ON THE 2-ADIC K-LOCALIZATIONS OF H-SPACES
A.K. BOUSFIELD
Abstract.We determine the 2-adic K-localizations for a large class of H-
spaces and related spaces. As in the odd primary case, these localizations
are expressed as fibers of maps between specified infinite loop spaces, a*
*llow-
ing us to approach the 2-primary v1-periodic homotopy groups of our space*
*s.
The present v1-periodic results have been applied very successfully to si*
*mply-
connected compact Lie groups by Davis, using knowledge of the complex,
real, and quaternionic representations of the groups. We also functoriall*
*y de-
termine the united 2-adic K-cohomology algebras (including the 2-adic KO-
cohomology algebras) for all simply-connected compact Lie groups in terms*
* of
their representation theories, and we show the existence of spaces realiz*
*ing a
wide class of united 2-adic K-cohomology algebras with specified operatio*
*ns.
1.Introduction
In [21], Mahowald and Thompson determined the p-adic K-localizations of the o*
*dd spheres at
an arbitrary prime p, expressing these localizations as homotopy fibers of maps*
* between specified
infinite loop spaces. Then working at an odd prime p in [8], we generalized th*
*is result to give
the p-adic K-localizations for a large class of H-spaces and related spaces. In*
* the present paper,
we obtain similar results for 2-adic K-localizations of such spaces, using our *
*preparatory work in
[10] and [11]. By a 2-adic K-localization, we mean a K=2*-localization (see [2]*
*, [3]), which is the
same as a K*(-; ^Z2)-localization since the K=2*-equivalences of spaces or spec*
*tra are the same as
the K*(-; ^Z2)-equivalences. Our localization results in this paper will apply *
*to many (but not all)
simply-connected finite H-spaces and to related spaces such as the spheres S4k-*
*1for k 1. We
show that these results allow computations of the v1-periodic homotopy groups (*
*see [13], [15]) of
our spaces from their united 2-adic K-cohomologies, and thus allow computations*
* of the v1-periodic
homotopy groups for a large class of simply-connected compact Lie groups from t*
*heir complex,
real, and quaternionic representation theories. The present results will be ext*
*ended in a subsequent
paper to cover the remaining simply-connected compact Lie groups and various sp*
*aces related to
the remaining odd spheres. This work has been applied very successfully by Davi*
*s [14] to complete
___________
Date: November 22, 2006.
2000 Mathematics Subject Classification. 55N15, 55P60, 55Q51, 55S25.
Key words and phrases. K-localizations, v1-periodic homotopy, 2-adic K-theor*
*y, united K-
theory, compact Lie groups.
1
2
his 13-year program (with Bendersky) of calculating the v1-periodic homotopy gr*
*oups of all simply-
connected compact Lie groups, and has also been applied by Bendersky, Davis, an*
*d Mahowald [1].
Throughout this paper, we work at the prime 2 and rely on the united 2-adic K*
*-cohomology
K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)}
of a space or spectrum X as in [10]. This combines the usual periodic cohomolo*
*gies with the
operations between them such as complexification and realification. For our H-s*
*paces and related
spaces X, the cohomology K*CR(X; ^Z2) is essentially determined by the 2-adic A*
*dams -module
eK-1(X; ^Z2) = {Ke-1(X; ^Z2), gKO-1(X; ^Z2), gKO-5(X; ^Z2)}
which combines the specified cohomologies with the additive operations among th*
*em (see 6.1). In
fact, for most simply-connected finite H-spaces X, we expect to have an isomorp*
*hism K*CR(X; ^Z2) ~=
^L(M) where M = {MC, MR, MH } is the submodule of primitives in Ke-1(X; ^Z2) an*
*d where ^L
is a functor that we introduce in 4.5 extending the 2-adic exterior algebra fun*
*ctor on complex
components. For a simply-connected compact Lie group G, the required 2-adic Ada*
*ms -module
may be obtained as the indecomposables ^QR G = {Q^RG, ^QRRG, ^QRH G} of the co*
*mplex, real,
and quaternionic representation ring R G = {RG, RRG, RH G} (see 10.1), and we *
*have:
Theorem 1.1. For a simply-connected compact Lie group G, there is a natural iso*
*morphism K*CR(G; ^Z2) ~=
^L(Q^R G) of algebras.
This will follow by Theorem 10.3. It extends results of Hodgkin [18], Seymour*
* [24], Minami [22],
and others on K*(G; ^Z2) and KO*(G; ^Z2). Our main result on K=2*-localization*
*s will apply to
a space X with K*CR(X; ^Z2) ~=^LM for a 2-adic Adams -module M that is strong *
*(7.11). This
technical algebraic condition seems relatively mild and holds for ^QR G when G*
* is a simply-connected
compact simple Lie group other_than_E6 or Spin(4k + 2) with k not a 2-power by *
*work of Davis
(see Lemma 10.5). For a strong 2-adic Adams -module M, we obtain two stable 2-*
*adic Adams
-modules (5.3) M~= {M~C, ~MR, ~MH} and ~ae~M= {M~C, ~MR+ ~MH, ~MR\ ~MH} where *
*M~C= MC,
~MR= im(MR ! MC), and M~H = im(MH ! MC); and we obtain two corresponding K=2*-l*
*ocal
spectra EM~ and Ea~e~Msuch that K-1(EM~; ^Z2) = ~M, K0(EM~; ^Z2) = 0, K-1(Ea~e~*
*M; ^Z2) = ~ae~M, and
K0(Ea~e~M; ^Z2) = 0 (see 8.1). Stated briefly, our main localization result is:
Theorem 1.2. If X is a connected space with K*CR(X; ^Z2) ~=^LM for a strong 2-a*
*dic Adams -
module M, then its K=2*-localization XK=2 is the homotopy fiber of a map from *
*1 EM~ to 1 Ea~e~M
with low dimensional modifications.
3
This will follow from Theorem 8.6. It will apply to simply-connected compact *
*simple Lie groups
with the above-mentioned exceptions, and it should apply to many other simply-c*
*onnected finite
H-spaces and related spaces; in fact, there must exist a great diversity of spa*
*ces with the required
united 2-adic K-cohomology algebras by:
Theorem 1.3. For each strong 2-adic Adams -module M, there exists a simply-con*
*nected space
X with K*CR(X; ^Z2) ~=^LM.
This will follow from Theorem 8.5. For our spaces X, we also obtain results o*
*n the 2-primary
v1-periodic homotopy groups v-11ss*X, which are naturally isomorphic to stable *
*homotopy groups
ss*o2 1X, where o2 1X is the 2-torsion part of the spectrum 1X obtained using *
*the v1-stabilization
functor 1 constructed in [4], [9], [16], and [19]. From this standpoint, the h*
*omotopy v-11ss*X is
essentially determined by the cohomology KO*( 1X; ^Z2) since there is an exact *
*sequence
3-9
. .-.! KOn-3( 1X; ^Z2) _---!KOn-3( 1X; ^Z2) -! (v-11ssnX)#
3-9
-! KOn-2( 1X; ^Z2) _---!KOn-2( 1X; ^Z2) -! . . .
where (-)# gives the Pontrjagin dual (see Theorem 9.2). A space X is called K=2*
**-durable (9.3)
when the K=2*-localization induces an isomorphism v-11ss*X ~=v-11ss*XK=2 or equ*
*ivalently 1X '
1XK=2. This condition holds for all connected H-spaces (and many other spaces)*
*, and our K=2*-
localization result implies:
Theorem 1.4. If X is a connected K=2*-durable space (e.g. H-space) with K*CR(X;*
* ^Z2) ~=^LM for a
strong 2-adic Adams -module M, then there is a (co)fiber sequence of spectra *
*1X ! EM~ ! Ea~e~M
with a KO*(-; ^Z2) cohomology exact sequence
2
0 -! KO-8( 1X; ^Z2) -! ~MC=(M~R+M~H) -~--!~MC=M~R -! KO-7( 1X; ^Z2) -! 0
2
-! ~MH=(M~R\M~H) -! KO-6( 1X; ^Z2) -! ~MR\M~H -~--!~MH-! KO-5( 1X; ^Z2)
2
-! 0 -! 0 -! KO-4( 1X; ^Z2) -! ~MC=(M~R \ ~MH) -~--!~MC=M~H
2
-! KO-3( 1X; ^Z2) -! (M~R + ~MH)=(M~R \ ~MH) -~--!~MR=(M~R \ ~MH)
2
-! KO-2( 1X; ^Z2) -! ~MR+ ~MH-~--!~MR-! KO-1( 1X; ^Z2) -! 0
This will follow from Theorem 9.5. It allows effective computations of 2-pri*
*mary v1-periodic
homotopy groups as shown by Davis [14], and its complex analogue implies that o*
*ur spaces X are
usually bK 1-good (9.6), which means that ^QKn(X; ^Z2)=~2 ~=Kn( 1X; ^Z2) for n *
*= -1, 0.
4
Theorem 1.5. If X is as in Theorem 1.4 with ~2: MC ! MC monic, then X is bK 1-g*
*ood.
This will be used in a subsequent paper to show that all simply-connected com*
*pact Lie groups
(and many other spaces) are bK 1-good, which is useful because the v1-periodic *
*homotopy groups
of bK 1-good spaces are often accessible by [10], even when our K=2*-localizati*
*on theorems do not
apply.
Throughout the paper, spaces and spectra will belong to the usual pointed sim*
*plicial or CW
homotopy categories. To provide a suitably precise setting for our main theorem*
*s and proofs, we
must devote considerable attention to developing the algebraic infrastructure o*
*f united 2-adic K-
cohomology theory. The paper is divided into the following sections:
1. Introduction
2. The united 2-adic K-cohomologies of spectra and spaces
3. The 2-adic OECR-algebras
4. The universal 2-adic OECR-algebra functor ^L
5. Stable 2-adic Adams operations and K=2*-local spectra
6. On the K=2*-localizations of infinite loop spaces
7. Strong 2-adic Adams -modules
8. On the K=2*-localizations of our spaces
9. On the v1-periodic homotopy groups of our spaces
10.Applications to simply-connected compact Lie groups
11.Proofs of basic lemmas for ^L
12.Proof of the Bott exactness lemma for ^L
13.Proofs for regular modules
14.Proof of the realizability theorem for ^LM
Although we have long been interested in the K-localizations and v1-periodic *
*homotopy groups
of spaces, we were prompted to develop the present results by Martin Bendersky *
*and Don Davis.
We thank them for their questions and comments.
2.The united 2-adic K-cohomologies of spectra and spaces
We now consider the united 2-adic K-cohomologies
K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)}
5
of spectra and spaces X, focusing on their basic structures as 2-adic CR-module*
*s or CR-algebras.
We first recall:
2.1. The 2-adic CR-modules. By a 2-adic CR-module, we mean a CR-module over the*
* category
of 2-profinite abelian groups (see [10, 4.1]). Thus, a 2-adic CR-module M = {MC*
*, MR} consists of
Z-graded 2-profinite abelian groups MC and MR with continuous additive operatio*
*ns
B :M*C~=M*-2C, t: M*C~=M*C, BR :M*R~=M*-8R,
j :M*R! M*-1R, c: M*R! M*C, r: M*C! M*R,
satisfying the relations
2j = 0, j3 = 0, jBR = BRj, jr = 0, cj = 0,
t2 = 1, tB = -Bt, rt = r, tc = c, cBR = B4c,
rB4 = BRr, cr = 1 + t, rc = 2, rBc = j2, rB-1c = 0.
For z 2 M*Cand x 2 M*R, the elements tz 2 M*Cand rB2cx 2 M*Rare sometimes writt*
*en as z* (or
_-1z) and ,x. For a spectrum or space X, the united 2-adic K-cohomology
K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)}
has a natural 2-adic CR-module structure with the usual periodicities B :K*(X; *
*^Z2) ~=K*-2(X; ^Z2),
and BR :KO*(X; ^Z2) ~=KO*-8(X; ^Z2), conjugation t: K*(X; ^Z2) ~=K*(X; ^Z2), Ho*
*pf operation
j :KO*(X; ^Z2) ! KO*-1(X; ^Z2), complexification c: KO*(X; ^Z2) ! K*(X; ^Z2), a*
*nd realification
r: K*(X; ^Z2) ! KO*(X; ^Z2).
2.2. Bott exactness. As in [10, 4.1], we say that a 2-adic CR-module M is Bott *
*exact when the
Bott sequence
-1 j
. .-.! M*+1R-j-!M*R-c-!M*CrB---!M*+2R--!. . .
is exact, and we note that the 2-adic CR-module K*CR(X; ^Z2) is always Bott exa*
*ct for a spectrum
or space X. To compare CR-modules, we shall often use:
Lemma 2.3. For Bott exact 2-adic CR-modules M and N, a map f :M ! N is an isomo*
*rphism
if and only if f :MC ! NC is an isomorphism.
Proof.For the "if" part, we note that f gives a map of Bott exact couples with *
*f :MC ~=NC. Using
the map of second derived couples with f :M(2)C~=N(2)C, we easily see that f :j*
*2MR ~=j2NR; then
using the map of first derived couples with f :M(1)C~=N(1)C, we easily see that*
* f :jMR ~=jNR; and
finally using the original map of exact couples, we easily see that f :MR ~=NR.
6
2.4. The free 2-adic CR-modules. For each integer n and L = C, R, there is a mo*
*nogenic free
2-adic CR-module FL (g, n) on a generator g 2 FL (g, n)nLhaving the universal p*
*roperty that, for
each 2-adic CR-module M and y 2 NnL, there is a unique map f :FL (g, n) ! M wit*
*h f(g) = y.
