KUNNETH THEOREMS AND UNSTABLE OPERATIONS IN 2-ADIC KO-COHOMOLOGY A.K. BOUSFIELD Abstract.We develop Kunneth theorems and obtain detailed results on un- stable operations in 2-adic KO-cohomology and, more generally, in united * *2- adic K-cohomology. These results are needed for work on the K-localizatio* *ns of H-spaces at the prime 2 and should be of independent interest. Our pro* *ofs of relations for unstable operations rely on Atiyah's Real K-theory and o* *n a convenient mod 2 simplification of 2-adic KO-cohomology. 1.Introduction In [9], we determined the K-localizations of finite H-spaces and related spac* *es at an odd prime, and we recently found that this work can be extended to the prime 2 with some m* *odifications and restrictions. For this, we must use various basic results on Kunneth theor* *ems and unstable operations, particularly in 2-adic KO-cohomology, that are not covered in the l* *iterature, and we provide the required background in this paper. In [2], Atiyah proved a Kunneth theorem for the K-cohomology of a product of * *finite CW- complexes but noted that the corresponding theorem fails for KO-cohomology beca* *use of difficulties at the prime 2. As suggested in [6] and [4], these difficulties may be overcom* *e by using united K-theory. For a space X, the united 2-adic K-cohomology Kb*CRTX = K*CRT(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2), KT*(X; ^Z2)} consists of the complex, real, and self-conjugate 2-adic K-cohomologies of X to* *gether with the realifi- cation, complexification, and other "stable KO-module operations" relating thes* *e cohomologies (see [5, Sections 2]). This has much better homological algebraic properties than 2-* *adic KO-cohomology alone, and we now obtain a Kunneth short exact sequence CRT *+1 * 0 -! bK*CRTX ^CRTbK*CRTY -! bK*CRT(XxY ) -! ^Tor1 (KbCRTX, bKCRTY ) -! 0 ___________ Date: September 20, 2005. 2000 Mathematics Subject Classification. 55N15, 55S25, 55U25. Key words and phrases. KO-cohomology operations, K-theory operations, Kunneth* * formula, 2-adic K-theory, united K-theory. 1 2 for arbitrary spaces X and Y in Theorem 2.12. This Kunneth theorem can often be* * simplified by eliminating the self-conjugate components and using the (partial) united 2-adic* * K-cohomology Kb*CRX = K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)} instead of bK*CRTX, as we show in Theorem 2.8 under the common condition that K* **CR(X; ^Z2) is CR-exact. In this case, KO*(X x Y ; ^Z2) depends on the 2-adic K-cohomology as * *well as the 2-adic KO-cohomology of X and Y . A major purpose of this paper is to study the unstable operations in the unit* *ed 2-adic KO- cohomology, K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)}, of a space X. The basic u* *nstable operations in K*(X; ^Z2) are just the exterior squares ` = -~2: K0(X; ^Z2) -! K0(X; ^Z2) ` = -~2: K-1(X; ^Z2) -! K-1(X; ^Z2) which give it the structure of a Z=2-graded 2-adic `-ring (see 3.4), and which * *combine with the stable Adams operations to give it the structure of a Z=2-graded 2-adic ~-ring (see [7* *, Theorem 6.2]). We find that the other basic unstable operations in {K*(X; ^Z2), KO*(X; ^Z2)} are * *the exterior squares ` = -~2: KOn(X; ^Z2) -! KO0(X; ^Z2) forn = 0, -4 ` = -~2: KOn(X; ^Z2) -! KO-1(X; ^Z2) forn = -1, -5 ` = -~2: KOn(X; ^Z2) -! KO1(X; ^Z2) forn = 1, -3 and the multiplicative operations OE: K0(X; ^Z2) -! KO0(X; ^Z2) OE: K-1(X; ^Z2) -! KO0(X; ^Z2) of Seymour [16], which are all constructed in [10, Section 6] using Atiyah's Re* *al K-theory. These operations satisfy a large assortment of algebraic relations which we present i* *n Section 3. Some of the relations are well-known or are easily proved using the methods of Real K-t* *heory, while others seem to defy those methods. To establish these difficult relations, we use a mo* *d 2 simplification j*X of the cohomology KO*(X; ^Z2) as explained below. For a space X, we let j*X denote the quotient of KO*(X; ^Z2) by the realifica* *tion ideal rK*(X; ^Z2). We find that j*X is an algebra over the 8-periodic exterior algebra j* = Z=2[j,* * BR, B-1R] with |j| = -1 and |BR| = -8, and we obtain a Kunneth isomorphism j*X ^j*j*Y ~= j*(X * *x Y ) in Theorem 5.2 for an arbitrary space Y when K*CR(X; ^Z2) is CR-exact with K*(X; ^* *Z2) torsion-free. 3 For many H-spaces X, this applies to show that j*X is actually a Hopf algebra o* *ver j*. In partic- ular, j*dKO_mand j*Kb_mare such Hopf algebras, where dKO_mand bK_mare the infin* *ite loop spaces m representing the cohomologies gKO (-; ^Z2) and eKm(-; ^Z2). To prove a relation* * ff = fi for unstable operations in {K*(X; ^Z2), KO*(X; ^Z2)}, we first check that the possible error* * ff - fi is additive and has trivial complexification. We then deduce that this error is represented by * *a primitive element ffl 2 Pj*dKO_mor ffl 2 Pj*Kb_m. We finally show that this ffl must vanish by us* *ing our knowledge of Pj*dKO_mand Pj*Kb_m, and we thereby deduce ff = fi. This method allows us to es* *tablish all of the difficult relations for our unstable operations. For a space X such as dKO_mor bK_m, we may approach j*X by first studying a m* *od 2 simplification h*X of K*(X; ^Z2), where hnX = ker(1 - t)= im(1 + t) for the conjugation t = _-* *1 on Kn(X; ^Z2). We find that h*X is an algebra over the 4-periodic algebra h* = Z=2[B2, B-2] an* *d that h*X has a natural differential @ :h*X ! h*-1X satisfying a Leibniz formula (see Theorem 5* *.9). Moreover, this differential vanishes if and only if K*CR(X; ^Z2) is CR-exact, which happens if* * and only if j*X is free over j*. In this case, we have j*X=j ~=h*X, and we may view h*X as a first appr* *oximation to the algebra j*X. For many H-spaces X, we can go even farther and view h*X as a firs* *t approximation to the Hopf algebra j*X. We use this approach to derive the needed properties * *of j*dKO_mand j*Kb_m, and we believe that the functors j* and h* are of independent interest. The paper is divided into the following sections: 1. Introduction 2. Kunneth theorems for united 2-adic K-cohomology 3. Unstable operations in united 2-adic K-cohomology 4. Proofs of the Kunneth theorems 5. The mod 2 K-cohomological functors j* and h* 6. On the j* groups of some infinite loop spaces 7. Proofs of relations for unstable operations The present work will allow us to determine the structure of the united 2-adi* *c K-cohomology K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)} for many H-spaces X, and this will all* *ow us to derive the promised 2-primary extensions of our results in [9] on K-localizations of space* *s. It will also allow us to complete the work of [10] on the v1-periodic homotopy theory of compact Lie * *groups. 