n the level
of groups by conjugation by S while it acts on F9[u 1] by Frobenius again. The *
*Fx9o C2-
invariants in the target of this monomorphism can therefore be identified with
Y2 * 1 FxoC2 * 1 C
H (CS2(Ei), F9[u ] 9 ~= H (CS2(E1), F9[u ]) 4 .
i=1
The preceding two paragraphs show that we can modify (if necessary) the element*
*s v1=22,
fi, ff, i and a35 of Corollary 19 by scalars in F9 so that they become invarian*
*t with respect
22 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
to the action of !OE. After having done this H*(G2, F9[u 1]) can be identified*
* with the
F3-subalgebra of F9[v1=22, fi] (ff, i, a35) generated by elements with the s*
*ame name as in
Corollary 19a and H*(G2, F9[u 1]) is a free module over F3[v21, fi] (i) on e*
*lements with
the same name as in Corollary 19b.
Now we are ready to compare the differentials in the two ANSS converging to L2V*
* (1) '
EhG22^ V (1) resp. EhN2^ V (1). We refer to them as the source SS resp. the tar*
*get SS.
For the target SS we deduce from section 2.3 that v29=2, fi, ff, i and a35 are *
*permanent
cycles and that the differentials are linear with respect to the algebra
R := F9[v29=2, fi] (ff, i, a35) .
To better compare with the spectral sequence of the source we should consider t*
*his SS as
one of modules over S := F9[v29, fi] (ff, i, a35). In fact, the E2-term of t*
*he target SS is
a free module over S on generators vk=22, k = 0, . .,.17.
We consider the E2-term of the ANSS of the source as a free module over
P := F3[v29, fi] (i)
(note that here we have taken the prime field) on the following generators (whe*
*re l =
0, 1, . .,.8):
o vl2, vl+1=22fi, vl2fia35, vl+1=22fia35
o vl2ff, vl+1=22ff, vl2ffa35, vl+1=22fiffa35
We have seen in section 1 and 2 that the first differential for the target SS i*
*s d5. It is
determined by
æ 0 ifk 0, 1, 2 mod 9
d5(vk=22) = (k-3)=2
ckv2 fffi2ifk 3, . .,.8 mod 9 .
This implies that the first differential of the source SS is also d5. Furtherm*
*ore by the
discussion above the nontrivial constants ck have to be in F3 and d5 in the sou*
*rce SS is
given by
æ 0 ifl 0, 1, 5 mod 9
d5(vl2) = l-2+1=2
v2 fffi2 ifl 2, 3, 4, 6, 7, 8 mod 9
æ 0 ifl 0, 4, 5 mod 9
d5(vl+1=22fi) = l-1
v2 fffi3 ifl 1, 2, 3, 6, 7, 8 mod 9
æ 0 ifl 0, 1, 5 mod 9
d5(vl2fia35) = l-2+1=2
v2 fffi3a35 ifl 2, 3, 4, 6, 7, 8 mod 9
æ 0 ifl 0, 4, 5 mod 9
d5(vl+1=22fia35) = l-1
v2 fffi3a35 ifl 1, 2, 3, 6, 7, 8 mod 9
d5(vl2ff) = d5(vl+1=22ff)i=f0l = 0, . .,.8 mod 9
d5(vl2ffa35) = d5(vl+1=22fiffa35)i=f0l = 0, . .,.8 mod 9 .
THE HOMOTOPY OF L2V (1) FOR THE PRIME 3 23
This yields the following E6-term (which is already presented in a form which i*
*s adapted to
the discussion of the next differential):
E6 ~=P {vl2}l=0,1,5 P {vl2ff}l=3,4,8 P=(fi3){vl2ff}l=0,1,2,5,6,7
P {vl+1=22fi}l=0,4,5 P {vl+1=22ff}l=3,7,8 P=(fi2){vl+1=22ff}l=0,1,2,4*
*,5,6
P {vl2fia35}l=0,1,5 P {vl2ffa35}l=3,4,8 P=(fi3){vl2ffa35}l=0,1,2,5,6,7
P {vl+1=22fia35}l=0,4,5 P {vl+1=22fiffa35}l=3,7,8 P=(fi2){vl+1=22fiff*
*a35}l=0,1,2,4,5,6
We know that the next differential in the target SS is d9 and is determined by
æ 0 ifk = 0, 1, 2, 3, 4, 5 mod 9
d9(vk=22ff) = k=2-3
c0kv2 fi5ifk = 6, 7, 8 mod 9
for suitable nontrivial constants c0k. Again by comparing with the spectral seq*
*uence of the
target we deduce that the next differential in the source SS is also d9, the co*
*nstants have to
be in F3 and d9 in the source SS is given by
æ 0 ifl 0, 1, 2, 5, 6, 7 mod 9
d9(vl2ff) = l-3
v2 fi5 ifl 3, 4, 8 mod 9
æ 0 ifl 0, 1, 2, 4, 5, 6 mod 9
d9(vl+1=22ff) = l-3+1=2
v2 fi5 ifl 3, 7, 8 mod 9
æ 0 ifl 0, 1, 2, 5, 6, 7 mod 9
d9(vl2ffa35) = l-3
v2 fi5a35 ifl 3, 4, 8 mod 9
æ 0 ifl 0, 1, 2, 4, 5, 6 mod 9
d9(vl+1=22fiffa35) = l-3+1=2
v2 fi6a35 ifl 3, 7, 8 mod 9
and we obtain the following E10-term
E10~=P=(fi5){vl2}l=0,1,5 P=(fi3){vl2ff}l=0,1,2,5,6,7
P=(fi4){vl+1=22fi}l=0,4,5 P=(fi2){vl+1=22ff}l=0,1,2,4,5,6
P=(fi4){vl2fia35}l=0,1,5 P=(fi3){vl2ffa35}l=0,1,2,5,6,7
P=(fi5){vl+1=22fia35}l=0,4,5 P=(fi2){vl+1=22fiffa35}l=0,1,2,4,*
*5,6.