The 2-adic CR modules FC (g, n) and FR (g, n) are given more explicitly by: FC *
*(g, n)n-2iC= ^Z2
^Z2= , FC (g, n)n-2i-1C= 0, FC (g, n)n-2iR= ^Z2= , FC (g, n)n*
*-2i-1R= 0,
FR (g, n)n-2iC= ^Z2 = , FR (g, n)n-2i-1C= 0, FR (g, n)n-8iR= ^Z2 = *
* ,
FR (g, n)n-8i-1R= Z=2 = , FR (g, n)n-8i-2R= Z=2 = , FR (g, n)n-8*
*i-4R= ^Z2= ,
and FR (g, n)n-8i-kR= 0 fork = 3, 5, 6, 7. We note that FC (g, n) and FR (g,*
* n) are Bott exact
for all n. In general, a free 2-adic CR-module on a finite set of generators ma*
*y be constructed as
a direct sum of the corresponding monogenic free 2-adic CR-modules. To test for*
* this freeness, we
may use:
Lemma 2.5. For a Bott exact 2-adic CR-module M (e.g. for some M = K*CR(X; ^Z2)*
*), if M*C
is a free module over ^K*= ^Z2[B, B-1] on the generators {cai}iq {bj}jq {b*j}j *
*for finite sets of
elements {ai}iin M*Rand {bj}j in M*C, then M is a free 2-adic CR-module on the *
*generators {ai}i
and {bj}j.
Proof.The canonical map to M from the specified 2-adic CR-module is an isomorph*
*ism by Lemma
2.3.
To describe the multiplicative structure of K*CR(X; ^Z2) for a space X, we in*
*troduce:
2.6. The 2-adic CR-algebras. By a 2-adic CR-algebra A = {AC, AR}, we mean a 2-a*
*dic CR-
module with continuous bilinear multiplications AmLxAnL! Am+nLand elements 1 2 *
*A0Lfor m, n 2 Z
and L = C, R such that:
(i)the multiplication in A*Cand A*Ris graded commutative and associative
with identity 1;
(ii)B(zw) = (Bz)w = z(Bw) and (zw)* = z*w* for z 2 AmCand w 2 AnC;
(iii)BR(xy) = (BRx)y = x(BRy), j(xy) = (jx)y = x(jy), and ,(xy) = (,x)y =
x(,y) for x 2 AmRand y 2 AnR;
(iv)c1 = 1 and c(xy) = (cx)(cy) for x 2 AmRand y 2 AnR;
(v)r((cx)z) = x(rz) and r(z(cx)) = (rz)x for x 2 AmRand z 2 AnC.
Equivalently, a 2-adic CR-algebra A consists of a 2-adic CR-module with a commu*
*tative associative
multiplication A^ CRA ! A with identity e_! A for e_= FR (1, 0) ~=K*CR(pt; ^Z2)*
*, where ^CR is the
(symmetric monoidal) complete tensor product for 2-adic CR-modules [11, 2.6].
7
2.7. Augmentations and nilpotency. For a 2-adic CR-algebra A, an augmentation i*
*s a map
A ! e_of 2-adic CR-algebras which is left inverse to the identity e_! A. When A*
* is augmented,
we let "A= {A"C, "AR} denote the augmentation ideal, and for m 1 we let "A(m)*
* denote the m-th
power of "Agiven by the image of the m-fold product "A^CR. .^.CR"A! "A. Thus, "*
*A(m)C is the
image of the m-fold product "A*C^. .^."A*C! "A*C, while "A(m)R is the image of *
*the m-fold product
"A*R^.^.".A*R! "A*Rplus the realification of "A(m)C. The indecomposables of A a*
*re given by the
2-adic CR-module ^QA = "A=A"(2). We call A nilpotent when "A(m) = 0 for suffici*
*ently large m and
call A pronilpotent when \m "A(m) = 0 or equivalently when A ~=limmA=A"(m). For*
* a space X, the
cohomology K*CR(X; ^Z2) has a canonical augmentation K*CR(X; ^Z2) ! e_induced b*
*y the basepoint
*
pt X with the usual augmentation ideal eK*CR(X; ^Z2) = {Ke*(X; ^Z2), gKO(X; ^Z*
*2)}. Moreover,
when X is connected, the cohomology K*CR(X; ^Z2) is pronilpotent since it is th*
*e inverse limit of the
cohomologies K*CR(Xff; ^Z2) for the finite connected subspaces Xff X, where ea*
*ch K*CR(Xff; ^Z2) is
nilpotent.
3. The 2-adic OECR-algebras
To capture some additional features of the 2-adic CR-algebras K*CR(X; ^Z2) fo*
*r spaces X, we now
introduce the 2-adic OECR-algebras. These structures are often surprisingly rig*
*id and will allow us
to construct convenient bases for K*CR(X; ^Z2) in some important general cases,*
* for instance, when
X is a simply-connected compact Lie group.
3.1. The 2-adic OECR-algebras. By a 2-adic OECR-algebra A, we mean a 2-adic CR-*
*algebra with
continuous functions OE: A0C! A0Rand OE: A-1C! A0Rsuch that:
(i)cOEa = a*a and cOEx = B-1x*x for a 2 A0Cand x 2 A-1C;
(ii)OE(a + b) = OEa + OEb + r(a*b) and OE(x + y) = OEx + OEy + rB-1(x*y) for
a, b 2 A0Cand x, y 2 A-1C;
(iii)OE(ab) = (OEa)(OEb), OE(ax) = (OEa)(OEx), and OEB-1(xy) = (OEx)(OEy) fo*
*r a, b 2
A0Cand x, y 2 A-1C;
(iv)OE(1) = 1, OE(ka) = k2OEa, OE(a*) = OEa, OE(kx) = k2OEx, and OE(x*) = *
*-OEx
for a 2 A0C, x 2 A-1C, and k 2 ^Z2.
For convenience, we extend the operation OE periodically to give OE: A2iC! A0Ra*
*nd OE: A2i-1C! A0R
with OEw = OEBiw for all i and elements w. For a space X, the cohomology K*CR(X*
*; ^Z2) has a natural
2-adic OECR-algebra structure with OE: K*(X; ^Z2) ! KO0(X; ^Z2) as in [11, Sect*
*ion 3]. In particular,
e_~=K*CR(pt; ^Z2) is a 2-adic OECR-algebra with OE(k1) = k21 for k 2 ^Z2. For a*
* 2-adic OECR-algebra
8
A, an augmentation is a map A ! e_of 2-adic OECR-algebras which is left inverse*
* to the identity,
and we retain the other notation and terminology of 2.7. Thus, for a space X, t*
*he OECR-algebra
K*CR(X; ^Z2) has a canonical augmentation and is pronilpotent whenever X is con*
*nected. To capture
some other needed features, we introduce:
3.2. Special 2-adic OECR-algebras. A 2-adic OECR-algebra A is called special wh*
*en:
(i)A is augmented and pronilpotent;
(ii)z2 = 0 for z 2 AnCwith n odd;
(iii)y2 = 0 for y 2 AnRwith n 1, -3 mod 8;
(iv)OEcx = 0 for x 2 AnRwith n -1, -5 mod 8.
For a connected space X, the cohomology K*CR(X; ^Z2) is a special 2-adic OECR-a*
*lgebra by [11,
Section 3].
3.3. Simple systems of generators. Let A be a special 2-adic OECR-algebra. By a*
* simple system
of generators of odd degree for A, we mean finite ordered sets of odd-degree el*
*ements {xi}iin "AR
and {zj}j in A"Csuch that AC is an exterior algebra over K^*= ^Z2[B, B-1] on th*
*e generators
{cxi}iq {zj}jq {z*j}j. Such a simple system determines associated products
xi1. .x.im(OEzj1) . .(.OEzjn) 2 AR,
(cxi1) . .(.cxim)(cOEzj1) . .(.cOEzjn)wk1. .w.kq2 AC
where: i1 < . .<.im with m 0; j1 < . .<.jn with n 0; k1 < . .<.kq with q *
*1; each wktis
zktor z*ktwith wkq= zkq; and {k1, . .,.kq} is disjoint from {j1, . .,.jn} in ea*
*ch complex product.
Proposition 3.4. If A is a Bott exact special 2-adic OECR-algebra with a simple*
* system of generators
of odd degree, then A is a free 2-adic CR-module on the associated products.
Proof.This follows by Lemma 2.5.
When the cohomology K*CR(X; ^Z2) of a connected space X has a simple system o*
*f generators of
odd degree, this result will determine the 2-adic CR-algebra structure of the c*
*ohomology, provided
that we can compute the squares of the real simple generators of degree -1, -*
*5 mod 8, since the
squares of the other simple generators and of their OE's must vanish. For a sim*
*ply-connected compact
Lie group G, we shall see that the cohomology K*CR(G; ^Z2) must always have a s*
*imple system of
generators of odd degree by Theorem 10.3 below.
9
4.The universal 2-adic OECR-algebra functor ^L
We must now go beyond simple systems of generators and develop functorial des*
*criptions of
cohomologies K*CR(X; ^Z2) using universal special 2-adic OECR-algebras. Our res*
*ults will apply, for
instance, when X is a suitable infinite loop space (Theorem 6.7) or a simply-co*
*nnected compact
Lie group (Theorem 10.3). We start by introducing the algebraic modules that wi*
*ll generate our
universal algebras.
4.1. The 2-adic -modules. By a 2-adic -module N = {NC, NR, NH }, we mean a tr*
*iad of
2-profinite abelian groups NC, NR, and NH with continuous additive operations
t: NC ~=NC, c: NR ! NC, r: NC ! NR,
c0:NH ! NC, q :NC ! NH
satisfying the relations
t2 = 1, cr = 1 + t, rc = 2, tc = c, rt = r,
c0q = 1 + t, qc0= 2, tc0= c0, qt = q
as in [10, 4.5]. For z 2 NC, the element tz is sometimes written as z* or _-1z*
*. For a 2-adic
CR-module N and integer n, we obtain a 2-adic -module nN = {NnC, NnR, Nn-4R} *
*with c0 =
B-2c: Nn-4R! NnCand q = rB2: NnC! Nn-4R. In particular, we obtain a 2-adic -m*
*odule
Kn (X; ^Z2) = nK*CR(X; ^Z2) for a space X. We say that a 2-adic -module N is *
*torsion-free when
NC, NR, and NH are torsion-free, and we say that N is exact when the sequence
0 1-t (r,q)
. .-.! NC -(r,q)---!NR NH -c-c---!NC ---! NC ----! NR NH -! . . .
is exact (see [10, 4.5]). It is straightforward to show:
Lemma 4.2. A 2-adic -module N = {NC, NR, NH } is torsion-free and exact if and*
* only if:
(i)c: NR ! NC and c0:NH ! NC are monic;
(ii)NC is torsion-free with ker(1 + t) = im(1 - t) for t: NC ! NC;
(iii)cNR + c0NH = ker(1 - t) and cNR \ c0NH = im(1 + t).
The 2-adic -module
K-1(X; ^Z2) = {K-1(X; ^Z2), KO-1(X; ^Z2), KO-5(X; ^Z2)}
of a space X has additional operations ` which we now include in:
10
4.3. The 2-adic ` -modules. By a 2-adic ` -module M = {MC, MR, MH }, we mean a *
*2-adic
-module with continuous additive operations `: MC ! MC, `: MR ! MR, and `: MH *
*! MR
satisfying the following relations for elements z 2 MC, x 2 MR, and y 2 MH :
`cx = c`x, `c0y = c`y, `tz = t`z, `qz = `rz, ``rz = `r`z.
In general, `rz may differ from r`z, and we let ~OE:MC ! MR be the difference o*
*peration with
~OEz = `rz - r`z for z 2 MC. Using the above relations, we easily deduce:
~OEcx = 0, ~OEc0x = 0, ~OEtz = ~OEz,
2~OEz = 0, c~OEz = 0, `O~Ez = 0.
For a space X, the cohomology K-1(X; ^Z2) has a natural 2-adic ` -module struct*
*ure by [11, Section
3] with the operations
` = -~2: K-1(X; ^Z2) -! K-1(X; ^Z2),
` = -~2: KO-1(X; ^Z2) -! KO-1(X; ^Z2),
` = -~2: KO-5(X; ^Z2) -! KO-1(X; ^Z2).
Moreover, this structure interacts with the 2-adic OECR-algebra structure of K**
*CR(X; ^Z2) in several
ways.
Lemma 4.4. For a space X, we have:
(i)jOEz = ~OEz for z 2 K-1(X; ^Z2);
(ii)x2 = j`x for x 2 KO-1(X; ^Z2);
(iii)y2 = BRj`y for y 2 KO-5(X; ^Z2).
Proof.This follows from [11, Section 3].
We shall take account of these relations in our universal algebras. For a 2-a*
*dic ` -module M
and a special 2-adic OECR-algebra A, an admissible map ff: M ! A consists of a *
*2-adic -module
map ff: M ! -1A"such that:
(i)jOEffz = ff~OEz in A-1Cfor each z 2 MC;
(ii)(ffx)2 = jff`x in A-2Rfor each x 2 MR;
(iii)(ffy)2 = BRjff`y in A-10Rfor each y 2 MH .
We say that a special 2-adic OECR-algebra A with an admissible map ff: M ! A is*
* universal if,
for each special 2-adic OECR-algebra B with admissible map g :M ! B, there exis*
*ts a unique
OECR-algebra map ~g:A ! B such that ~gff = g.