4 2.Kunneth theorems for united 2-adic K-cohomology In this section, we first discuss the 2-adic version of Atiyah's original Kun* *neth theorem [3] and then develop our Kunneth theorems for the united 2-adic K-cohomology theories K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)}, K*CRT(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2), KT*(X; ^Z2)}, of spectra or spaces X. We note that the Kunneth theorem for K*CRT(-; ^Z2) is t* *he strongest, but may be somewhat cumbersome since it involves the self-conjugate theory KT*(-; ^* *Z2). For simplicity, we shall focus mainly on the version for K*CR(-; ^Z2). The results of this sect* *ion immediately extend to an odd prime p, but these extensions add nothing valuable beyond the Kunneth* * theorem for K*(-; ^Zp) since K*CR(X; ^Zp) and K*CRT(X; ^Zp) are naturally determined by K*(* *X; ^Zp) as in [5, Theorem 4.6]. 2.1. The 2-profinite abelian groups. Recall that a 2-profinite abelian group is* * the topological inverse limit of an inverse system of finite 2-torsion abelian groups. For ins* *tance, each finitely generated ^Z2-module is a 2-profinite abelian group with the canonical topology* *. In general, the 2-profinite abelian groups correspond to the 2-torsion abelian groups by Pontrj* *agin duality, and they form an abelian category A^b. For 2-profinite abelian groups A, B 2 A^b, * *let A^ B 2 A^b be the complete tensor product with A^ B = limff,fiAff Bfiwhere Affand Bfirange o* *ver the finite 2-torsion quotients of A and B (see e.g. [15, p.184]). Note that A^ B ~=A ^Z2B* * when A and B are finitely generated ^Z2-modules. The operation ^ gives a symmetric monoidal stru* *cture to the abelian category ^Ab. For A 2 ^Ab, the functor A^ -: ^Ab ! ^Ab is right exact, and we c* *all A flat when A^ - is exact. Using Pontrjagin duality, we see that the following conditions on A a* *re equivalent: (i)A is flat in ^Ab; (ii)A is torsion-free; (iii)A is projective in ^Ab. Moreover, there are enough projectives in A^b, and the complete tensor product * *^ has left derived functors ^Torn(A, B) 2 ^Ab for A, B 2 ^Ab and n 0. As usual, ^Tor0(A, B) ~=A^* * B and ^Torn(A, B) = 0 for n > 1. 2.2. The 2-adic K*-modules. By a 2-adic K*-module M, we mean a Z-graded 2-profi* *nite abelian group with operation B :M* ~=M*-2. We let ^K*denote the abelian category of the* *se modules. For a spectrum or space X, we obtain a 2-adic K*-module K*(X; ^Z2) using the Bott p* *eriodicity B and 5 the limit topology on K*(X; ^Z2) ~=limffK*(Xff; ^Z2) where Xffranges over the f* *inite subcomplexes of X. For M, N, P 2 ^K*, a K*-pairing f :(M, N) ! P consists of homomorphisms f* * :Mm ^Nn ! Pm+n for m, n 2 Z such that Bf = f(B ^1) = f(1^ B) on each Mm ^Nn. The complet* *e tensor product M ^K*N 2 ^K*may be defined as the target of the universal K*-pairing (M* *, N) ! M ^K*N, and we note that (M ^K*N)0 ~= (M0^ N0) (M1^ N1), (M ^K*N)1 ~= (M1^ N0) (M0^ N1). The operation ^K* in the abelian category ^K*inherits the basic properties of ^* * in A^b (see 2.1), K* * * and ^K* has left derived functors ^Torn(M, N) 2 K^ for M, N 2 K^ and n 0. As* * usual, * K* ^TorK0(M, N) ~=M ^K*N and ^Torn(M, N) = 0 for n > 1. For spectra X and Y , the * *product homomorphisms ~: Km (X; ^Z2)^ Kn(Y ; ^Z2) -! Km+n (X ^ Y ; ^Z2) give a product homomorphism ~: K*(X; ^Z2)^ K*K*(Y ; ^Z2) -! K*(X ^ Y ; ^Z2) of 2-adic K*-modules. This also holds for spaces X and Y when X ^ Y is replaced* * by X x Y . We can now state a 2-adic version of Atiyah's Kunneth theorem [2], where we write * *K*(X; ^Z2) as bK*X. Theorem 2.3. For spectra X and Y , there is a natural short exact sequence K* *+1 * 0 -! bK*X ^K*bK*Y -~-!bK*(X ^ Y ) -! ^Tor1(Kb X, bKY ) -! 0 of 2-adic K*-modules. This also holds for spaces X and Y when X ^ Y is replaced* * by X x Y . This will be proved in 4.7, and we now develop our Kunneth theorems for the u* *nited 2-adic K-cohomology theories. 2.4. The 2-adic CR-modules. By a 2-adic CR-module, we mean a CR-module over A^b* * in the sense of [10, Section 4.1]. Thus, a 2-adic CR-module consists of Z-graded 2-pro* *finite abelian groups MC and MR with operations 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, 6 t2 = 1, tB = -Bt, rt = r, tc = c, cBR = B4c, rB4 = BRr, cr = 1 + t, rc = 2, rBc = j2, rB-1c = 0. These operations are patterned after the usual periodicity, conjugation, Hopf, * *complexification, and realification operations in K-theory. For z 2 M*C, the element tz is sometimes * *written as _-1z or z*. We let ^CR denote the abelian category of 2-adic CR-modules. For a spectrum* * or space X, we obtain a 2-adic CR-module K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)} using the operations coming from the standard maps of the spectra K and KO [5, * *Section 1.9] and using the limit topology on K*CR(X; ^Z2) ~=limffK*CR(Xff; ^Z2) where Xffran* *ges over the finite subcomplexes of X. 2.5. Bott exactness and CR-exactness. As in [5, Section 4.7] or [10, Section 4.* *1], we say that a 2-adic CR-module M 2 ^CR is Bott exact when the Bott sequence -1 j . .-.! M*+1R-j-!M*R-c-!M*CrB---!M*+2R--!. . . is exact, and we say that M is CR-exact when it is Bott exact and the chain com* *plex . .-.! M*+1R=r -j-!M*R=r -j-!M*-1R=r -j-!. . . is exact. For a short exact sequence 0 ! L ! M ! N ! 0 of 2-adic CR-modules, we* * note that if any two of the three modules are Bott exact or CR-exact, then so is the third. * *We also note that a Bott exact CR-module M is automatically CR-exact when MC is concentrated in e* *ven or odd degrees. Thus, for a spectrum or space X, the 2-adic CR-module K*CR(X; ^Z2) is * *always CR-exact when "K*(X; ^Z2) is concentrated in even or odd degrees. 