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. 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