11
Lemma 4.5. For each 2-adic ` -module M, there exists a universal special 2-adic*
* OECR-algebra
^LM with admissible map ff: M ! ^LM.
This will be proved later in 11.6. By universality, ^LM is unique up to isomo*
*rphism and is natural
in M, so that we have a functor ^Lfrom the category of 2-adic ` -modules to the*
* category of special
2-adic OECR-algebras. We believe that the OECR-algebra ^LM can be given canoni*
*cal operations
` satisfying all the formulae of [11, Section 3] and that this provides a stren*
*gthened version of
^Lthat is right adjoint to -1e(). However, for simplicity, we rely on the pre*
*sent basic functor
^L. We can describe the algebra (^LM)C explicitly using the 2-adic exterior al*
*gebra ^MC with
^MC = limfi^MCfiwhere MCfiranges over the finite 2-adic quotients of MC (ignori*
*ng `).
Lemma 4.6. For a 2-adic ` -module M, the canonical map ^MC ! (^LM)C is an algeb*
*ra isomor-
phism.
This will be proved later in 11.8. We must impose extra conditions on M to en*
*sure that ^LM is
Bott exact and hence topologically relevant.
4.7. The robust 2-adic ` -modules. We say that a 2-adic ` -module M is profinit*
*e when it is
the inverse limit of an inverse system of finite 2-adic ` -modules, and we let *
*M=~OEdenote the 2-adic
-module {MC, MR=~OEMC, MH }. We call M robust when:
(i)M is profinite;
(ii)M=~OEis torsion-free and exact (4.1);
(iii)ker~OE= cMR + c0MH + 2MC.
When M is obtained from K-1(X; ^Z2) for a space X, the profiniteness condition *
*will usually hold
automatically since K-1(X; ^Z2) = limff,iK-1(Xff; ^Z2)=2i for the system of fin*
*ite subcomplexes
Xff X and i 1. The following key lemma will be proved later in 12.2.
Lemma 4.8. If M is a robust 2-adic ` -module, then the special 2-adic OECR-alge*
*bra ^LM is Bott
exact; in fact, ^LM is the inverse limit of an inverse system of finitely gener*
*ated free 2-adic CR-
modules.
This leads to a crucial comparison theorem.
Theorem 4.9. For a connected space X and a robust 2-adic ` -module M, suppose t*
*hat g :M !
eK-1(X; ^Z2) is a 2-adic ` -module map that induces an isomorphism ^MC ~=K*(X; *
*^Z2). Then g
induces an isomorphism ^LM ~=K*CR(X; ^Z2) of special 2-adic OECR-algebras.
12
Proof.Since g gives an admissible map M ! K*CR(X; ^Z2) by Lemma 4.4, the result*
* follows by
Lemmas 2.3, 4.6, and 4.8.
When M is finitely generated in this theorem, we may easily choose a simple s*
*ystem of odd-
degree generators (3.3) for K*CR(X; ^Z2) from MC, MR, and MH . However, the pre*
*sent description
of K*CR(X; ^Z2) as ^LM is more natural and includes the full multiplicative str*
*ucture. To check
whether such a description is possible for a given space X, we may use:
4.10. Determination of M from K*CR(X; ^Z2). For a connected space X, we may tak*
*e the inde-
composables ^QK*CR(X; ^Z2) as in 2.7 with the operations ` of 4.3 to produce a *
*2-adic ` -module
Q^K-1(X; ^Z2) = {Q^K-1(X; ^Z2), ^QKO-1(X; ^Z2), ^QKO-5(X; ^Z2)}
together with a natural quotient map eK-1(X; ^Z2) i ^QK-1(X; ^Z2). Now by Lemma*
* 4.11 below,
whenever Theorem 4.9 applies to X, there is a canonical isomorphism M ~=^QK-1(X*
*; ^Z2) and the
map g :M ! eK-1(X; ^Z2) in the theorem corresponds to a splitting of eK-1(X; ^Z*
*2) i ^QK-1(X; ^Z2).
When X is an H-space, we may often obtain the required splitting by mapping ^QK*
*-1(X; ^Z2) to
the primitives in eK-1(X; ^Z2). For instance, this applies when X is a suitable*
* infinite loop space or
simply-connected compact Lie group (see Theorems 6.7 and 10.3). Finally, we not*
*e that the 2-adic
` -module ^QK-1(X; ^Z2) will automatically be robust by Proposition 3.4 wheneve*
*r K*CR(X; ^Z2)
has a simple system of odd-degree generators with no real generators of degree *
* 1, -3 mod 8. We
have used:
Lemma 4.11. For a ` -module M, the canonical map M ! -1Q^^LM is an isomorphism.
This will be proved later in 11.10.
5. Stable 2-adic Adams operations and K=2*-local spectra
We now bring stable Adams operations into our united 2-adic K-cohomology theo*
*ry and use this
theory to classify the needed K=2*-local spectra. We first recall some terminol*
*ogy from [8, 2.6].
5.1. Stable 2-adic Adams modules. By a finite stable 2-adic Adams module A, we *
*mean a finite
abelian 2-group with automorphisms _k: A ~=A for the odd k 2 Z such that:
(i)_1 = 1 and _j_k = _jkfor the odd j, k 2 Z;
(ii)when n is sufficiently large, the condition j k mod 2n implies _j = _k.
13
By a stable 2-adic Adams module A, we mean the topological inverse limit of an *
*inverse system
of finite stable 2-adic Adams modules. Such an A has an underlying 2-profinite *
*abelian structure
with continuous automorphisms _k: A ~=A for the odd k 2 Z (and in fact for k 2 *
*^Zx2). We note
that the operations _-1 and _3 on A determine all of the other stable Adams ope*
*rations _k as
in [5, 6.4]. Our main examples of stable 2-adic Adams modules are the cohomolog*
*ies Kn(X; ^Z2)
and KOn(X; ^Z2) for a spectrum or space X and integer n with the usual stable A*
*dams operations
_k. We let A^denote the abelian category of stable 2-adic Adams modules, and fo*
*r i 2 Z we let
~Si:^A! ^Abe the functor with ~SiA equal to A as a group but with _k on ~SiA eq*
*ual to ki_k on A
for the odd k 2 Z. We note that ~SiA = A in ^Afor all i when 2A = 0.
5.2. Stable 2-adic Adams CR-modules. By a stable 2-adic Adams CR-module M, we m*
*ean a
2-adic CR-module consisting of stable 2-adic Adams modules {M*C, M*R} such that*
* the operations
B :~SM*C~=M*-2C, t: M*C~=M*C, BR :~S4M*R~=M*-8R, j :M*R! M*-1R, c: M*R! M*C*
*, and
r: M*C! M*Rare all maps in A^, where _-1 = t in M*Cand _-1 = 1 in M*R. For a sp*
*ectrum or
space X, the 2-adic KCR -cohomology
K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)}
has a natural stable 2-adic Adams CR-module structure with the usual operations.
5.3. Stable 2-adic Adams -modules. By a stable 2-adic Adams -module N, we mea*
*n a 2-
adic -module consisting of stable 2-adic Adams modules {NC, NR, NH } such that*
* the operations
t: NC ~=NC, c: NR ! NC, r: NC ! NR, c0:NH ! NC, and q :NC ! NH are all maps in *
*^A,
where _-1 = t in NC and _-1 = 1 in both NR and NH . For a stable 2-adic Adams C*
*R-module M
and integer n, we obtain a stable 2-adic Adams -module
nM = {MnC, MnR, ~S-2Mn-4R}
as in 4.1. Thus, for a spectrum or space X and integer n, we now obtain a stabl*
*e 2-adic Adams
-module
Kn (X; ^Z2) = nK*CR(X; ^Z2) = {Kn(X; ^Z2), KOn(X; ^Z2), ~S-2KOn-4(X; ^Z2)}.
To give another example, we say that a 2-profinite abelian group G with involut*
*ion t: G ~=G is
positively torsion-free when G is torsion-free with ker(1 + t) = im(1 - t). By *
*[5, Proposition 3.8],
this is equivalent to saying that G factors as a (possibly infinite) product of*
* ^Z2's with t = 1 and
^Z2 t^Z2's. For a positively torsion-free stable 2-adic Adams module A, we may*
* use the operation
14
_-1: A ~=A to construct a torsion-free exact stable 2-adic Adams -module {A, A*
*+, A+} with
A+ = ker(1 - _-1), A+ = coker(1 - _-1), t = _-1, c = 1, r = 1 + _-1, c0= 1 + _-*
*1, and q = 1.
We let ^ACR(resp. A^ ) denote the abelian category of stable 2-adic Adams CR-*
*modules (resp.
-modules), and we note that the functor n: ^ACR! ^A for n 2 Z has a left adj*
*oint CRn: ^A !
^ACRwith CRn(N)nC= NC, with CRn(N)n-1C= 0, and with
8
>>>NR for i = 0
>>>N =r for i = 1
>>> R
><~SNC=c0 for i = 2
CRn(N)n-iR= > 0 for i = 3, 7
>>>~2
>>>SNH for i = 4
>>>~S2NH =qfor i = 5
: ~3
S NC=c for i = 6
as in [10, 4.10]. We easily see that CRn(N) is Bott exact whenever N is torsion*
*-free and exact. Our
next lemma will often allow us to work in the simpler category ^A instead of ^*
*ACR.
Lemma 5.4. For n 2 Z, the adjoint functors CRn: ^A ! A^CR and n: ^ACR! ^A re*
*strict
to equivalences between the full subcategories of all torsion-free exact N 2 A^*
* and all Bott exact
M 2 ^ACRwith MnCpositively torsion-free and Mn-1C= 0.
Proof.For M 2 ^ACRas above, we see that nM is a torsion-free exact -module by*
* [10, 4.4 and
4.7] with an adjunction isomorphism CRn nM ! M by Lemma 2.3. The corresponding *
*result for
N 2 ^A is obvious.
When E is a spectrum with Kn(E; ^Z2) positively torsion-free and Kn-1(E; ^Z2)*
* = 0 for some n, we
now have K*CR(E; ^Z2) ~=CRn(N) in ^ACRfor the torsion-free exact module N = nK*
**CR(E; ^Z2) in
^A , and we have the following existence theorem for such spectra in the stable*
* homotopy category.
Theorem 5.5. For each torsion-free exact N 2 ^A and n 2 Z, there exists a K=2**
*-local spectrum
EnN with K*CR(EnN; ^Z2) ~=CRn(N) in A^CR. Moreover, EnN is unique up to (nonca*
*nonical)
equivalence.
Proof.This follows by Lemma 5.4 and [10, Theorem 5.3].
The spectrum EnN in the theorem will be endowed with an isomorphism K*CR(EnN;*
* ^Z2) ~=
CRn(N) in A^CR. Thus, for an arbitrary spectrum E, a map g :E ! EnN induces a *
*map
g*: CRn(N) ! K*CR(E; ^Z2) in ^ACR. Each algebraic map of this sort must come f*
*rom a topo-
logical map by:
15
Theorem 5.6. For a torsion-free exact N 2 ^A , n 2 Z, and an arbitrary spectrum*
* E, if fl :CRn(N) !
K*CR(E; ^Z2) is a map in ^ACR, then there exists a map of spectra g :E ! EnN wi*
*th g* = fl.
Proof.Let o2E denote the 2-torsion part of E given by the homotopy fiber of its*
* localization away
from 2. By Pontrjagin duality [10, Theorem 3.1], the map fl corresponds to an A*
*CR-module map
KCR*(o2E) ! KCR*(o2EnN) in the sense of [5], where KCR*(o2EnN) is CR-exact with*
* K*(o2EnN)
divisible. This ACR-module map prolongs canonically to an ACRT-module map KCRT**
*(o2E) !
KCRT*(o2EnN) by [5, Theorem 7.14], and the results of [5, 9.8 and 7.11] now sho*
*w that this prolonged
algebraic map must come from a topological map o2E ! o2EnN, which gives the des*
*ired g :E !
EnN.
The map g in this theorem is generally not unique (see [10, 5.4]).
6.On the united 2-adic K-cohomologies of infinite loop spaces
In preparation for our work on K=2*-localizations of spaces, we functorially *
*determine the united
2-adic K-cohomologies of the needed infinite loop spaces (see Theorem 6.7). We *
*must first introduce:
6.1. The 2-adic Adams -modules. By a 2-adic Adams -module M, we mean a 2-adic*
* ` -
module (4.3) consisting of stable 2-adic Adams modules {MC, MR, MH } such that *
*the operations
t: MC ~=MC, c: MR ! MC, r: MC ! MR, c0:MH ! MC, q :MC ! MH , `: MC ! MC,
`: MR ! MR, and `: MH ! MR are all maps in ^A, where _-1 = t in MC and _-1 = 1 *
*in both
MR and MH . We let M^ denote the abelian category of 2-adic Adams -modules. W*
*e say that
M is `-nilpotent when it has `i = 0 for sufficiently large i, and we say that M*
* is `-pronilpotent
when it is the inverse limit of an inverse system of `-nilpotent 2-adic Adams *
*-modules. Thus, M is
`-pronilpotent if and only if M ~=limiM=`iwhere M=`iis the quotient module of M*
* in M^ with
(M=`i)C = MC=`iMC,
(M=`i)R = MR=(`iMR + `iMH + r`iMC),
(M=`i)H = MH =q`iMC
for i 1. More simply, M is `-pronilpotent if and only if \i`iMC = 0 and \i`iM*
*R = 0. It is not
hard to show that whenever M is `-pronilpotent, M must be profinite (i.e. M mus*
*t be the inverse
limit of an inverse system of finite 2-adic Adams -modules). For a space X, th*
*e cohomology
Ke-1(X; ^Z2) = {Ke-1(X; ^Z2), gKO-1(X; ^Z2), ~S-2gKO-5(X; ^Z2)}
has a natural 2-adic Adams -module structure by 4.3, 5.3, and [11, 3.16] where:
16
Lemma 6.2. If X is a connected space with H1(X; ^Z2) = 0, then the 2-adic Adams*
* -module
eK-1(X; ^Z2) is `-pronilpotent.