2.6. Complete tensor products of 2-adic CR-modules. For 2-adic CR-modules M, N,* * P 2 ^CR, a CR-pairing f :(M, N) ! P consists of homomorphisms fC :MmC^NnC ! Pm+nC a* *nd fR :MmR^NnR! Pm+nRfor m, n 2 Z such that: (i)BfC = fC(B ^1) = fC(1^ B) on each MmC^NnC; (ii)BRfR = fR(BR ^1) = fR(1^ BR) on each MmR^NnR; (iii)jfR = fR(j ^1) = fR(1^ j) on each MmR^NnR; (iv)tfC = fC(t^ t) on each MmC^NnC; (v)cfR = fC(c^ c) on each MmR^NnR; (vi)rfC(c^ 1) = fR(1^ r) on each MmR^NnCand rfC(1^ c) = fR(r ^1) on each MmC^NnR. 7 The complete tensor product M ^CRN 2 ^CR may be defined as the target of the un* *iversal CR-pairing (M, N) ! M ^CRN. We note that (M ^CRN)C ~=MC ^K*NC while (M ^CRN)R depends on b* *oth real and complex components; it is generated by terms MmR^NnRtogether with term* *s r(MmC^NnC). The operation ^CR gives a symmetric monoidal structure to the abelian category * *^CR. For M 2 ^CR, the functor M ^CR: ^CR ! ^CR is right exact, and we call M flat when M ^CR is e* *xact. Lemma 2.7. For a 2-adic CR-module M, the following are equivalent: (i)M is flat in ^CR; (ii)M is CR-exact with M*Ctorsion-free; (iii)M is projective in ^CR. The abelian category ^CR has enough projectives. This will be proved in 4.3. The complete tensor product ^CR in ^CR now has le* *ft derived func- CR CR tors ^Torn(M, N) 2 ^CR for M, N 2 ^CR and n 0. As usual, ^Tor0(M, N) ~=M ^CR* *N and ^TorCRn(M, N) = 0 for n > 1 when M or N is CR-exact. For spectra X and Y , the* * product homomorphisms ~: Km (X; ^Z2)^ Kn(Y ; ^Z2) -! Km+n (X ^ Y ; ^Z2) ~: KOm (X; ^Z2)^ KOn(Y ; ^Z2) -! KOm+n (X ^ Y ; ^Z2) give a CR-pairing by [5, Section 1], and they consequently determine a product * *homomorphism ~: KmCR(X; ^Z2)^ CRKnCR(Y ; ^Z2) -! Km+nCR(X ^ Y ; ^Z2) of 2-adic CR-modules. This also holds for spaces X and Y when X ^ Y is replaced* * by X x Y . We can now state our Kunneth theorem for the cohomology K*CR(-; ^Z2), which we wri* *te as bK*CR(-). Theorem 2.8. For spectra X and Y with ^K*CRX (or bK*CRY ) CR-exact, there is a * *natural short exact sequence CR *+1 * 0 -! bK*CRX ^CRbK*CRY -~-!bK*CR(X ^ Y ) -! ^Tor1(KbCRX, bKCRY ) -! 0 of 2-adic CR-modules. This also holds for spaces X and Y when X ^ Y is replaced* * by X x Y . This will be proved in 4.6. Finally, we develop an unrestricted general Kunn* *eth theorem for K*CRT(-; ^Z2). Since this is basically similar to the version over CR, we shall* * be quite brief and shall assume familiarity with [5]. 8 2.9. The 2-adic CRT-modules. A 2-adic CRT-module M = {MC, MR, MT} consists of Z* *-graded 2-profinite abelian groups MC, MR, and MT with the operations and relations of * *[5, Section 2.1], but with cohomological indexing. We let ^CRTdenote the abelian category of 2-ad* *ic CRT-modules. For a spectrum or space X, we obtain a 2-adic CRT-module K*CRT(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2), KT*(X; ^Z2)} using the operations coming from the standard maps of the spectra K, KO, and KT* * [5, Section 1.9]. We say that a 2-adic CRT-module M is CRT-exact when the three chain complexes o* *f [5, Section 2.3] (including the Bott sequence) are exact for M. For a short exact sequence 0 ! L* * ! M ! N ! 0 of 2-adic CRT-modules, if any two of the modules are CRT-exact, then so is the thi* *rd. For a spectrum or space X, the 2-adic CRT-module K*CRT(X; ^Z2) is always CRT-exact. 2.10. Complete tensor products of 2-adic CRT-modules. For 2-adic CRT-modules M,* * N, P 2 ^CRT, a CRT-pairing f :(M, N) ! P consists of homomorphisms fC :MmC^NnC! Pm+nC,* * fR :MmR^NnR! Pm+nR, and fT :MmT^NnT! Pm+nT, for m, n 2 Z such that: (i)BfC = fC(B ^1) = fC(1^ B) on each MmC^NnC; (ii)BRfR = fR(BR ^1) = fR(1^ BR) on each MmR^NnR; (iii)BTfT = fT(BT ^1) = fT(1^ BT) on each MmT^NnT; (iv)jfR = fR(j ^1) = fR(1^ j) on each MmR^NnR; (v)jfT = fT(j ^1) = fT(1^ j) on each MmT^NnT; (vi)tfC = fC(t^ t) on each MmC^NnC; (vii)tTfT = fT(tT ^tT) on each MmT^NnTwhere tT = _-1T; (viii)fflfR = fT(ffl^ ffl) on each MmR^NnR; (ix)ifT = fC(i ^i) on each MmT^NnT; (x)ofT(1^ ffl) = fR(o ^1) on each MmT^NnRand ofT(ffl^ 1) = (-1)m fR(1^ o) on each MmR^NnT; (xi)flfC(1^ i) = fT(fl ^1) on each MmC^NnTand flfC(i ^1) = (-1)m fT(1^ fl) on each MmT^NnC; (xii)fflofT = fT(fflo ^1) + (-1)m fT(1^ fflo) + jfT on each MmT^NnT. The complete tensor product M ^CRTN 2 ^CRTmay be defined as the target of the u* *niversal CRT- pairing (M, N) ! M ^CRTN. The operation ^CRT gives a symmetric monoidal structu* *re to the abelian category ^CRT. For M 2 ^CRT, the functor M ^CRT: ^CRT! ^CRTis right exa* *ct, and we call M flat when M ^CRT is exact. 9 Lemma 2.11. For a 2-adic CRT-module M, the following are equivalent: (i)M is flat in ^CRT; (ii)M is CRT-exact with M*Ctorsion-free; (iii)M is projective in ^CRT. The abelian category ^CRThas enough projectives. This will be proved in 4.10. The complete tensor product ^CRT in ^CRT now has* * left derived CRT CRT functors ^Torn (M, N) 2 ^CRTfor M, N 2 ^CRTand n 0. As usual, ^Tor0 (M, N) ~=* *M ^CRTN CRT and ^Torn (M, N) = 0 for n > 1 when M or N is CRT-exact. For spectra X and Y , the product homomorphisms ~: Km (X; ^Z2)^ Kn(Y ; ^Z2) -! Km+n (X ^ Y ; ^Z2) ~: KOm (X; ^Z2)^ KOn(Y ; ^Z2) -! KOm+n (X ^ Y ; ^Z2) ~: KTm (X; ^Z2)^ KTn(Y ; ^Z2) -! KTm+n (X ^ Y ; ^Z2) give a CRT-pairing, where conditions (i)-(xi) follow by [5, Section 1] and (xii* *) follows by Lemma 4.11 below, and they consequently determine a product homomorphism ~: KmCRT(X; ^Z2)^ CRTKnCRT(Y ; ^Z2) -! Km+nCRT(X ^ Y ; ^Z2) of 2-adic CRT-modules. This also holds for spaces X and Y when X ^ Y is replace* *d by X x Y . We can finally state our Kunneth theorem for K*CRT(-; ^Z2), where we write K*CRT(X* *; ^Z2) as bK*CRTX. Theorem 2.12. For spectra X and Y , there is a natural short exact sequence CRT *+1 * 0 -! bK*CRTX ^CRTbK*CRTY -~-!bK*CRT(X^Y ) -! ^Tor1 (KbCRTX, bKCRTY ) -! 0 of 2-adic CRT-modules. This also holds for spaces X and Y when X ^ Y is replace* *d by X x Y . This will be proved in 4.12. 3.Unstable operations in united 2-adic K-cohomology In this section, we discuss the unstable operations in the united 2-adic K-co* *homology K*CR(X; ^Z2) = {K*(X; ^Z2), KO*(X; ^Z2)} of a space X and determine their algebraic relations.