At this point there is no more room for further nontrivial differentials and we*
* arrive at the
desired result below in which the names of the generators are chosen so as to d*
*escribe their
image in the E2-term of the ANSS for ß*(EhN2^ V (1)).
Theorem 20. As a module over P = F3[v29, fi] (i) there is an isomorphism
ß*L2V (1) ~=P=(fi5){vl2}l=0,1,5 P=(fi3){vl2ff}l=0,1,2,5,6,7
P=(fi4){vl+1=22fi}l=0,4,5 P=(fi2){vl+1=22ff}l=0,1,2,4,5,6
P=(fi4){vl2fia35}l=0,1,5 P=(fi3){vl2ffa35}l=0,1,2,5,6,7
P=(fi5){vl+1=22fia35}l=0,4,5 P=(fi2){vl+1=22fiffa35}l=0,1,2,4*
*,5,6.
We finish by observing that this matches with Shimomura's result (if his parame*
*ter is taken
to be k = 1!).
24 PAUL GOERSS, HANS-WERNER HENN AND MARK MAHOWALD
References
[D] P. Deligne, Courbes elliptiques: Formulaire (d'apr`es J. Tate), in: Modul*
*ar Functions of One Variable
IV, Lecture Notes in Math 476 (1975), Springer Verlag.
[DH1] E. Devinatz and M. Hopkins, The action of the Morava stabilizer group on *
*the Lubin-Tate moduli
space of lifts, Amer. J. Math. 117 (1995), 669-710.
[DH2] E. Devinatz and M. Hopkins, Homotopy fixed point spectra for closed subgr*
*oups of the Morava
stabilizer groups, preprint.
[GS] V. Gorbounov and P. Symonds, Towards the homotopy groups of the higher re*
*al K theory EO2,
Homotopy theory via algebraic geometry and group representations. Contemp*
*. Math. 220 (1998),
103-115.
[GSS] V. Gorbounov, S. Siegel and P. Symonds, Cohomology of the Morava stabiliz*
*er group S2 at the
prime 3, Proc. Amer. Math. Soc. 126 (1998), 933-941.
[H] H.-W. Henn, Centralizers of elementary abelian p-subgroups and mod-p coho*
*mology of profinite
groups, Duke Math. J. 91 (1998), 561-585.
[LT] J. Lubin and J. Tate, Formal moduli for one-parameter formel Lie groups, *
*Bull. Soc. Math. France
94 (1966), 49-60.
[N] L. Nave, On the nonexistence of Smith-Toda complexes, preprint (http://ww*
*w-math.mit.edu/ nave/
papers/).
[Ra1] D. Ravenel, The Cohomology of the Morava Stabilizer Algebras, Math. Zeit.*
* 152 (1977), 287-297.
[Ra2] D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Acad*
*emic Press, 1986.
[Ra3] D. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of*
* Math. Studies, Prince-
ton University Press, 1992.
[Re] C. Rezk, Notes on the Hopkins-Miller theorem, Homotopy theory via algebra*
*ic geometry and group
representations. Contemp. Math. 220 (1998), 313-366.
[Sh1] K. Shimomura, The homotopy groups of the L2-localized Toda-Smith complex *
*V (1) at the prime 3,
Trans. Amer. Math. Soc. 349 (1997), 1821-1850.
[Sh2] K. Shimomura, The homotopy groups of the L2-localized mod-3 Moore spectru*
*m at the prime 3, J.
Math. Soc. Japan 52 (2000), 65-90.
[SY] K. Shimomura and A. Yabe, The homotopy groups i*(L2S0), Topology 34 (1995*
*), 261-289.
[SW] K. Shimomura and X. Wang, The homotopy groups i*(L2S0) at the prime 3, pr*
*eprint.
[St] N. Strickland, Gross-Hopkins duality, Topology 39 (2000), 1021-1033.
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evansto*
*n, IL,
60208-2730, U.S.A.
Institut de Recherche Math'ematique Avanc'ee, C.N.R.S. - Universit'e Louis Past*
*eur, 7 rue
Ren'e Descartes, F-67084 Strasbourg, France
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evansto*
*n, IL,
60208-2730, U.S.A.