Proof.The condition \i`ieK0( X; ^Z2) = 0 holds by [6, 5.4 and 5.5] since H2( X;*
* ^Z2) = 0, and a
0
similar proof shows \i`igKO ( X; ^Z2) = 0 since H1( X; Z=2) = 0. This proof use*
*s the fact that
0
the ~-ideal gKO Y is fl-nilpotent for a connected finite CW complex Y by [10, T*
*heorem 6.7] and the
fact that the real line bundles over Y are classified by H1(Y ; Z=2).
6.3. The functor F". We shall construct a functor F":A^ ! M^ where A^ is the*
* abelian
category of stable 2-adic Adams -modules (5.3) and M^ is that of 2-adic Adams*
* -modules
(6.1). This functor will carry each N 2 ^A to a universal `-pronilpotent targe*
*t module "FN 2 M^ .
For N 2 A^ , we first let NRH 2 A^denote the pushout of NR --r NC -q-!NH with *
*a map
~c:NRH ! NC induced by c and c0, and with a map ~r:NC ! NRH induced by r or q. *
*We also let
NC+ 2 ^Adenote NC=(1 - t)NC and let NCOE2 ^Adenote NC=(cNR + c0NH + 2NC). We ne*
*xt let
aeN = {NC, NRH NCOE, NC+ }
be the stable 2-adic Adams -module with operations given by tz = tz, c(x, w) =*
* ~cx, rz = (~rz, [z]),
c0[z] = (1 + t)z, and qz = [z]. We then obtain a stable 2-adic Adams -module
F"N = N x aeN x aeN x . . .
with components
"FCN = NC x NC x NC x . .,.
F"RN = NR x NRH x NCOEx NRH x NCOEx . .,.
"FHN = NH x NC+ x NC+ x . . .
We finally define operations `: "FCN ! "FCN, `: "FRN ! "FRN, and `: "FHN ! "FRN*
* respectively
by the formulae
`(z1, z2, z3, . .).= (0, z1, z2, z3, . .).,
`(x1, x2, z2, x3, z3, . .).= (0, [x1], 0, x2, 0, x3, 0, . .).,
`(y1, z2, z3, . .).= (0, [y1], 0, ~rz2, 0, ~rz3, 0, . .)..
This gives a natural 2-adic Adams -module "FN and hence a functor "F:A^ ! M^ *
*. We let
': N ! F"N be the map in A^ with 'C(z) = (z, 0, 0, . .)., 'R(x) = (x, 0, 0, . *
*.)., and 'H (y) =
(y, 0, 0, . .)., and we show:
17
Theorem 6.4. For a stable 2-adic Adams -module N 2 ^A , the 2-adic Adams -mod*
*ule "FN 2
^M is `-pronilpotent and the map ': N ! F"N has the universal property that, *
*for each `-
pronilpotent M 2 M^ and map f :N ! M in A^ , there exists a unique map ~f:"FN *
*! M
in M^ with ~f' = f.
Proof."FN is `-pronilpotent since it is the inverse limit of its quotient modul*
*es
F"N=`i+1 ~=N x aeN x . .x.aeN.
For i 1, we define a map f(i):aeN ! M in ^A by
f(i)C= `ifC :NC -! MC,
f(i)R= (`ifR, `ifH ) + ~OE`i-1fC :NRH NCOE-! MR,
f(i)H= q`ifC :NC+ -! MH .
We then define ~f:"FN ! M as the inverse limit of the maps
f + f(1)+ . .+.f(i):N x aeN x . .x.aeN -! M=`i+1
in M^ , and we check that ~f' = f. The uniqueness condition for ~ffollows since*
* the 2-adic Adams
-modules "FN=`i+1= N x aeN x . .x.aeN are generated by 'N.
To show the robustness (4.7) of "FN for suitable N, we need:
6.5. The functor ~ae:^A ! ^A . For N 2 ^A , we let ~aeN = {NC, NRH , NC+ } be *
*the stable 2-adic
Adams -module with operations given by tz = z, cx = ~cx, rz = ~rz, c0[z] = *
*(1 + t)z, and
qz = [z]. Thus, ~aeN is the quotient of aeN = {NC, NRH NCOE, NC+ } by NCOE. I*
*f N is torsion-free
and exact, then ~aeN is also torsion-free and exact by Lemma 4.2 since it is is*
*omorphic to the module
{NC, NR + NH , NR \ NH } with c and c0treated as inclusions.
Lemma 6.6. If N 2 ^A is torsion-free and exact, then "FN 2 M^ is robust.
Proof.We check that ~OE:"FCN ! "FRN is given by
~OE(z1, z2, z3, . .).= (0, 0, [z1], 0, [z2], 0, . .).
for zi2 NC and [zi] 2 NCOE. Thus, ker~OE= cF"RN +c0"FHN +2F"CN and "FN=~OE~=N x*
*~aeN x~aeN x. . .
Hence, "FN=~OEis torsion-free and exact by 6.5 as required.
Our main result in this section is:
18
Theorem 6.7. If E is a 0-connected spectrum with H1(E; ^Z2) = 0 = H2(E; ^Z2), w*
*ith K0(E; ^Z2) =
0, and with K-1(E; ^Z2) positively torsion-free (5.3), then there is a natural *
*isomorphism ^L"FK-1(E; ^Z2) ~=
K*CR( 1 E; ^Z2).
Proof.Since eK-1( 1 E; ^Z2) is `-pronilpotent by Lemma 6.2, the infinite suspen*
*sion map oe :K-1(E; ^Z2) !
eK-1( 1 E; ^Z2) induces a map ~oe:"FK-1(E; ^Z2) ! eK-1( 1 E; ^Z2) in M^ , where*
* "FK-1(E; ^Z2) is
robust by Lemmas 5.4 and 6.6. Thus, ~oeinduces an isomorphism ^L"FK-1(E; ^Z2) ~*
*=K*CR( 1 E; ^Z2)
by Theorem 4.9, since it induces an isomorphism of the complex components by [6*
*, Theorem 8.3].
7.Strong 2-adic Adams -modules
Our main results on K=2*-localizations in Section 8 will involve a space X wi*
*th K*CR(X; ^Z2) ~=
^LM for a 2-adic Adams -module M that is strong (7.11) in the sense that it is*
* robust (7.1), _3-
splittable (7.2), and regular (7.8). In this section, we provide the required a*
*lgebraic definitions and
explanations of these notions. We first recall:
7.1. The robust modules. We say that a 2-adic Adams -module M is robust when i*
*t is robust
in the sense of 4.7, ignoring stable Adams operations. When M is robust, the un*
*derlying 2-adic
-module M=~OEsatisfies the conditions of Lemma 4.2 and may be factored as a (p*
*ossibly infinite)
product of monogenic free 2-adic -modules
FC (z) = {^Z2 t^Z2, ^Z2, ^Z2} = { , , },
FR (x) = {^Z2, ^Z2, ^Z2} = {, , },
FH (y) = {^Z2, ^Z2, ^Z2} = {, , }
by an argument using the factorization of positively torsion-free groups in 5.3*
*. We let genCM,
genRM, and genHM respectively denote the number of complex, real, and quaternio*
*nic monogenic
free factors of M=~OE. These numbers do not depend on the factorization since t*
*hey respectively equal
the Z=2-dimensions of the Pontrjagin duals (MCOE)# , (MR=(~OEMC + rMC))# , and *
*(MH =qMC)# .
Using the factorization of M=~OE, we find that
genMC = 2 genCM + genRM + genHM
where genMC denotes the number of ^Z2factors in the 2-profinite abelian group M*
*C.
19
7.2. The _3-splittable modules. For a 2-adic Adams -module M 2 M^ , we conside*
*r the stable
2-adic Adams -module M~= M=~OE2 ^A , and we say that M is _3-splittable when t*
*he quotient
map M i M~ has a right inverse s: M~ ! M in A^ . We call such a map s a _3-spl*
*itting of
M, and we note that it corresponds to a left inverse s0:MR=rMC ! ~OEMC of the c*
*anonical map
~OEMC ! MR=rMC in the category A^of stable 2-adic Adams modules (5.1), or equiv*
*alently in the
category of profinite Z=2-modules with automorphisms _3. We deduce that M is a*
*utomatically
_3-splittable in some important cases:
Lemma 7.3. If M is a robust 2-adic Adams -module with genCM = 0 or genRM = 0, *
*then M
is _3-splittable.
Proof.Since MC is positively torsion-free, the map cr = 1 + t: MC+ ! MC is moni*
*c, and hence
c: rMC ! MC is also monic. Thus, ~OEMC \ rMC = 0 and there is a short exact seq*
*uence
0 -! ~OEMC -! MR=rMC -! MR=(~OEMC + rMC) -! 0
in ^A. Since genCM = 0 or genRM = 0, this has ~OEMC = 0 or MR=(~OEMC + rMC) = 0*
*, and hence
the map ~OEMC ! MR=rMC has an obvious left inverse in ^A.
We shall use the _3-splittability condition to construct:
7.4. `-resolutions of modules. Let M 2 M^ be a 2-adic Adams -module that is `*
*-pronilpotent,
robust, and _3-splittable. These conditions will hold when M is strong (7.11). *
*For a _3-splitting
s: ~M! M in ^A , we shall construct an associated `-resolution
d~ ~s
0 -! "F~ae~M--!"F~M--!M -! 0
of M in M^ , with ~ae~M= {M~C, ~MRH, ~MC+} as in 6.5, where ~s:"F~M! M is induc*
*ed by s via
Theorem 6.4. To specify ~d, we use the commutative square
a~e~M-`---!~M
?? ?
yoe ?ys
(1)
aeM~--s--!M
in ^A with aeM~ = {M~C, ~MRH ~MCOE, ~MC+} as in 6.3, where s(1)is given by th*
*e proof of Theorem
6.4, where ` = {`, (`, `), q`}, and where oe = {1, (1, `OE), 1}, using the map *
*`OE:~MRH! ~MCOE= MCOE
given by the composition of the sequence
~MRH-s-!MRH -(`,`)---!MR ~= M~R MCOE-proj---!MCOE
20
in which the isomorphism is the inverse of (s, ~OE): ~MR MCOE~=MR. The commuta*
*tive square now
gives a map
d = (`, -oe, 0, 0, . .).:~ae~M! "F~M
in ^A with ~sd = 0, and this induces the required map ~d:"F~ae~M! "F~Min M^ w*
*ith ~s~d= 0.
Lemma 7.5. If M 2 M^ is `-pronilpotent and robust with a _3-splitting s: M~! M*
*, then the
~d ~s
`-resolution 0 ! "F~ae~M-!"F~M-!M ! 0 is exact in M^ .
Proof.We easily check that 0 ! ~OE(F"~ae~M)C ! ~OE(F"~M)C ! ~OEMC ! 0 is exact *
*and that ~s=~OE:"F~M=~OE!
M=~OEis onto. Hence, it suffices to show that the map "F~ae~M=~OE! ker(~s=~OE) *
*is an isomorphism. This
follows by [10, Lemma 4.8] since the 2-adic -modules "F~ae~M=~OEand ker(~s=~OE*
*) are exact by Lemma
6.6 and [10, 4.8] and since the map (F"~ae~M=~OE)C ! ker(~s=~OE)C is clearly an*
* isomorphism.
To formulate our regularity condition for M, we use:
7.6. The 2-adic Adams modules. These are the unstable versions of the stable mo*
*dules in 5.1,
previously discussed in [8, 2.8]. By a finite 2-adic Adams module A, we mean a *
*finite abelian 2-group
with endomorphisms _k: A ! A for k 2 Z such that:
(i)_1 = 1 and _j_k = _jkfor j, k 2 Z;
(ii)when n is sufficiently large, the condition j k mod 2n implies _j = _k.
By a 2-adic Adams module A, we mean the topological inverse limit of an inverse*
* system of finite
2-adic Adams modules. Such an A has an underlying 2-profinite abelian group wi*
*th continuous
endomorphisms _k: A ! A for k 2 Z (and in fact for k 2 ^Z2). For a space X, the*
* cohomology
K1(X; ^Z2) is a 2-adic Adams module with the usual Adams operations _k for k 2 *
*Z as in [6,
Example 5.2]. We note that the operations _2 and _k, for k odd, in K1(X; ^Z2) *
*correspond via
Bott periodicity to ` and to k-1_k in K-1(X; ^Z2). In general, for a `-pronilpo*
*tent 2-adic Adams
-module M, we obtain a 2-adic Adams module MC having the same group as MC but *
*having
_0 = 0 and having _k2iequal to k-1_k`ion MC for k odd and i 0.