* * We first consider: 10 3.1. The CR-algebra operations. For a space X, the cohomology K*CR(X; ^Z2) has * *commuta- tive associative multiplication K*CR(X; ^Z2)^ CRK*CR(X; ^Z2) ! K*CR(X; ^Z2) ind* *uced by the di- agonal X ! X x X using the homomorphism ~ of Theorem 2.8. In particular, for e* *lements x 2 KOm (X; ^Z2), y 2 KOn(X; ^Z2), z 2 Ks(X; ^Z2), and w 2 Kt(X; ^Z2), there ar* *e products xy 2 KOm+n (X; ^Z2) and zw 2 Ks+t(X; ^Z2) satisfying the usual relations includ* *ing: (i)xy = (-1)mnyz and zw = (-1)stwz; (ii)c(xy) = (cx)(cy); (iii)r((cx)z) = x(rz); (iv)(zw)* = z*w*. For odd dimensional elements, we obtain additional squaring properties which * *will be refined in Proposition 3.11 below. Proposition 3.2. For a space X and elements x 2 KOm (X; ^Z2) and z 2 Ks(X; ^Z2)* *, we have: (i)x2 = 0 when m 1, -3 mod 8; (ii)x4 = 0 when m -1, -5 mod 8; (iii)z2 = 0 when s is odd. Proof.Part (i) follows by [11, p.66]. Part (ii) follows since x4 = (j~2x)2 = j* *2(j~2~2x) = 0 by [11, p.67] when m = -1 and by [12, Proposition 2.2] when m = -5. Part (iii) fol* *lows by, e.g., [7, Theorem 1.11]. The algebraic structure of K*CR(X; ^Z2) is shaped by two important families o* *f unstable operations which we call the OE's and `'s. 3.3. The operations OE. By [10, Theorem 6.5], for a space X, there are natural * *operations OE: K0(X; ^Z2) -! KO0(X; ^Z2) OE: K-1(X; ^Z2) -! KO0(X; ^Z2) such that the following relations hold for elements a, b 2 K0(X; ^Z2) and x, y * *2 K-1(X; ^Z2): (i)OE(a + b) = OEa + OEb + r(a*b); (ii)OE(x + y) = OEx + OEy + rB-1(x*y); (iii)OE(a*) = OE(a) and OE(x*) = -OE(x); (iv)OE(ab) = (OEa)(OEb); (v)OE(ax) = (OEa)(OEx); 11 (vi)OEB-1(xy) = (OEx)(OEy); (vii)OE(1) = 1; (viii)OE(ka) = k2OEa and OE(kx) = k2OEx for k 2 ^Z2; (ix)cOE(a) = a*a and cOE(x) = B-1(x*x). For convenience, we extend the operation OE periodically to give OE: K2i(X; ^Z2* *) ! KO0(X; ^Z2) and OE: K2i-1(X; ^Z2) ! KO0(X; ^Z2) with OEw = OEBiw for all i 2 Z and elements w. * *This corrects an inconsistent extension of OE used in [10, Theorem 6.5]. We note that the operat* *ion OE on K-1(X; ^Z2) was implicitly introduced by Seymour [16]. We now consider: 3.4. The complex operations `. For a space X, we let ` = -~2: K0(X; ^Z2) -! K0(X; ^Z2) ` = -~2: K-1(X; ^Z2) -! K-1(X; ^Z2) be the specified exterior power operations and let _2: K0(X; ^Z2) ! K0(X; ^Z2) * *be the Adams operation with _2(a) = a2 + 2`a. As in [7], we have the following `-ring relat* *ions for elements a, b 2 K0(X; ^Z2) and x, y 2 K-1(X; ^Z2): (i)`(a + b) = `a + `b - ab; (ii)`(x + y) = `x + `y; (iii)`(a*) = (`a)* and `(x*) = (`x)*; (iv)`(ab) = (`a)b2+ a2(`b) + 2(`a)(`b); (v)`(ax) = (_2a)(`x); (vi)`(B-1xy) = B-1(`x)(`y); (vii)`(1) = 0; k (viii)`(ka) = k(`a) - 2 a2 and `(kx) = k(`x) for k 2 ^Z2. For convenience, we extend the operation ` periodically to give `: K2i(X; ^Z2) * *! K2j(X; ^Z2) and `: K2i-1(X; ^Z2) ! K2j-1(X; ^Z2) with `w = B-j`Biw for all i, j 2 Z and element* *s w. We note that K0(X; ^Z2) and K-1(X; ^Z2) have additional exterior power operations, but * *these are captured by the action of the stable 2-adic Adams operations as shown in [7]. Since we a* *re primarily interested in the unstable operations, we now turn to: 3.5. The real operations `. For a space X, we let ` = -~2: KOn(X; ^Z2) -! KO0(X; ^Z2) forn = 0, -4 ` = -~2: KOn(X; ^Z2) -! KO-1(X; ^Z2) forn = -1, -5 12 ` = -~2: KOn(X; ^Z2) -! KO1(X; ^Z2) forn = 1, -3 be the specified exterior power operations of [10, Section 6]. To simplify form* *ulae, we now treat the cohomology KO*(X; ^Z2) as Z=8-graded with BR = 1, and we periodically exten* *d the above operations ` to KO*(X; ^Z2). We likewise treat the cohomology K*(X; ^Z2) as Z=* *8-graded with B4 = 1, and we periodically extend the preceding operations OE and ` to K*(X; ^* *Z2). For convenience, we also use the Adams operations _2: KOm-4i(X; ^Z2) -! KOm (X; ^Z2) defined for m 0, 2 mod 8 and i 2 Z by ( 2 _2(x) = x + 2`x for m 0 mod 8 r`cx for m 2 mod 8 where the target dimensions of the complex operations ` are determined by the c* *ontext. There are elementary relations: (i)_2(x + y) = _2x + _2y for m 0, -2 mod 8, (ii)_2(x + y) = _2x + _2y + j2xy for m 2 mod 8, (iii)c_2x = _2cx for m 0, 2 mod 8, which may be derived using the `-ring formula _2a = `a - `(-a) and the CR-modul* *e formula j2 = rBc. We devote the rest of this section to a detailed study of the algebra* *ic relations involving the real 2-adic operations `, starting with sum and product formulae. Proposition 3.6. For a space X and elements x, y 2 KOn(X; ^Z2), we have 8 ><`x + `y - xy for n 0, -4 mod 8 `(x + y) = > `x + `y for n -1, -5 mod 8 : `x + `y + jxyfor n 1, -3 mod.8 Proof.This follows from [10, Theorem 6.4] since ` = -~2. Proposition 3.7. For a space X and elements x 2 KOm (X; ^Z2) and y 2 KOn(X; ^Z2* *), we have 8 >>>(`x)y2+ x2(`y) + 2(`x)(`y)for m, n 0, -4 mod 8 >>> 2 < (_ x)(`y) for m even and n odd `(xy) = > (`x)(_2y) for m odd and n even >>> >>:(`x)(`y) for m, n odd and m + n 0, -4 mod 8 r(`cx)(`cy)) - x2y2 for m, n 2 mod.8 This will be proved in 7.7. For brevity, we have not specified the target di* *mensions of the operations _2 and ` in these formulae, since they are sufficiently determined b* *y the context. 13 Proposition 3.8. For a space X and elements x 2 KOn(X; ^Z2), k 2 ^Z2, and 1 2 K* *O0(X; ^Z2), we have `(1) = 0 and ( k 2 `(kx) = k`x - 2 x for n 0, -4 mod 8 k`x for n odd. Proof.This follows from [10, Theorem 6.4] since ` = -~2. Proposition 3.9. For a space X and element x 2 KOn(X; ^Z2), we have 8 > j`x for n 1, -3 mod 8 : 0 for n 2 mod.8 This will be proved in 7.3. In the case n 0, -4 mod 8, we can equivalently * *write `(jx) = jx2 since _2x = x2+ 2`x. We next consider the operation , = rB2c: KO*(X; ^Z2) -! KO*-4(X; ^Z2). Proposition 3.10. For a space X and element x 2 KOn(X; ^Z2), we have ( 2 `(,x) = 2`x - x for n 0, -4 mod 8 2`x for n odd. This will be proved in 7.5. The operations ` play an important role in formulae for squares of elements i* *n KO*(X; ^Z2). Proposition 3.11. For a space X and element x 2 KOn(X; ^Z2), we have 8 > 0 for n 1, -3 mod 8 : -r`cx for n 2 mod.8 This will be proved in 7.2 using results of Crabb and Minami. A similar resul* *t is: Proposition 3.12. For a space X and element x 2 KOn(X; ^Z2), we have 8 >>>x2 for n 0, -4 mod 8 < 0 for n -1, -5 mod 8 OEcx = > >>:j`x for n 1, -3 mod 8 r`cx for n 2 mod.8 This will be proved in 7.4. Our last three propositions will give formulae fo* *r the commutation of ` with the operations c, r, and OE. Proposition 3.13. For a space X and element x 2 KOn(X; ^Z2) with n 6 2 mod 8,* * we have `cx = c`x. 14 Proof.This follows by [10, Theorem 6.4]. Proposition 3.14. For a space X and element z 2 Kn(X; ^Z2), we have 8 > r`z + jOEzfor n -1, -5 mod 8 : r`z for n 1, -3 mod.8 Proof.This follows by [10, Theorems 6.4 and 6.5]. Proposition 3.15. For a space X and element z 2 Kn(X; ^Z2), we have ( 2 * * `OEz = r(z `z + (`z)(`zf))or n = 0 OE`z for n = -1. This will be proved in 7.6. Although we are primarily interested in unstable * *operations, we finally consider: 3.16. The stable 2-adic Adams operations. For a space X, the cohomology K*CR(X;* * ^Z2) of 3.1 is endowed with the usual stable Adams operations _k: K*(X; ^Z2) ! K*(X; ^Z2) and * *_k: KO*(X; ^Z2) ! KO*(X; ^Z2) for units k 2 Zx(2)or more generally for units k 2 ^Zx2(see, e.g., * *[10]). Moreover, for elements x 2 KOm (X; ^Z2), y 2 KOn(X; ^Z2), z 2 Ks(X; ^Z2), and w 2 Kt(X; ^Z2),* * we have the usual relations including: _k(xy) = (_kx)(_ky), _k(wz) = (_kw)(_kz), _kBRx = k* *4BR_kx, _kBz = kB_kz, _kcx = c_kx, _krz = r_kz, and _kjx = j_kx. We also have the foll* *owing formulae for the commutation of _k with the operations ` and OE: (i)_k`x = `_kx for m = 0, -1, 1; (ii)_k`x = `(k-2_kx) for m = -4, -5, -3; (iii)_k`z = `_kz for s = 0, -1; (iv)_kOEz = OE_kz for s = 0 and _kOEz = k-1OE_kz for s = -1. These will be proved in 7.8. 4.Proofs of the Kunneth theorems The rest of this paper will be devoted to proving results of Sections 2 and 3* *. In this section, we prove the Kunneth theorems 2.3, 2.8, and 2.12, together with the lemmas 2.7 and* * 2.11 for united 2-adic K-theory. We start with the CR-results. To use the work of [5], we need: 15 4.1. Pontrjagin duality for 2-adic CR-modules. Recall that Pontrjagin duality g* *ives contravari- ant equivalences (-)]: "Abopo ^Ab : (-)] between the category "Ab of 2-torsion * *abelian groups and the category ^Ab of 2-profinite abelian groups. As in [10, Theorem 3.1], this p* *rolongs to contravariant equivalences (-)]: "CRopo ^CR : (-)]between the category "CR of 2-torsion CR-mo* *dules [5, Section 4.7] and the category ^CR of 2-adic CR-modules, where the former are indexed ho* *mologically. More explicitly, a 2-torsion CR-module M 2 "CR corresponds to a 2-adic CR-module M] * *2 ^CR with (M])nC= (MCn)] and (M])nR= (MRn+4)] for n 2 Z, where the operations B,t, BR,j, * *c, and r in M] correspond respectively to B, t, BR, j, rB2, and B-2c in M. We note that M is B* *ott exact or CR-exact if and only if its Pontrjagin dual M] has this property. For a spectru* *m X, there are now natural isomorphisms of 2-adic CR-modules K*CR(X; ^Z2) ~= KCR*(X; Z21)] ~=KCR*-1(o2X)] by [10, Theorem 3.1], where o2X is the 2-torsion part of X given by the homotop* *y fiber of X ! X[1=2]. To prove Lemma 2.7 on projectivity and flatness in ^CR, we need: 4.2. Free 2-adic CR-modules. For a 2-profinite abelian group G 2 A^b and intege* *r n, there are free 2-adic CR-modules FC(G, n) and FR(G, n) whose maps into a 2-adic CR-module* * M correspond to the maps G ! MnCand G ! MnR. We may obtain FC(G, n) and FR(G, n) explicitly * *by tensoring G with the free CR-modules of [5, Secton 2.4]. When G is torsion-free, FC(G, n)* * and FR(G, n) are projective in ^CR since G is projective in ^Ab, and they are flat in ^CR since * *G is flat in ^Ab and since (FC(G, n)^ CRM)*C ~=(G^ M*-nC) (G^ M*-nC) (FC(G, n)^ CRM)*R~= G^ M*-nC (FR(G, n)^ CRM)*C ~=G^ M*-nC (FR(G, n)^ CRM)*R~= G^ M*-nR for 2-adic CR-modules M 2 ^CR. 4.3. Proof of Lemma 2.7. There are enough projectives in ^CR given by finite di* *rect sums of FC(G, n)'s and FR(G, n)'s for various torsion-free G 2 ^Ab and n 2 Z. The impli* *cation (iii) ) (i) now follows since each projective M 2 ^CR is a quotient (and hence direct summa* *nd) of a flat projective module of the above sort. The implication (i) ) (ii) follows by appl* *ying M ^CR to the exact sequences -1c] [* *j] . .-.! FR(^Z2, n-1) -[j]--!FR(^Z2, n) -[r]--!FC(^Z2, n) -[B----!FR(^Z2, n-2) --* *-!. . . 0 -! FC(^Z2, n) -[2]--!FC(^Z2, n). 16 The remaining implication (ii) ) (iii) follows from the Pontrjagin dual result * *in "CR, which in turn follows from [5, Proposition 4.9]. The proof of the Kunneth theorem 2.8 will depend on two lemmas. Lemma 4.4. If X and Y are spectra with bK*CRX (or bK*CRY ) flat in ^CR, then ~:* * bK*CRX ^CRbK*CRY ! bK*CR(X ^ Y ) is an isomorphism. Proof.When Y is the sphere spectrum S, this follows since bK*CRS = FR(^Z2, 0) i* *s the unit for ^CR. When Y is a finite spectrum, it follows by induction on the number of cells in * *Y since bK*CRX is flat. The general result then follows by a limit argument. Lemma 4.5. For a spectrum X, there exists a spectrum P and a map f :X ! P such * *that bK*CRP is flat in ^CR and f*: bK*CRP ! bK*CRX is onto. Proof.Let f :X ! P be a map from X to a (possibly infinite) product P of spectr* *a iKZ^2and iKO^Z2such that f*: bK*CRP ! bK*CRX is onto. This P is a finite wedge of suspe* *nsions of spectra Q KG and KOG for various products G = ^Z2, and we must show that bK*CRcarries s* *uch KG and KOG to flat modules in ^CR. For this, it suffices by 2.5 and Lemma 2.7 to show * *that K*(KG; ^Z2) and K*(KOG; ^Z2) are torsion-free and concentrated in even degrees. This follow* *s by 4.1 since o2KG and o2KOG are (possibly infinite) wedges of spectra -1KZ21 and -1KOZ21 which * *have suitable K-homologies by [5, Theorem 8.2]. 4.6. Proof of Theorem 2.8. By Lemmas 4.5 and 2.7, we may choose a spectrum P an* *d a map f :X ! P with homotopy cofiber P0 such that f*: bK*CRP ! bK*CRX is onto and suc* *h that bK*CRP and bK*CRP0 are flat in ^CR. Thus, by Lemma 4.4, we obtain a short exact Kunnet* *h sequence from the long exact bK*CR-sequence of X ^ Y ! P ^ Y ! P0^ Y , and we deduce the requ* *ired naturality by the argument of Atiyah [2]. The result for spaces follows easily from the re* *sult for spectra. 