7.7. Linear and strictly nonlinear modules. As in [8, Section 4] and [7, Sectio*
*n 2], a 2-adic
Adams module H is called linear when it has _k = k for all k 2 Z, and H is call*
*ed quasilinear when
2H _2H. Each 2-adic Adams module A has a largest linear quotient module
LinA = A=((_2- 2)A + (_-1 + 1)A + (_3- 3)A)
and also has a largest quasilinear submodule Aql A by Lemma 13.1 below. A 2-ad*
*ic Adams module
A is called strictly nonlinear when Aql= 0. This implies that A is torsion-free*
* with \i(_2)iA = 0,
21
and A will be strictly nonlinear by 13.2 and [7, 2.5] whenever it is torsion-fr*
*ee with (_2)iA 2i+1A
for some i 1.
7.8. Regular modules. As in [8, 4.4], we say that a 2-adic Adams module A is re*
*gular when the
kernel of A ! LinA is strictly nonlinear. This implies that \i(_2)iA = 0, and A*
* will be regular
whenever it is an extension of a strictly nonlinear submodule by a linear quoti*
*ent module. We
also say that a 2-adic Adams -module M is regular when it is `-pronilpotent wi*
*th MC regular
as a 2-adic Adams module. Thus, for a connected space X with H1(X; ^Z2) = 0, th*
*e 2-adic Adams
-module eK-1(X; ^Z2) is regular if and only if eK1(X; ^Z2) is regular as a 2-a*
*dic Adams module. The
following two lemmas will often guarantee regularity for our modules.
Lemma 7.9. Let X be a connected space with H1(X; ^Z2) = 0, with Hm (X; ^Z2) = 0*
* for suffi-
ciently large m, and with eK1(X; ^Z2) torsion-free. Then eK1(X; ^Z2) is regular*
* with _2: eK1(X; ^Z2) !
eK1(X; ^Z2) monic, and hence eK-1(X; ^Z2) is regular with `: eK-1(X; ^Z2) ! eK-*
*1(X; ^Z2) monic.
The proof is in 13.5.
Lemma 7.10. For a regular 2-adic Adams module A, each submodule is regular, and*
* each torsion-
free quotient module is regular when A is finitely generated over ^Z2.
The proof is in 13.4. Combining the preceding definitions, we finally introdu*
*ce:
7.11. Strong modules. We say that a 2-adic Adams -module M 2 M^ is strong whe*
*n:
(i)M is robust (7.1);
(ii)M is _3-splittable (7.2);
(iii)M is regular (7.8);
Such an M is automatically `-pronilpotent (and hence profinite) since it is reg*
*ular.
8. On the K=2*-localizations of our spaces
We recall that the K=2*-localizations of spaces or spectra are the same as th*
*e K*(-; ^Z2)-
localizations since the K=2*-equivalences are the same as the K*(-; ^Z2)-equiva*
*lences. In this sec-
tion, we give our key result (Theorem 8.6) on the K=2*-localization of a connec*
*ted space X with
K*CR(X; ^Z2) ~=^LM for a strong 2-adic Adams -module M. We first consider:
22
8.1. Building blocks for K=2*-localizations. For a torsion-free exact stable 2-*
*adic Adams -
module N 2 ^A , we let EN denote the K=2*-local spectrum E-1N of Theorem 5.5 wi*
*th an isomor-
phism K*CR(EN; ^Z2) ~=CR-1N in the category ^ACRof stable 2-adic Adams CR-modul*
*es. As in [8,
3.5], we let "EN ! EN ! ~P2EN denote the Postnikov fiber sequence of spectra wi*
*th ssi"EN ~=ssiEN
for i > 2, with ssi"EN = 0 for i < 2, and with ss2"EN ~=^t2ss2EN, where ^t2ss2"*
*EN ss2EN denotes
the Ext-2-completion of the torsion subgroup of ss2EN. We now obtain a simply-c*
*onnected infinite
loop space 1 "EN which is K=2*-local by [8, Theorem 3.8]. These 1 "EN, with t*
*heir companions
1 "E~aeN, will serve as our building blocks for K=2*-localizations of spaces, *
*where ~aeN denotes the
torsion-free exact stable 2-adic Adams -module ~aeN = {NC, NR + NH , NR \ NH }*
* of 6.5.
8.2. Strict homomorphisms and isomorphisms. For a 2-adic Adams -module M 2 M^ *
* and
a connected space X, a strict homomorphism (resp. strict isomorphism) ^LM ! K**
*CR(X; ^Z2) is
a homomorphism (resp. isomorphism) of special 2-adic OECR-algebras induced by *
*a map M !
eK-1(X; ^Z2) of 2-adic Adams -modules. For instance, there is a strict isomorp*
*hism
L^"FN ~=K*CR( 1 "EN; ^Z2)
for each torsion-free exact stable 2-adic Adams -module N 2 ^A by Theorem 6.7*
*, and we have:
Lemma 8.3. For a torsion-free exact module N 2 ^A and a connected space X with*
* H1(X; ^Z2) =
0 = H2(X; ^Z2), each strict homomorphism ^L"FN ! K*CR(X; ^Z2) is induced by a (*
*possibly non-
unique) map X ! 1 "EN.
Proof.A strict homomorphism ^L"FN ! K*CR(X; ^Z2) corresponds successively to: a*
* map "FN !
eK-1(X; ^Z2) in M^ , a map N ! eK-1(X; ^Z2) in A^ , and a map CR-1N ! K*CR( 1 X*
*; ^Z2) in
^ACR. By Theorem 5.6, this last map is induced by a map 1 X ! EN, which lifts *
*uniquely to a
map 1 X ! "EN, and we now check that the adjoint map X ! 1 "EN induces the or*
*iginal strict
homomorphism.
8.4. The key construction. For a strong 2-adic Adams -module M 2 M^ , we may t*
*ake a
`-resolution (7.4)
d~ ~s
0 -! "F~ae~M--!"F~M--!M -! 0
using the torsion-free exact module ~M= M=~OE2 ^A . We may then apply Lemma 8.3*
* to give a map
f : 1 "E~M! 1 "E~ae~Minducing the K*CR(-; ^Z2)-homomorphism f* = ^L~d:^L"F~ae~*
*M! ^L"FM. Any
such f will be called a companion map of M, and its homotopy fiber Fibf will be*
* K=2*-local since
23
1 "EM and 1 "E~aeM are. As in [8, 4.6] and 8.1, we let
gFibf -! Fibf -! ~P2Fibf
denote the Postnikov fiber sequence with ssigFibf ~=ssiFibf for i > 2, with ssi*
*gFibf = 0 for i < 2, and
with ssigFibf ~=^t2ss2gFibf. We note that ~P2Fibf is an infinite loop space whi*
*ch is K=2*-local by [8,
Theorem 3.8], and we conclude that gFibf is also K=2*-local. Moreover, we have *
*K*CR(gFibf; ^Z2) ~=
^LM by:
Theorem 8.5. For a strong 2-adic Adams -module M 2 M^ and any companion map f*
* : 1 "E~M!
1 "E~ae~M, there is a strict isomorphism ^LM ~=K*CR(gFibf; ^Z2).
Thus, ^LM is topologically realizable for each strong M 2 M^ . This theorem w*
*ill be proved in
14.7 and leads immediately to our key result on K=2*-localizations of spaces.
Theorem 8.6. If X is a connected space with a strict isomorphism ^LM ~=K*CR(X; *
*^Z2) for a strong
2-adic Adams -module M 2 M^ , then there is an equivalence XK=2 ' gFibf for so*
*me compan-
ion map f : 1 "E~M! 1 "E~ae~Mof M, where the equivalence induces the canonical*
* isomorphism
K*CR(gFibf; ^Z2) ~=^LM ~=K*CR(X; ^Z2). Moreover, H1(X; ^Z2) = 0 = H2(X; ^Z2).
*
* d~
Proof.The last statement follows by [6, 5.4]. For the first, we take a `-resolu*
*tion 0 ! "F~ae~M-!
"F~M~s-!M ! 0 of M and apply Lemma 8.3 to give a map h: X ! 1 "E~Mwith h* = ^L*
*~s:^L"F~M!
^LM. We then apply Lemma 8.3 again to give a map k :Cofh ! 1 "E~ae~Mwith
k* = ^L~d:^L"F~ae~M-! K*CR(Cofh; ^Z2) ^L"F~M.
Composing k with the cofiber map, we obtain a companion map f : 1 "E~M! 1 "E~a*
*e~Mof M such
that h lifts to a map u: X ! gFibf which is a K=2*-equivalence by Theorem 8.5. *
*Since gFibf is
K=2*-local, this gives the desired equivalence XK=2' gFibf.
In this theorem, M is uniquely determined by the space X since there is a can*
*onical isomorphism
M ~=^QK-1(X; ^Z2) in M^ by 4.10 and [11, Section 3].
9.On the v1-periodic homotopy groups of our spaces
The p-primary v1-periodic homotopy groups v-11ss*X of a space X at a prime p *
*were defined by
Davis and Mahowald [15] and have been studied extensively (see [13]). In this s*
*ection, we apply the
preceding result (Theorem 8.6) on the K=2*-localizations of our spaces to appro*
*ach their v1-periodic
homotopy groups at p = 2 using:
24
9.1. The functor 1. As in [4], [9], [16], and [19], there is a v1-stabilizatio*
*n functor 1 from the
homotopy category of spaces to that of spectra such that:
(i)for a space X, there is a natural isomorphism v-11ss*X ~=ss*o2 1X where
o2 1X is the 2-torsion part of 1X (given by the fiber of its localization
away from 2);
(ii) 1X is K=2*-local for each space X;
(iii)for a spectrum E, there is a natural equivalence 1( 1 E) ' EK=2;
(iv) 1 preserves fiber squares.
Various other properties of 1 are described in [10, Section 2], and the isomor*
*phism v-11ss*X ~=
ss*o2 1X may be applied as in [10, Theorem 3.2] to show:
Theorem 9.2. For a space X, there is a natural long exact sequence
3-9
. .-.! KOn-3( 1X; ^Z2) _---!KOn-3( 1X; ^Z2) -! (v-11ssnX)#
3-9
-! KOn-2( 1X; ^Z2) _---!KOn-2( 1X; ^Z2) -! . . .
where (-)# gives the Pontrjagin dual.
This may be used to calculate v-11ss*X from KO*( 1X; ^Z2) up to extension. T*
*o approach
KO*( 1X; ^Z2) or K*( 1X; ^Z2), we work with:
9.3. K=2*-durable spaces. Following [8, 7.8], we say that a space X is K=2*-dur*
*able when the
K=2*-localization X ! XK=2 induces an equivalence 1X ' 1XK=2 (or equivalently*
* induces an
isomorphism v-11ss*X ~=v-11ss*XK=2), and we recall that each connected H-space *
*is K=2*-durable.
For such X, we may apply our key result on K=2*-localizations (Theorem 8.6) to *
*deduce:
Theorem 9.4. If X is a connected K=2*-durable space (e.g. H-space) with a stric*
*t isomorphism
^LM ~=K*CR(X; ^Z2) for a strong module M 2 M^ , then there is a (co)fiber seque*
*nce of spectra
1X ! EM~ ffl-!Ea~e~Msuch that ffl*: K*CR(Ea~e~M; ^Z2) ! K*CR(EM~; ^Z2) is give*
*n by CR-1`: CR-1~ae~M!
CR-1M~.
Here, the map `: ~ae~M! ~Mis given by
` = (`, `, `): {M~C, ~MR+ ~MH, ~MR\ ~MH} -! {M~C, ~MR, ~MH}
in A^ . This theorem will be proved below in 9.9 and may be used to calculate K*
**( 1X; ^Z2) and
KO*( 1X; ^Z2) since it immediately implies:
25
Theorem 9.5. For X as in Theorem 9.4, there is a K*(-; ^Z2) cohomology exact se*
*quence
0 -! K-2( 1X; ^Z2) -! ~MC-`-!~MC-! K-1( 1X; ^Z2) -! 0,
and there is a KO*(-; ^Z2) cohomology exact sequence
0 -! KO-8( 1X; ^Z2) -! ~MC=(M~R+M~H) -`-!~MC=M~R -! KO-7( 1X; ^Z2) -! 0
-! ~MH=(M~R\M~H) -! KO-6( 1X; ^Z2) -! ~MR\M~H -`-!~MH-! KO-5( 1X; ^Z2)
-! 0 -! 0 -! KO-4( 1X; ^Z2) -! ~MC=(M~R \ ~MH) -`-!~MC=M~H
-! KO-3( 1X; ^Z2) -! (M~R + ~MH)=(M~R \ ~MH) -`-!~MR=(M~R \ ~MH)
-! KO-2( 1X; ^Z2) -! ~MR+ ~MH-`-!~MR-! KO-1( 1X; ^Z2) -! 0
In these sequences, ` may be replaced by ~2 = -`. Also, for i, k 2 Z with k od*
*d, the Adams
operation _k in K2i-1( 1X; ^Z2), K2i-2( 1X; ^Z2), KO2i-1( 1X; ^Z2), or KO2i-2( *
*1X; ^Z2) agrees
with k-i_k in the adjacent M~terms.
Thus, for X as in Theorem 9.4, we may essentially calculate v-11ss*X from M~(*
*up to extension
problems) using Theorems 9.2 and 9.5. By [10, 7.6], this approach to v-11ss*X m*
*ay be extended to
various other important spaces X using:
9.6. The bK 1-goodness condition. For a space X, we let 1: eK*CR(X; ^Z2) -! K**
*CR( 1X; ^Z2)
denote the v1-stabilization homomorphism of [10, 7.1], and we recall that it in*
*duces a homomorphism
1: ^QKn (X; ^Z2)=` -! Kn ( 1X; ^Z2) in A^ for n = -1, 0 by [10, 7.4], where ^*
*QKn (X; ^Z2)=` is
as in 4.10 and 6.1. Following [10, 7.5], we say that a space X is bK 1-good whe*
*n the complex v1-
stabilization homomorphism 1: ^QKn(X; ^Z2)=` ! Kn( 1X; ^Z2) is an isomorphism *
*for n = -1, 0.
Our next theorem will provide initial examples of bK 1-good spaces from which o*
*ther examples may
be built.
Theorem 9.7. If X is a connected K=2*-durable space (e.g. H-space) with a stric*
*t isomorphism
^LM ~=K*CR(X; ^Z2) for a strong module M 2 M^ such that `: M~C! M~C is monic, *
*then X is
bK 1-good with K0( 1X; ^Z2) = 0, with K-1( 1X; ^Z2) = ~MC=`, and with K-1( 1X; *
*^Z2) ~=~M=`.
To prove Theorems 9.4 and 9.7, we first consider the spectrum "EN for a torsi*
*on-free exact module
N 2 ^A and note that 1 1 "EN ' (E"N)K=2' EN.
Lemma 9.8. The space 1 "EN is bK 1-good, and the v1-stabilization gives a natu*
*ral isomorphism
1: ^QK-1( 1 "EN; ^Z2)=` ~= K-1(EN; ^Z2)
26
Proof.By [10, 7.1], the homomorphism 1: K-1( 1 "EN; ^Z2) ! K-1(EN; ^Z2) is lef*
*t inverse to the
infinite suspension homomorphism, and the lemma now follows by Theorem 6.7 toge*
*ther with 4.11,
and 6.3.
9.9. Proof of Theorem 9.4. Applying the functor 1 to the fiber sequence of The*
*orem 8.6, we
obtain a (co)fiber sequence of spectra
1XK=2- ! 1 1 "E~M-1f---! 1 1 "E~ae~M
for some companion map f of M. We then deduce that 1f corresponds to a map EM~*
* ! Ea~e~M
having the desired properties by Lemmas 9.8 and 5.4.
9.10. Proof of Theorem 9.7. The results on K*( 1X; ^Z2) and K-1( 1X; ^Z2) follo*
*w from Theo-
rem 9.5. Since K*(X; ^Z2) ~=^MC by Lemma 4.6, we obtain isomorphisms ^QK0(X; ^Z*
*2)=` = 0 and
^QK-1(X; ^Z2)=` ~=MC=`, and we deduce that 1: ^QKn(X; ^Z2)=` ~=Kn( 1X; ^Z2) fo*
*r n = -1, 0
by Lemma 9.8 and naturality.
10.Applications to simply-connected compact Lie groups
We now apply the preceding results to a simply-connected compact Lie group G.*
* We first use
the representation theory of G to functorially determine the united 2-adic K-co*
*homology ring
K*CR(G; ^Z2) = {K*(G; ^Z2), KO*(G; ^Z2)} in Theorem 10.3. Then, with slight re*
*strictions on the
group, we show that G is bK 1-good, and we use the representation theory of G t*
*o give expressions
for the K=2*-localization GK=2, for the v1-stabilization 1G, and for the cohom*
*ology KO*( 1G; ^Z2).
Our results are summarized in Theorem 10.6 and permit calculations of the 2-pri*
*mary v1-periodic
homotopy v-11ss*G using Theorem 9.2 as accomplished very successfully by Davis *
*[14]. In this sec-
tion, we assume some general familiarity with the representation rings of our L*
*ie groups as described
in [12, Sections II.6 and VI.4] and [14, Theorem 2.3].
10.1. The representation ring R G. For a simply-connected compact Lie group G,*
* we let RG
be the complex representation ring and let RRG, RH G RG be the real and quate*
*rnionic parts
of RG with the usual ~-ring structures on RG and RRG RH G. We also let t = _*
*-1: RG ~=
RG, c: RRG RG, r: RG ! RRG, c0:RH G RG, and q :RG ! RH G be the usual
operations satisfying the -module relations of 4.1. These structures are compa*
*tible in the expected
ways and combine to give a ~-ring R G = {RG, RRG, RH G} in the sense of [10, *
*6.2]. We let
"RG = {R"G, "RRG, "RHG} be the augmentation ideal of R G given by the kernel "*
*RG of the complex
27
augmentation dim :RG ! Z, where "RRG = RRG \ "RG and "RHG = RH G \ "RG. We als*
*o let
QR G = {QRG, QRRG, QRH G} be the indecomposables of R G given by
QRG = R"G=(R"G)2,
QRRG = R"RG=((R"RG)2+ (R"HG)2+ r(R"G)2),
QRH G = R"HG=((R"RG)(R"HG) + q(R"G)2).
It is straightforward to show that "RG and QR G inherit ~-ring structures (wi*
*thout identities)
from R G. Since QR G is a ~-ring with trivial multiplication, it is equippe*
*d with additive
operations t: QRG ~=QRG, c: QRRG ! QRG, r: QRG ! QRRG, c0:QRH G ! QRG,
q :QRG ! QRH G, ` = -~2: QRG ! QRG, ` = -~2: QRRG ! QRRG, ` = -~2: QRH G !
QRRG, _k: QRG ! QRG, _k: QRRG ! QRRG, and _k: QRH G ! QRH G for the odd
k 2 Z. We now let ^QR G = {Q^RG, ^QRRG, ^QRH G} be the 2-adic completion of QR*
* G with the
induced additive operations on the components ^QRG = ^Z2 QRG, ^QRRG = ^Z2 QRR*
*G, and
^QRH G = ^Z2 QRH G.
Lemma 10.2. For a simply-connected compact Lie group G, Q^R G is a robust 2-ad*
*ic Adams
-module.
This will be proved later in 10.9. To determine the cohomology ring K*CR(G; ^*
*Z2) = {K*(G; ^Z2), KO*(G; ^Z2)}
from the representation theory of G, we now let fi :^QR G ! eK-1(G; ^Z2) be th*
*e 2-adic Adams
-module homomorphism induced by the composition of the canonical homomorphisms*
* "R G !
eK0(BG; ^Z2) ! eK-1(G; ^Z2).
Theorem 10.3. For a simply-connected compact Lie group G, there is a natural st*
*rict isomorphism
~fi:^L(Q^R G) ~=K*CR(G; ^Z2).
Proof.This follows by Lemma 10.2 and Theorem 4.9 since fi :^QRG ! K-1(G; ^Z2) i*
*nduces an
isomorphism ^(Q^RG) ~=K*(G; ^Z2) by [18].
We note that K*CR(G; ^Z2) has a simple system of generators (3.3) consisting *
*of the fi"zfl2
K-1(G; ^Z2), the fi"xff2 KO-1(G; ^Z2), and the fi"yfi2 KO-5(G; ^Z2) obtained fr*
*om the analysis
of ^QR G below in 10.8. Thus, by Proposition 3.4, K*CR(G; ^Z2) is a free 2-adi*
*c CR-module on the
associated products. However, our description of K*CR(G; ^Z2) as ^L(Q^R G) is*
* more natural and
includes the full multiplicative structure. Moreover, it will let us apply our *
*main results to G.
28
Lemma 10.4. For a simply-connected compact Lie group G, the 2-adic Adams -modu*
*le ^QR G is
regular with `: ^QRG ! ^QRG monic.
Proof.This follows by Lemmas 7.9 and 7.10 since fi :^QRG ! eK-1(G; ^Z2) is moni*
*c by Theorem
10.3.
Thus, ^QR G is strong (robust, _3-splittable, and regular) if and only if it*
* is _3-splittable, and
this is usually the case by:
Lemma 10.5. For a simply-connected compact simple Lie group G, the 2-adic Adams*
* -module
^QR G is _3-splittable (and hence strong) if and only if G is not_E6 or Spin(4*
*k + 2) with k not a
2-power.
This will be proved later in 10.11 using work of Davis [14]. For a simply-co*
*nnected compact
Lie group G, we now let ^Q = {Q^, ^QR, ^QH} briefly denote the associated stab*
*le 2-adic Adams
______
-module Q^ RG = (Q^ RG)=~OE. This agrees with the notation of [10, 9.2] and [*
*14], since our
^Q = {Q^, ^QR, ^QH} is the 2-adic completion of their Q = {Q, QR, QH }. Our ma*
*in results now give
the following omnibus theorem, whose four parts may be expanded in the obvious *
*ways to match
the cited theorems.
Theorem 10.6. Let G be a simply-connected compact Lie group such that the 2-adi*
*c Adams -
module ^QRG is _3-splittable (see Lemma 10.5), and let ^Q = {Q^, ^QR, ^QH} be t*
*he associated stable
2-adic Adams -module. Then:
(i)the K=2*-localization GK=2 is the homotopy fiber of a map 1 "E^Q!
1 "E~ae^Qwith low dimensional modifications as in Theorem 8.6;
(ii)the 2-adic v1-stabilization 1G is the homotopy fiber of a map of spectra
EQ^ ! Ea~e^Qas in Theorem 9.4;
(iii)there is an exact sequence
0 -! KO-8( 1G; ^Z2) -! ^Q=(Q^R+ ^QH) -`-!^Q=Q^R-! . . .
continuing as in Theorem 9.5;
(iv)G is bK 1-good at the prime 2 as in Theorem 9.7.
This permits calculations of the 2-primary v1-periodic homotopy v-11ss*G usin*
*g Theorem 9.2.
29
Remark 10.7. Slightly extending the present work, it is straightforward to show*
* that the above
conclusions remain valid for the exotic 2-compact group DI(4) of Dwyer-Wilkerso*
*n [17]. In particu-
lar, parts (i)-(iv) remain valid with DI(4) in place of G, with ^QK0 (BDI(4); ^*
*Z2) in place of ^QRG,
and with ^QK0 (BDI(4); ^Z2)=~OEin place of ^Q.
We devote the rest of this section to proving Lemmas 10.2 and 10.5 and start *
*by giving:
10.8. Generators for representation rings. For a simply-connected compact Lie g*
*roup G, stan-
dard results summarized in [14, Theorem 2.3] show that RG is a finitely generat*
*ed polynomial
ring Z[zfl, z*fl, xff, yfi]fl,ff,fion certain basic complex representations zfl*
*together with their conjugates
z*fl= tzfl, certain basic real representations xff, and certain basic quaternio*
*nic representations yfi.
Moreover, in terms of these generators, the Z=2-graded ring {RRG, RH G} is char*
*acterized by the fact
that its quotient {RRG=rRG, RH G=qRG} is a Z=2-graded polynomial algebra Z=2[xf*
*f, ~OEzfl, yfi]ff,fl,fi
on the real generators xffand ~OEzfl(with c~OEzfl= z*flzfl) and the quaternioni*
*c generators yfi. Conse-
quently, the indecomposables QR G = {QRG, QRRG, QRH G} may be expressed as
QRG = Z{"zfl, "z*fl, c"xff, c0"yfi}fl,ff,fi,
QRRG = Z{r"zfl, "xff, rc0"yfi}fl,ff,fi Z=2{~OE"zfl}fl,
QRH G = Z{q"zfl, qc"xff, "yfi}fl,ff,fi
where "wdenotes w-dimw for w 2 RG. Thus, the 2-adic indecomposables ^QR G = {Q*
*^RG, ^QRRG, ^QRH G}
may be expressed similarly using ^Z2in place of Z, and the stable 2-adic indeco*
*mposables ^Q =
{Q^, ^QR, ^QH} may be expressed as
Q^ = ^Z2{"zfl, "z*fl, c"xff, c0"yfi}fl,ff,fi,
Q^R = ^Z2{r"zfl, "xff, rc0"yfi}fl,ff,fi,
^QH= ^Z2{q"zfl, qc"xff, "yfi}fl,ff,fi.
10.9. Proof of Lemma 10.2. Since QR G is a ~-ring with trivial multiplication*
*, it is straightfor-
ward to check all of the required relations for operations (see 4.3 and 6.1) . *
*In particular, we deduce
``r = `r` from the relations ~4r = r~4+ ~OE~2, ~4 = -~2~2, ~OE= ~2r - r~2, 2~OE*
*= 0, and ` = -~2,
which hold generally in ~-rings with trivial multiplication [10, 6.2]. We next*
* observe that ^QRG,
^QRRG, and ^QRH G are stable 2-adic Adams modules by [6, 6.2], since QRG and QR*
*RG QRH G are
fl-nilpotent and finitely generated abelian (because they have trivial multipli*
*cations and have finite
generating sets of elements "wfor representations w). Thus, ^QR G is a 2-adic *
*Adams -module,
and it must be robust by the analysis of 10.8.
30
To check the _3-splittability of ^QR G, we let hG = ker(1-t)= im(1+t) be the*
* augmented algebra
over Z=2 obtained from RG using the involution t = _-1: RG ~=RG. This is a poly*
*nomial algebra
hG ~=Z=2[c"xff, "z*fl"zfl, c0"yfi]ff,fl,fiwhich is Z=2-graded since there is an*
* isomorphism
c + c0:RRG=rRG RH G=qRG ~= hG,
and we let QRhG ~=Z=2{c"xff, "z*fl"zfl}ff,fldenote the real (degree 0) indecomp*
*osables. We define a
homomorphism s: QRG ! QRhG by s[u] = [u*u] for u 2 "RG and note that sQRG = Z=2*
*{"z*fl"zfl}fl.