4.7. Proof of Theorem 2.3. Lemmas 4.4 and 4.5 remain valid when bK*CR, ^CR, and* * ^CR , are replaced by bK*, ^K*, and ^K*. Hence, Theorem 2.3 follows as in 4.6. We now proceed to prove the corresponding CRT-results assuming some familiari* *ty with the work of [5] and [10]. 4.8. Pontrjagin duality for 2-adic CRT-modules. As in [10, Theorem 3.1], the Po* *ntrjagin duality equivalences (-)]: "Abopo ^Ab : (-)] prolong to equivalences (-)]: "CRT* *opo ^CRT: (-)] between the category "CRT of 2-torsion CRT-modules [5, Section 2.1] and the cat* *egory ^CRT of 17 2-adic CRT-modules, where the former are indexed homologically. More explicitl* *y, a 2-torsion CRT-module M 2 "CRTcorresponds to a 2-adic CRT-module M] 2 ^CRTwith (M])nC= (MC* *n)], (M])nR= (MRn+4)], and (M])nT= (MTn+3)] for n 2 Z, where the operations B, t, BR* *, BT, tT, j, ffl, o, i, and fl in M] correspond to B, t, BR, BT, tT, j, o, ffl, flB2, and B-2i in* * M. We note that M is CRT-exact if and only if its Pontrjagin dual M] has this property. For a spe* *ctrum X, there are now natural isomorphisms of 2-adic CRT-modules K*CRT(X; ^Z2) ~= KCRT*(X; Z21)] ~=KCRT*-1(o2X)] by [10, Theorem 3.1]. 4.9. Free 2-adic CRT-modules. For a 2-profinite abelian group G 2 A^b and integ* *er n, there are free 2-adic CRT-modules FC(G, n), FR(G, n), FT(G, n) whose maps into a 2-ad* *ic CRT-module M correspond to the maps G ! MnC, G ! MnR, and G ! MnTas in 4.2. Moreover, when* * G is torsion-free, the modules FC(G, n), FR(G, n), and FT(G, n) are projective and f* *lat in ^CRT. 4.10. Proof of Lemma 2.11. The above proof of Lemma 2.7 is easily modified to p* *rove Lemma 2.11 using 4.8, 4.9, and [5, Theorem 3.3]. Before proving the general Kunneth theorem 2.12 for united 2-adic K-cohomolog* *y, we note the following technical lemma which was used to construct the product homomorphism * *in that theorem. Lemma 4.11. The relation (fflo) O ~ = ~ O (fflo ^ 1) + ~ O (1 ^ fflo) + j O ~ holds in [KT ^ KT, KT]1 where ~: KT ^ KT ! KT is the multiplication map. This will be proved in 4.15. 4.12. Proof of Theorem 2.12. Lemmas 4.4 and 4.5 remain valid when bK*CR, ^CR, a* *nd ^CR are replaced by bK*CRT, ^CRT, and ^CRT . Hence, Theorem 2.12 follows as in 4.6. To prove Lemma 4.11, we must show that the difference map D = (fflo) O ~ - ~ O (fflo ^ 1) - ~ O (1 ^ fflo) - j O ~ is trivial in [KT ^ KT, KT]1, and we first consider: 18 4.13. The action of fflo on ss*KT. By [5, Section 1.6], the graded commutative * *ring ss*KT has ss4iKT = Z = , ss4i+1KT = Z=2 = , ss4i+3KT = Z = , and ss4i+* *2KT = 0 for i 2 Z, where the product is determined by the relations j2 = 0, j! = 0, and !2 * *= 0. Moreover, by [5, Sections 1.6 and 1.7], the map fflo 2 [KT, KT]1 induces a homomorphism fflo* **: ss*KT ! ss*+1KT with fflo*(BiT) = jBiTfor i even, with fflo*(BiT) = 0 for i odd, with fflo*(jBi* *T) = 0 for all i, and with fflo*(BiT!) = 2Bi+1Tfor all i. Lemma 4.14. The compositions D O (1 ^ e) and q O D are trivial where 1 ^ e: KT * *^ S ! KT ^ KT is the canonical map and where q :KT ! KTQ is the rationalization. Proof.It is straightforward to verify that D O (1 ^ e) is trivial since (fflo) * *O e = j 2 ss1KT by 4.13. To verify that q O D is trivial, it suffices to show that D*(x ^ y) is t* *rivial in ssi+j+1KT for each x 2 ssiKT and y 2 ssjKT. This follows by a calculation using 4.13 tog* *ether with the relation (f ^ g)*(x ^ y) = (-1)nif*x ^ g*y in ssi+j+m+n(KT ^ KT) for maps f 2 [* *KT, KT]m and g 2 [KT, KT]n. 4.15. Proof of Lemma 4.11. As in [5, Section 1.2], we let C(j2) = S [j2e3 and r* *ecall that the unit map S ! KT may be extended to a map w: C(j2) ! KT that induces a KO-mo* *dule equivalence KO ^ C(j2) ' KT. Since fflo 2 [KT, KT]1 is a KO-module map, the des* *ired triviality of D 2 [KT ^ KT, KT]1 will follow from the triviality of D O (w ^ w) 2 [C(j2) ^* * C(j2), KT]1. By [5, Section 1.3], we have C(j2) _ 3C(j2) ' C(j2) ^ C(j2) [C(j2), KT]1 ~=Z=2 Z [ 3C(j2), KT]1 ~=Z Z where the wedge summand C(j2) is mapped by 1 ^ e: C(j2) ! C(j2) ^ C(j2). Conseq* *uently, the triviality of D O(w ^w) will follow from the triviality of the maps D O(w ^w)O(* *1^e) 2 [C(j2), KT]1 and q O D O (w ^ w) 2 [C(j2) ^ C(j2), KTQ]1, which in turn follows from Lemma 4* *.14. 5.The mod 2 K-cohomological functors j* and h* In preparation for our proofs of relations for unstable operations, we now in* *troduce mod 2 sim- plifications, j*X and h*X, for the cohomologies KO*(X; ^Z2) and K*(X; ^Z2) of a* * spectrum or space X. These simplifications will help us to handle the purely real parts of these * *cohomologies. We 19 show that j*X and h*X have convenient multiplicative properties, and we obtain * *Kunneth theo- rems for them. We also show that h*X has a natural differential satisfying a Le* *ibniz formula and show that this differential vanishes if and only if K*CR(X; ^Z2) is CR-exact. A* *lthough we approach j*X and h*X directly, we note that they may equivalently be derived from the Bo* *tt exact couple for KO*(X; ^Z2) and K*(X; ^Z2) (see Proposition 5.7). 5.1. The functor j*. For a spectrum or space X, we let j*X be the graded profin* *ite Z=2-module j*X = KO*(X; ^Z2)=rK*(X; ^Z2) ~= jKO*+1(X; ^Z2). We may view j*X as a module over the graded commutative Z=2-algebra j* = j*S wi* *th j-8k = Z=2 = and j-8k-1 = Z=2 = for k 2 Z and with jn = 0 for n 6 0, -1 * *mod 8. For spectra X and Y , there is a product homomorphism ~: j*X ^j*j*Y -! j*(X ^ Y ) induced by the KO-cohomological product homomorphisms jm X ^jnY ! jm+n (X ^ Y )* * for m, n 2 Z. Likewise, for spaces X and Y , there is a product homomorphism ~: j*X ^j*j*Y ! j*(X x Y ) and j*X has a commutative graded algebra structure over j*. For a spectrum or s* *pace X, we see from 2.5 that K*CR(X; ^Z2) is CR-exact if and only if j*X is free as a j*-modul* *e. Theorem 5.2. If X and Y are spectra with K*CR(X; ^Z2) CR-exact and K*(X; ^Z2) t* *orsion-free (or with K*CR(Y ; ^Z2) CR-exact and K*(Y ; ^Z2) torsion-free), then ~: j*X ^j*j*Y !