We view s as a homomorphism of _3-modules (abelian groups with endomorphisms _3*
*) as in [14,
2.4].
Lemma 10.10. For a simply-connected compact Lie group G, ^QR G is _3-splittabl*
*e if and only if
the _3-submodule sQRG QRhG is a direct summand.
Proof.By 7.2 and the proof of Lemma 7.3, ^QR G is _3-splittable if and only if*
* the _3-submodule
~OE^QRG Q^RRG=rQ^RG ( or equivalently ~OEQRG QRRG=rQRG) is a direct summand*
*. The
lemma now follows since ~OEQRG corresponds to sQRG under the isomorphism c: QRR*
*G=rQRG ~=
QRhG.
10.11. Proof of Lemma 10.5. By Lemma 10.10 and Davis [14, Theorem 1.3], the fol*
*lowing condi-
tions are equivalent: ^QR G is _3-splittable; the _3-submodule sQRG QRhG is *
*a direct summand;
G satisfies the Technical Condition of [14, Definition 2.4]; G is not_E6 or Spi*
*n(4k + 2) with k not a
2-power.
11.Proofs of basic lemmas for ^L
We shall prove Lemmas 4.5, 4.6, and 4.11 showing the basic properties of the *
*functor ^L:` M^od !
OECR ^Alg, where ` M^od is the category of 2-adic ` -modules (4.3) and OECR ^Al*
*g is that of special
2-adic OECR-algebras (3.2). We first introduce an intermediate category of modu*
*les.
11.1. The 2-adic j -modules. By a 2-adic j -module N = {NC, NR, NH , NS}, we me*
*an a
2-adic -module {NC, NR, NH }, with operations t, c, r, c0, and q as in 4.1, to*
*gether with a 2-
profinite abelian group NS and continuous additive operations ~OE:NC ! NR, j :*
*NR ! NS,
()[2]:NR ! NS, and ()[2]:NH ! NS satisfying the following relations for elemen*
*ts z 2 NC,
x 2 NR, and y 2 NH :
~OEcx = 0, ~OEc0y = 0, ~OEtz = ~OEz, 2~OEz = 0, c~OEz = 0,
(~OEz)[2]= 0, 2jx = 0, jrz = 0, (qz)[2]= (rz)[2]= jO~Ez.
31
We let j M^od denote the category of 2-adic j -modules.
11.2. A functorial interpretation of admissible maps. Let J :` M^od ! j M^od be*
* the
functor carrying a 2-adic ` -module M to the 2-adic j -module JM = {MC, MR, MH *
*, MR=rMC}
having the original operations t, c, r, c0, q, and ~OEtogether with operations *
*j :MR ! MR=rMC,
()[2]:MR ! MR=rMC, and ()[2]:MH ! MR=rMC given by jx = [x], x[2]= [`x], and y[2*
*]= [`y]
for x 2 MR and y 2 MH . Let I :OECR ^Alg ! j M^od be the functor carrying a spe*
*cial 2-adic OECR-
algebra A to the 2-adic j -module IA = {A"-1C, "A-1R, "A-5R, "A-2R} having the *
*operations t, c, r, c0, and
q of -1A"(see 4.1) together with operations ~OE:"A-1C! "A-1R, j :"A-1R! "A-2R,*
* ()[2]:"A-1R! "A-2R,
and ()[2]:"A-5R! "A-2Rgiven by ~OEz = jOEz, jx = jx, x[2]= x2, and y[2]= B-1R*
*y2 for z 2 A-1C,
x 2 A-1R, and y 2 A-5R. We now easily see:
Lemma 11.3. For M 2 ` M^od and A 2 OECRA^lg, an admissible map f :M ! A (see 4.*
*4) is
equivalent to a map f :JM ! IA in j M^od.
To construct the functor ^L, we need:
Lemma 11.4. The functor I :OECR ^Alg ! j M^od has a left adjoint ^V:j M^od ! OE*
*CR ^Alg.
Proof.This follows by the Special Adjoint Functor Theorem (see [20]) since I pr*
*eserves small limits
and since OECR ^Alg has a small cogenerating set by Lemma 11.5 below.
A special 2-adic OECR-algebra A will be called finite when the groups "AmCand*
* "AmRare finite for
all m.
Lemma 11.5. Each special 2-adic OECR-algebra A is the inverse limit of its fini*
*te quotients in
OECR ^Alg.
Proof.This is similar to the corresponding result for topological rings in [23,*
* 5.1.2]. For a 2-adic
CR-submodule G "Awith "A=G finite, we must obtain a special 2-adic OECR-ideal*
* H of A with
H G and "A=H finite. We first obtain an ideal M of AR (closed under BR, B-1R,*
* j, and ,) with
M GR and "AR=M finite as in [23]. We next obtain an ideal N of AC (closed und*
*er B, B-1, and
t) with N GC \ r-1M \ OE-1M0 and "AC=N finite as in [23]. The desired ideal H*
* is now given by
HC = N and HR = M \ c-1N.
32
11.6. Proof of Lemma 4.5. Using Lemmas 11.3 and 11.4, we obtain the desired uni*
*versal algebra
^LM from the functor ^L= ^VJ :` M^od ! OECR M^od.
A 2-adic j -module N is called sharp when j :NR=rNC ! NS is an isomorphism, a*
*nd we may
now derive the properties of ^Lfrom the corresponding properties of ^Von such s*
*harp modules,
Lemma 11.7. For a sharp 2-adic j -module N, the canonical map ^NC ! (V^N)C is a*
*n algebra
isomorphism.
Proof.Let W :OECR ^Alg ! CA^lg be the forgetful functor carrying each A 2 OECR *
*^Alg to its complex
part AC 2 CA^lg where CA^lg is the category of special 2-adic C-algebras, which*
* are defined similarly
to special 2-adic OECR-algebras (3.2) but using only complex terms and their op*
*erations. The functor
W has a right adjoint H :CA^lg ! OECR ^Alg where (HX)C = X and (HX)R = {x 2 X|t*
*x = x}
with c = 1, r = 1 + t, j = 0, OEz = z*z for z 2 X0, and OEw = B-1w*w for w 2*
* X-1. For each
N 2 j M^od and each X 2 CA^lg, a map N ! IHX in j M^od corresponds to a map NC *
*! "X-1
respecting t, which in turn corresponds to a map ^NC ! X in CA^lg. Hence, sinc*
*e WV^ is left
adjoint to IH, the canonical map ^NC ! WV^N is an isomorphism.
11.8. Proof of Lemma 4.6. For a 2-adic ` -module M, the canonical map ^MC ! (^L*
*M)C is an
isomorphism by 11.6 and Lemma 11.7.
Let ^Q:OECR ^Alg ! OECR M^od be the functor carrying each A 2 OECR ^Alg to it*
*s indecomposables
^QA 2 OECR M^od where OECR M^od is the category of special 2-adic OECR-modules,*
* which may be
defined as the augmentation ideals of the special 2-adic OECR-algebras having t*
*rivial multiplication.
Lemma 11.9. For a sharp 2-adic j -module N, the canonical map {NC, NR, NH } ! *
*-1Q^^VN
is an isomorphism.
Proof.The functor ^Qhas a right adjoint E :OECR M^od ! OECR ^Alg where EX = e_ *
* X. Since
^Q^V:j M^od ! OECR M^od is left adjoint to IE, a detailed analysis shows that ^*
*Q^VN is a special
2-adic OECR-module with (Q^^VN)-1C= NC, (Q^^VN)-1R= NR, and (Q^^VN)-5R= NH .
11.10. Proof of Lemma 4.11. For a 2-adic ` -module M, the canonical map M ! -1*
*Q^^LM is
an isomorphism by 11.6 and Lemma 11.9.
12.Proof of the Bott exactness lemma for ^L
We must now prove Lemma 4.8 showing Bott exactness of ^LM for a robust 2-adic*
* ` -module M.
This lemma will follow easily from the corresponding result for j -modules (Lem*
*ma 12.1), whose
33
proof will extend through most of this section. We say that a 2-adic j -module *
*N is profinitely
sharp when it is the inverse limit of an inverse system of finite sharp 2-adic *
*j -modules. This
obviously implies that N is sharp. We call N robust when:
(i)N is profinitely sharp;
(ii)the 2-adic -module {NC, NR=~OENC, NH } is torsion-free and exact (4.1);
(iii)ker~OE= cNR + c0NH + 2NC.
Lemma 12.1. If N is a robust 2-adic j -module, then the special 2-adic OECR-alg*
*ebra ^VN is
Bott exact; in fact, ^VN is the inverse limit of an inverse system of finitely *
*generated free 2-adic
CR-modules.
This will be proved in 12.9.
12.2. Proof of Lemma 4.8. For a robust 2-adic ` -module M, the 2-adic j -module*
* JM is also
robust, and hence ^LM has the required properties by 11.6 and Lemma 12.1.
Before proving Lemma 12.1, we must analyze the robust 2-adic j -modules, and *
*we start with:
12.3. The complex 2-adic j -modules. The functor (-)C :j M^od ! A^b from the 2-*
*adic
j -modules to the 2-profinite abelian groups has a left adjoint C :A^b ! j M^od*
* with C(G)C =
G G = G tG, C(G)R = G G=2 = rG ~OEG, C(G)H = G = qG, and C(G)S = G=2 = *
*(~OEG)[2]
for G 2 A^b. If G is torsion-free, then C(G) is obviously robust. For an arbitr*
*ary N 2 j M^od
and G 2 A^b, we may describe the possible maps N ! C(G) as follows. Let f :NC *
*! G and
g :NS ! G=2 be maps such that the diagram
0
NR NH fc+fc----!G
?? ?
y ()[2] ?y1
NS --g--!G=2
commutes. Then there is a map F(f, g): N ! C(G) with components (f, ft): NC ! *
*G G,
(fc, gj): NR ! G G=2, fc0:NH ! G, and g :NS ! G=2. Moreover, each map N ! C(G*
*) is of
the above form for some f and g. When N is robust, the compatibility condition *
*on f :NC ! G
and g :NS ! G=2 may be expressed by the commutativity of the diagram
N+C --f--! G
?? ?
yss ?y1
NS --g--!G=2
34
` *
* +
where N+C= {z 2 NC|tz = z} and ss is the composition of (c, c0): NR=~OENC NC N*
*H ~=NC and
` -
()[2]:NR=~OENC NC NH ! NS. Letting NC = {z 2 NC|tz = -z}, we now have:
Lemma 12.4. If "N N is an inclusion of robust 2-adic j -modules such that NC=N*
*"Cis torsion-
free and "N-C= N-C, then each map "N! C(G) for G 2 ^Ab may be extended to a map*
* N ! C(G)
of 2-adic j -modules.
Proof.For a given map F(f", "g): "N! C(G), we first extend "g:"NS! G=2 to a map*
* g :NS ! G=2.
Since N"C=N"+C~=N"-C, NC=N+C~= N-C, and N"-C= N-C, we see that NC is the pusho*
*ut of the
inclusions N+C "N+C! "NC. Thus, the maps gss :N+C! G=2 and [f"]: "NC! G=2 indu*
*ce a map
f0: NC ! G=2, and we obtain a commutative diagram
"NC---"f-! G
?? ?
y ?y1
0
NC --f--! G=2
Since NC=N"Cis projective in ^Ab, we may now choose a lifting f :NC ! G in the *
*diagram, and this
gives the desired extension F(f, g): NC ! C(G) of F(f", "g).
Lemma 12.5. For a robust 2-adic j -module N, there exists a decomposition N ~=C*
*(G) P where
G is torsion-free and P is robust with t = 1 on PC.
Proof.By the factorization of positively torsion-free groups in 5.3, there exis*
*ts a decomposition
NC ~=(G tG) H with t = 1 on H, and we let i: C(G) ! N be the induced map. *
*Then i
is monic since i: {G tG, G, G} ! {NC, NR=~OENC, NH } is monic by [10, Lemma 4*
*.8], and since
i: G=2 ! ~OENC and j :~OENC ! NS are monic by the proof of Lemma 7.3. Thus, i: *
*C(G) ! N has
a left inverse by Lemma 12.4, and the result follows.
12.6. The t-trivial robust 2-adic j -modules. A robust 2-adic j -module N will *
*be called
t-trivial when t = 1 on NC. Such an N must have ~OE= 0: NC ! NR since NC = cNR *
*+ c0NH by
the exactness of {NC, NR=~OENC, NH }. Moreover, it must also have (rNC)[2]= 0, *
*(qNC)[2]= 0, and
c + c0:NR=rNC NH =qNC ~=NC=2 by [10, Lemma 4.7]. Hence, the operations ()[2]:*
*NR ! NS
and ()[2]:NH ! NS induce operations ~`:NR=rNC ! NR=rNC and ~`:NH =qNC ! NR=rNC,
where the ~`-module NR=rNC is profinite since N is profinitely sharp. In this w*
*ay, a t-trivial robust
2-adic j -module N corresponds to a torsion-free group G 2 A^b together with a *
*decomposition
(G=2)R (G=2)H = G=2 equipped with operations ~`:(G=2)R ! (G=2)R and ~`:(G=2)H *
*! (G=2)R
35
such that the ~`-module (G=2)R is profinite. We say that a 2-adic j -module N i*
*s of finite type
when NC, NR, NH , and NS are finitely generated over ^Z2, and we now easily ded*
*uce:
Lemma 12.7. A t-trivial robust 2-adic j -module may be expressed as the inverse*
* limit of an
inverse system of t-trivial robust quotient modules of finite type.
A similar result obviously holds for the robust 2-adic j -modules C(G) with G*
* torsion-free, and
this will allow us to focus on the robust modules of finite type since we have:
Lemma 12.8. If a 2-adic j -module N is the inverse limit of an inverse system {*
*Nff}ffof quotient
modules, then ^VN ~=limff^VNff.
Proof.For a finite special 2-adic OECR-algebra F, there is a canonical isomorph*
*ism Hom(limff^VNff, F) ~=
Hom(V^N, F). Hence, the map ^VN ! limff^VNffis an isomorphism by Lemma 11.5.
12.9. Proof of Lemma 12.1. It now suffices to show that ^VN is a free 2-adic CR*
*-module when
N = C(G) P for a finitely generated free ^Z2-module G and a t-trivial robust 2-*
*adic j -module P of
finite type. By 7.1, we may choose finite ordered sets of elements {zk}k in G, *
*{xi}iin PR, and {yj}j
in PH such that G is a free ^Z2-module on {zk}k and {PC, PR, PH } is a free 2-a*
*dic -module on {xi}i
and {yj}j. Since PS is a free Z=2-module on the generators {jxi}i, there are ex*
*pressions x[2]i= ri
and y[2]j= sj for each i and j where the riand sj are Z=2-linear combinations o*
*f these generators.
We may now obtain ^VN as the free augmented 2-adic CR-algebra on the generators*
* xi2 (g^VN)-1R,
yj 2 (g^VN)-5R, zk 2 (g^VN)-1C, and OEzk 2 (g^VN)0Rsubject to the relations x2*
*i= ri, y2j= BRsj,
z2k= 0, z*kzk = BcOEzk, and (OEzk)2 = 0 for each i, j, and k. It follows by a s*
*traightforward analysis
that ^VN is a free 2-adic CR-module on the associated products (3.3) of {xi}i, *
*{yj}j, and {zk}k.
13.Proofs for regular modules
We first show that our strict nonlinearity condition (7.7) for 2-adic Adams m*
*odules agrees with
that of [7, 2.4], and we then prove Lemmas 7.9 and 7.10 for regular modules. Fo*
*r a 2-adic Adams
module A, we let TA A be given by the pullback square
TA ----! (A=_2A)\2
?? ?
y ?y
A ----! A=_2A
where (A=_2A)\2 is the kernel of 2: A=_2A ! A=_2A. Since the square is also a p*
*ushout, A is
quasilinear if and only if TA = A. Now let T1 A be the intersection of the subm*
*odules TiA A for
i > 0.
36
Lemma 13.1. T1 A is the largest quasilinear submodule of A, and hence Aql= T1 A.
Proof.Using the inverse limit of the pullback squares for TiA with i > 0, we fi*
*nd that T1 A contains
each quasilinear submodule of A and that T(T1 A) = T1 A.
13.2. Strict nonlinearity conditions. Our definition of strict nonlinearity in *
*7.7 is equivalent to
our earlier definition in [7, 2.3 and 2.4]. In fact, for a 2-adic Adams module *
*A, the largest quasilinear
submodule Aqlremains unchanged in the earlier category of 2-adic _2-modules, si*
*nce it is still given
by T1 A. To prove Lemma 7.10, we need:
Lemma 13.3. For a strictly nonlinear 2-adic Adams module A, each submodule is s*
*trictly nonlin-
ear. Moreover, when A is finitely generated over ^Z2, each torsion-free quotien*
*t module is strictly
nonlinear.
Proof.The first statement is clear, and we shall prove the second by working in*
* the earlier category
^Nof 2-adic _2-modules that are _2-pronilpotent. Let 0 ! "A! A ! ~A! 0 be a sh*
*ort exact
sequence in ^Nwith A strictly nonlinear and finitely generated over ^Z2and with*
* ~Atorsion-free. To
show that ~Ais strictly nonlinear, it suffices to show that Hom ^N(H, ~A) = 0 f*
*or each torsion-free
quasilinear H 2 N^ that is finitely generated over ^Z2. Since ~Ais torsion-fre*
*e, it now suffices to
show that Hom ^N(H, ~A) is finite for such H. Hence, since Hom ^N(H, A) = 0 by *
*strict nonlinearity,
it suffices to show that Ext1^N(H, "A) is finite for such H. This finiteness f*
*ollows using the exact
sequence
0 -! HomN^(H, "A) -! HomA^b(H, "A) -! HomA^b(H, "A) -! Ext1^N(H, "A) -! 0
with HomN^(H, "A) = 0 by strict nonlinearity, where ^Ab is the category of 2-pr*
*ofinite abelian groups.
13.4. Proof of Lemma 7.10. This result follows easily from 7.8 and Lemma 13.3.
13.5. Proof of Lemma 7.9. By [8, Lemma 5.5], there is an exact sequence
0 -! eK1(X=X3; ^Z2) -! eK1(X; ^Z2) -! H3(X; ^Z2)
of 2-adic Adams modules with H3(X; ^Z2) linear and eK1(X=X3; ^Z2) torsion-free,*
* where X3 is the
3-skeleton of X. Hence, it suffices to show that eK1(X=X3; ^Z2) is strictly no*
*nlinear with monic
_2. Since Hm (X; ^Z2) = 0 for sufficiently large m, the map eK1(X=X3; ^Z2) ! eK*
*1(Xm =X3; ^Z2) is
monic for such m. Thus by skeletal induction, the operator _2 on Q eK1(X=X3; *
*^Z2) is annihilated
by the polynomial f(x) = (x - 22)(x - 23) . .(.x - 2k) for sufficiently large k*
*. It follows that
37
Q eK1(X=X3; ^Z2) is the direct sum of the eigenspaces Eiof _2 with eigenvalue*
*s 2ifor 2 i k,
and hence _2 is monic on eK(X=X3; ^Z2) as desired. Moreover, the projection to*
* Ei is given by
the operator fi(_2)=fi(2i) on Q eK1(X=X3; ^Z2) where fi(x) = f(x)=(x - 2i). T*
*his implies that
L k
2vKe1(X=X3; ^Z2) is contained in i=2Ei\Ke1(X=X3; ^Z2) where 2v is the highest*
* power of 2 dividing
an integer fi(2i) for some i. Since the above direct sum is strictly nonlinear,*
* so is 2vKe1(X=X3; ^Z2)
by Lemma 13.3, and hence so is eK1(X=X3; ^Z2).
14.Proof of the realizability theorem for ^LM
We shall prove Theorem 8.5 giving a strict isomorphism ^LM ~=K*CR(gFibf; ^Z2)*
* for a companion
map f : 1 "E~M! 1 "E~ae~Mof a strong 2-adic Adams -module M. For this, it wi*
*ll suffice by
our comparison theorem (4.9) to obtain an isomorphism ^MC ~= K*(gFibf; ^Z2) of *
*the complex
components. We do this by adapting our proof of the corresponding odd primary r*
*esult (Theorem
4.7) in [8]. First, to determine the 2-adic K-cohomology of the loops on 1 "E~*
*Mor 1 "E~ae~M, we
may replace Theorem 11.2 of [8] by the following two theorems.
Theorem 14.1. If X = 1 E for a 1-connected spectrum E with H2(E; ^Z2) = 0, wit*
*h K0(E; ^Z2) =
0, and with K1(E; ^Z2) torsion-free, then K1( X; ^Z2) = 0 and K0( X; ^Z2) is to*
*rsion-free.
Proof.This follows by [6, Theorem 8.3].
Theorem 14.2. If X is a 1-connected H-space with K1( X; ^Z2) = 0 and K0( X; ^Z2*
*) torsion-free,
then oe :U(Q^K1(X; ^Z2) # H3(X; ^Z2)) ~=K0( X; ^Z2).
Proof.This follows from [7, Theorem 10.2].
When X is 1 "E~Mor 1 "E~ae~M, we shall determine H3(X; ^Z2) from the united*
* 2-adic K-
-1
cohomology of X. For any 1-connected space X, we let ffR :gKO (X; ^Z2) ! H3(X;*
* ^Z2) be the
homomorphism induced by the Postnikov section KO^Z2! P4KO^Z2. Using the indecom*
*posables
^QKO*(X; ^Z2) of 2.7 and 4.10, we have:
-1
Lemma 14.3. If X is a 1-connected space with H2(X; ^Z2) = 0, then ffR :gKO (X;*
* ^Z2) !
H3(X; ^Z2) factors through ^QKO-1(X; ^Z2) and vanishes on the following subgrou*
*ps: ~OEeK-1(X; ^Z2),
-1 -5 -1
(` - 2)gKO (X; ^Z2), (` - rB-2c)gKO (X; ^Z2), and (_3- 9)gKO (X; ^Z2).
Proof.The map ffR factors through ^QKO-1(X; ^Z2) by a suspension argument using*
* the isomor-
phism H3(X; ^Z2) ~=H2( X; ^Z2). Since X is 1-connected with H2(X; ^Z2) = 0, the*
*re is a natural
38
isomorphism H3(X; ^Z2) ~=(ss2(o2X))# by [8, Lemma 11.4]. Thus, it suffices by n*
*aturality to prove
the desired vanishing results when X is S2 [2ke3 for k 1, and these results n*
*ow follow from the
elementary case X = S3 since the collapsing map S2 [2ke3 ! S3 induces epimorphi*
*sms of the
-1 -5
cohomologies eK-1(-; ^Z2), gKO (-; ^Z2), and gKO (-; ^Z2).
For a 1-connected space X with H2(X; ^Z2) = 0, the above ffR now induces a ho*
*momorphism
~ffR:Lin^QK-1(X; ^Z2) ! H3(X; ^Z2) where ^QK-1(X; ^Z2) is the 2-adic Adams -mo*
*dule of in-
decomposables given by 4.10 and 6.1, and where Lin carries a 2-adic Adams -mod*
*ule M to the
group
Lin M = MR=(~OEMC + (` - 2)MR + (` - rc0)MH + (_3- 9)MR).
To determine H3(X; ^Z2) when X is 1 "E~Mor 1 "E~ae~M, we may replace Proposit*
*ion 11.3 of [8] by:
Proposition 14.4. If N is a torsion-free exact stable 2-adic Adams -module, th*
*en
~ffR:Lin^QK-1( 1 "EN; ^Z2) ~=H3( 1 "EN; ^Z2).
Proof.Since there is a stable isomorphism ~ffR:KO-1(E"N; ^Z2)=(_3 - 9) ~=H3(E"N*
*; ^Z2) by [10,
Theorem 3.2] and [8, Lemma 11.4], the proposition follows using Theorem 6.7 and*
* Lemma 4.11.
For any `-pronilpotent 2-adic Adams -module M, we obtain a homomorphism r: M*
*C !
Lin M of 2-adic Adams modules with MC as in 7.6 and Lin M linear. Such a homomo*
*rphism is
called properly torsion-free [7, 4.5] when its source is torsion-free and its k*
*ernel is strictly nonlinear
(7.7). We shall need:
Lemma 14.5. If M is a strong 2-adic Adams -module, then r: MC ! Lin M is prop*
*erly
torsion-free.
Proof.Since M is strong, MC is torsion-free and ker(MC ! LinMC ) is strictly no*
*nlinear. Using
the maps r: LinMC ! Lin M and c: Lin M ! LinMC with cr = 2, we see that 2 ker(M*
*C !
Lin M) is contained in ker(MC ! LinM). Thus, ker(MC ! Lin M) is strictly nonli*
*near by
Lemma 13.3.
As in [8, Section 11], for a strong 2-adic Adams -module M and a companion m*
*ap f, we obtain
a ladder of p-complete fiber sequences
gFibf----! X --"f--! Y
?? ? ?
y ?y ?y
Fibf ----! 1 "E~M--f--! 1 "E~ae~M
39
such that:
(i)X and Y satisfy the hypotheses of Theorems 14.1 and 14.2;
(ii)the vertical maps from X and Y are K*(-; ^Z2)-equivalences;
(iii)H3(Y ; ^Z2) = 0 and the sequence H3( 1 "E~ae~M; ^Z2) ! H3( 1 "E~M; ^Z2) !
H3(X; ^Z2) ! 0 is exact.
Lemma 14.6. There is a canonical isomorphism H3(X; ^Z2) ~=Lin M.
Proof.Since f*: K*CR( 1 "E~ae~M; ^Z2) ! K*CR( 1 "E~M; ^Z2) is equivalent to ^L~*
*d:^L"F~ae~M! ^L"F~Mfor
the `-resolution map ~d, the homomorphism f*: H3( 1 "E~ae~M; ^Z2) ! H3( 1 "E~M;*
* ^Z2) is equivalent
to Lin ~d:Lin "F~ae~M! Lin "F~Mby Proposition 14.4. Hence, there is an isomorph*
*ism of cokernels
H3(X; ^Z2) ~=Lin M.
14.7. Proof of Theorem 8.5. The proof of Theorem 4.7 in [8] is now easily adapt*
*ed to give
Theorem 8.5. In more detail, Propositions 11.5 and 11.6 of [8] remain valid in *
*our setting using
Lemmas 14.5 and 14.6 together with the short exact sequence
0 -! (F"MC # 0) -! (F"MC # Lin M) -! (MC # Lin M) -! 0
induced by the `-resolution. Propositions 11.7 and 11.8 likewise remain valid, *
*and thus ^MC ~=
K*(gFibf; ^Z2), so that Theorem 8.5 follows by Theorem 4.9.
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Department of Mathematics, University of Illinois at Chicago, Chicago, Illino*
*is 60607
E-mail address: bous@uic.edu