* * j*(X ^ Y ) is an isomorphism. This also holds for spaces X and Y when X ^ Y is replaced by X x Y* * . Proof.We may assume that X and Y are spectra. Then ~: K*CR(X; ^Z2)^ CRK*CR(Y ; ^Z2) -! K*CR(X ^ Y ; ^Z2) is an isomorphism by Theorem 2.8, and it suffices to show that the natural map (MR=r)^ j*(NR=r) -! (M ^CRN)R=r is an isomorphism whenever M and N are 2-adic CR-modules with M CR-exact and MC* * torsion- free. By 4.3, we may assume that M is FR(G, n) or FC(G, n) for some torsion-fre* *e 2-profinite G and integer n, and the result then follows easily using 4.2. 20 5.3. The functor h*. For a spectrum or space X, we let h*X be the graded profin* *ite Z=2-module with hnX = ker(1_-_t)im(1 + t) for the involution t = _-1: Kn(X; ^Z2) ! Kn(X; ^Z2). We may view h*X as a modul* *e over the graded commutative Z=2-algebra h* = h*S with h-4k= Z=2 = for k 2 Z and wi* *th hn = 0 for n 6 0 mod 4. For spectra X and Y , there is a product homomorphism ~: h*X ^h*h*Y -! h*(X ^ Y ) induced by the K-cohomological product homomorphisms ~: hm X ^hnY - ! hm+n (X ^* * Y ) for m, n 2 Z. Likewise, for spaces X and Y , there is a product homomorphism ~: h*X ^h*h*Y -! h*(X x Y ) and h*X has a commutative graded algebra structure over h*. Theorem 5.4. If X and Y are spectra with K*(X; ^Z2) (or K*(Y ; ^Z2)) torsion-fr* *ee, then ~: h*X ^h*h*Y -! h*(X ^Y ) is an isomorphism. This also holds for spaces X and Y when X ^Y is re* *placed by X xY . Proof.We may assume that X and Y are spectra. Then ~: K*(X; ^Z2)^ K*(Y ; ^Z2) -! K*(X ^ Y ; ^Z2) is an isomorphism by Theorem 2.3, and we may proceed algebraically. Let ^Inv be* * the category of 2-profinite abelian groups with involution t (i.e., with endomorphism t having * *t2 = 1). Then, by [5, Proposition 3.8] and 4.1, each torsion-free N 2 ^Inv has a direct sum decomposi* *tion M ~=I J (G tG) for 2-profinite abelian groups I, J, and G, where t = 1 on I and t = -1 on J. T* *his may be used to decompose K*(X; ^Z2). We also note that, for each N 2 ^Inv, the object (G tG)* *^ N 2 ^Inv, with diagonal t action, is isomorphic to (G^ N) t(G^ N) 2 ^Inv. This allows us to * *ignore the G tG components of K*(X; ^Z2), and the result follows easily. 5.5. The complexification and realification maps. For a spectrum or space X, we* * let c: j*X ! h*X and r0:h*X ! j*+3X be the complexification and realification maps given by * *c[x] = [cx] for x 2 KO*(X; ^Z2) and by r0[z] = [y] for z 2 K*(X; ^Z2) with tz = z and for y 2 K* *O*+3(X; ^Z2) with jy = rB-1z, where such a y exists by Bott exactness since c(rB-1z) = (1+t)B-1z * *= 0. For spectra X and Y with m, n 2 Z, the maps c and r0have the usual multiplicative propertie* *s: c~ = ~(c^ c): jm X ^jnY -! hm+n (X ^ Y ) 21 ~(1^ r0) = r0~(c^ 1): jm X ^hnY -! jm+n+3(X ^ Y ) ~(r0^1) = r0~(1^ c): hm X ^jnY -! jm+n+3(X ^ Y ). These also hold for spaces X and Y when X ^ Y is replaced by X x Y . Moreove* *r, the map c: j*X ! h*X is an algebra homomorphism, while r0:h*X ! j*+3X is a left or righ* *t j*X-module homomorphism. Other basic properties are given by: Proposition 5.6. For a spectrum or space X, the operations c: j*X ! h*X and r0:* *h*X ! j*+3X satisfy: (i)cBR = B4c and BRr0= r0B4; (ii)cj = 0, r0c = 0, and jr0= 0; (iii)r0B2c = j; (iv)B2cr0= cr0B2. Proof.The relations (i)-(iii) follow easily from the CR-module relations in 2.4* *. The relation (iv) requires additional input from united K-theory and will be proved in 5.14. Proposition 5.7. For a spectrum or space X, the sequence 0 j . .-.! j*+1X -j-!j*X -c-!h*X -r-!j*+3X --! . . . is exact. Proof.This sequence is equivalent to the derived couple of the Bott exact couple -1 j . .-.! KO*+1(X; ^Z2) -j-!KO*(X; ^Z2) -c-!K*(X; ^Z2) -rB----!KO*+2(X; ^Z2) --!. * *. . 5.8. The differential in h*X. For a spectrum or space X, we define a differenti* *al @ :h*X ! h*-1X by @ = B2cr0 = cr0B2. Of course, this is equivalent to the differential * *cr0 of the above derived couple. It has the properties @2 = 0, B2@ = @B2, @c = 0, and r0@ = 0 by* * Proposition 5.6, and it obeys a Leibniz rule by: Theorem 5.9. For spectra X and Y and for elements x 2 hm X and y 2 hnY , we hav* *e @(xy) = (@x)y + x(@y) in hm+n-1(X ^ Y ). This also holds for spaces X and Y when X ^ Y * *is replaced by X x Y , and holds for a space X = Y when X ^ Y is replaced by X. This will be proved in 5.15. Our main interest in @ stems from its connectio* *n with the CR- exactness condition. 22 Proposition 5.10. For a spectrum or space X, the cohomology K*CR(X; ^Z2) is CR-* *exact if and only if @ = 0 in h*X. This happens if and only if j*X is free as a j*-module. Proof.The following conditions are successively equivalent: (i) K*CR(X; ^Z2) is* * CR-exact; (ii) kerj imj in j*X; (iii) imr0 kerc in j*X; (iv) cr0= 0 in h*X; and (v) @ = 0 in h*X. * *The proposition now follows from 5.1. When the cohomology K*CR(X; ^Z2) is CR-exact, we may view h*X as a first appr* *oximation to j*X by: Proposition 5.11. If X is a spectrum or space such that K*CR(X; ^Z2) is CR-exac* *t, then the com- plexification map c + B-2c: (j*X)=j (j*-4X)=j -! h*X is an isomorphism. Proof.There is a short exact sequence 0 0 -! (j*X)=j -c-!h*X -r-!(j*+3X)\j -! 0 by Proposition 5.7 where (j*+3X)\j denotes the kernel of j :j*+3X ! j*+2X. When* * K*CR(X; ^Z2) is CR-exact, there is an isomorphism j :(j*+4X)=j ~=(j*+3X)\j which factors as * *the composition of B2c: (j*+4X)=j ! h*X and r0:h*X ! (j*+3X)\j by 5.6 (iii). This gives a split* *ting of the short exact sequence, and the result follows using BR :j*+4X ~=j*-4X. We devote the rest of this section to proving 5.6(iv) and Theorem 5.9 using i* *nputs from united K-theory. For a spectrum or space X, we let h*TX denote the graded profinite Z=* *2-module h*TX = KT*(X; ^Z2)=(flK*-1(X; ^Z2) + fflrK*(X; ^Z2)), and we let i :h*TX ! h*X be the homomorphism induced by i :KT*(X; ^Z2) ! K*(X; * *^Z2) using the relations ti = i, ifl = 0, and ifflr = 1 + t of [5, Section 1.9]. Lemma 5.12. For a spectrum or space X, i :h*TX ! h*X is an isomorphism. Proof.The homomorphism i :KT*(X; ^Z2)=fl -! K*(X; ^Z2)\(1 - t) is an isomorphism since K*CRT(X; ^Z2) is CRT-exact by 2.9. The lemma now follo* *ws since the homomorphism fflr: K*(X; ^Z2) -! KT*(X; ^Z2)=fl composes with the above isomorp* *hism i to give 1 + t. 23 We may now replace h*X by h*TX in the exact sequence of Proposition 5.7. Let * *ffl: j*X ! h*TX and oB-1T:h*TX ! j*+3X be the homomorphisms induced by ffl: KO*(X; ^Z2) ! KT*(X* *; ^Z2) and oB-1T:KT*(X; ^Z2) ! KO*+3(X; ^Z2) using the relations oB-1Tfl = rB-2 and oB-1Tf* *fl = 0. Lemma 5.13. The composition of ffl: j*X ! h*TX with i :h*TX ~=h*X is c: j*X ! h* **X, and the composition of i :h*TX ~=h*X with r0:h*X ! j*+3X is oB-1T:h*TX ! j*+3X. Proof.This follows from the equality iffl = c: KO*(X; ^Z2) ! K*(X; ^Z2) and the* * equality rB-1i = joB-1T:KT*(X; ^Z2) ! KO*+2(X; ^Z2) of [5]. 5.14. Proof of 5.6(iv). By Lemma 5.13, the operations ffloB-1Tand BT in h*TX co* *rrespond to the operations cr0 and B2 in h*X under the isomorphism i :h*TX ~=h*X. Thus, it* * suffices to show BT(ffloB-1T) = (ffloB-1T)BT in h*TX. This follows from the relations BTffl* *o = ffloBT + jBT and j = flBi in KT*(X; ^Z2). 5.15. Proof of Theorem 5.9. As in 5.14, the operation fflo in h*TX corresponds * *to the operation @ in h*X under the isomorphism i :h*TX ~=h*X. Thus, for spectra X and Y and for* * elements x 2 hmTX and y 2 hnTY , it suffices to show fflo(xy) = (fflox)y +x(ffloy) in hm* *+n-1T(X ^Y ). This follows since, for elements z 2 KTm (X; ^Z2) and w 2 KTn(Y ; ^Z2), we have fflo(zw) = (ffloz)w + (-1)m z(fflow) + jzw in KTm+n-1 (X ^ Y ; ^Z2) by Lemma 4.11. 6.On the j* groups of some infinite loop spaces In this section, we complete our preparations for proofs of relations for uns* *table operations. For a spectrum E and integer m, we let E_mdenote the infinite loop space 10 m E gi* *ven by the base component of 1 m E. Thus, for a connected space X, there is a natural isomorp* *hism eEm(X) ~= m [X, E_m] which specializes to isomorphisms gKO (X; ^Z2) ~=[X, dKO_m] and eKm(X;* * ^Z2) ~=[X, bK_m] where dKO and bKdenote the 2-adic completions of KO and K. Our main results in * *this section (Theorems 6.1 and 6.2) will show the sparseness of primitive elements in the j** * groups of dKO_mand bK_m. After stating these results, we turn to their proofs, which require some * *extensive preliminaries and need not detain the reader. Theorem 6.1. For each integer m, the cohomologies K*CR(dKO_m; ^Z2) and K*CR(Kb_* *m; ^Z2) are CR- exact with K*(dKO_m; ^Z2) and K*(Kb_m; ^Z2) torsion-free. 24 The proof is in 6.5. Letting Lm denote the infinite loop space dKO_mor bK_m, * *we may now apply the Kunneth theorem for j* to show that j*Lm is a 2-profinite Hopf algebra over* * j* = j*(pt). Using the spectrum B1 Lm , we let oe :j*B1 Lm ! Pj*Lm denote the infinite cohomology * *suspension from j*B1 Lm to the primitives in j*Lm . We now give the needed sparseness results f* *or Pj*KO__mand Pj*K_m. Theorem 6.2. We have: (i)PjidKO_0= 0 for i 6 0, -1 mod 8 and j :PjidKO_0! Pji-1dKO_0is monic for i 0 mod 8 with j = 0 on Pji-1dKO_0; (ii)PjidKO_1= 0 for i 6 1, 0 mod 8 and j :PjidKO_1! Pji-1dKO_1is monic for i 1 mod 8 with j = 0 on Pji-1dKO_1; (iii)PjidKO_2= 0 for i 6 2, 1 mod 8 and j :PjidKO_2! Pji-1dKO_2is monic for i 2 mod 8 with j = 0 on Pji-1dKO_2; in fact, oe :j*B1 dKO_2~= Pj*dKO_2; (iv)PjidKO_3= 0 for i 6 3, 2, -1, -2 mod 8 and j :PjidKO_3! Pji-1dKO_3is monic for i 3, -1 mod 8 with j = 0 on Pji-1dKO_3; (v)PjidKO_-4= 0 for i 6 0, -1, -4, -5 mod 8 and j :PjidKO_-4! Pji-1dKO_-4 is monic for i 0, -4 mod 8 with j = 0 on Pji-1dKO_-4; (vi)PjidKO_-3= 0 for i 6 1, 0, -3, -4 mod 8 and j :PjidKO_-3! Pji-1dKO_-3 is monic for i 1, -3 mod 8 with j = 0 on Pji-1dKO_-3; (vii)PjidKO_-2= 0 for i 6 -2, -3 mod 8 and j :PjidKO_-2! Pji-1dKO_-2is monic for i -2 mod 8 with j = 0 on Pji-1dKO_-2; in fact, oe :j*B1 dKO_-* *2~= Pj*dKO_-2; (viii)PjidKO_-1= 0 for i 6 -1, -2 mod 8 and j :PjidKO_-1! Pji-1dKO_-1is monic for i -1 mod 8 with j = 0 on Pji-1dKO_-1; (ix)PjibK_-1= 0 for i 6 1, 0, -1 mod 8 and ff*: PjibK_-1~=PjiS1 for i 1 mod 8 where ff: S1 ! bK_-1is obtained from the inclusion U(1) U; (x)PjibK_0= 0 for i 6 0, -1 mod 8 and j :PjibK_0! Pji-1bK_0is monic for i 0 mod 8 with j = 0 on Pji-1bK_0. The proof is in 6.10. In preparation, we first recall a result from [7], usin* *g notation and terminology of that paper. For a 2-profinite abelian group G, we let "IG = G x G x G x . .d* *.enote the 2-adic _2-module with _2(x1, x2, x3, . .).= (0, x1, x2, . .).. 25 Theorem 6.3. Let E be a 0-connected spectrum with K*(E; ^Z2) torsion-free. Then* * K*( 1 E; ^Z2) is torsion-free, and if K0(E; ^Z2) = 0, then there is a natural isomorphism K*( 1 E; ^Z2) ~= ^"IK1(E; ^Z2)^ ^H1(E; ^Z2)^ JH2(E; ^Z2) of Z=2-graded 2-adic ~-rings and Hopf algebras. Also, if K1(E; ^Z2) = 0 and H1(* *E; ^Z2) = 0, then K1( 1 E; ^Z2) = 0 and there is a natural isomorphism K0( 1 E; ^Z2) ~= W(K0(E; ^Z2) # H2(E; ^Z2)) of 2-adic ~-rings and Hopf algebras. Proof.This is obtained from [7, Theorem 8.3]. The above result applies to the spectra B1 dKO_mand B1 bK_mby: Theorem 6.4. For each m, K*(B1 dKO_m; ^Z2) and K*(B1 bK_m; ^Z2) are torsion-fre* *e with Km+1 (B1 dKO_m; ^Z2) = 0 and Km+1 (B1 bK_m; ^Z2) = 0. Moreover, for each m, Km (B1 dKO_m; ^Z2) has _-* *1 = 1 with c: KOm (B1 dKO_m; ^Z2) ~=Km (B1 dKO_m; ^Z2), and Km (B1 bK_m; ^Z2) is of the fo* *rm G _-1G (under the _-1 action) for some 2-profinite abelian group G. Proof.Since all Postnikov spectra and all uniquely 2-divisible spectra are K=2** *-acyclic, the maps B1 dKO_m! m dKO m KO and B1 bK_m! m ^K m K are K=2*-equivalences. Hence, * *they are also KZ^*2-equivalences and KO^Z*2-equivalences. Thus, it suffices to prov* *e the version of the theorem with m replaced by 0, with B1 dKO_mreplaced by KO, and with B1 bK_mrepl* *aced by K. This version follows using our knowledge of KCR*(KOZ21) and KCR*(KZ21) from [5,* * Theorem 8.2] and using the Pontrjagin duality of [10, Theorem 3.1] between KCR*(EZ21) and K** *CR(E; ^Z2) for a spectrum E. 6.5. Proof of Theorem 6.1. Let Lm denote dKO_mor bK_m. Then K*(Lm ; ^Z2) is tor* *sion-free by Theorems 6.3 and 6.4. Moreover, for m even, K*CR(Lm ; ^Z2) is CR-exact since K1* *(Lm ; ^Z2) = 0 by the above theorems. Now let m be odd, and let "Lmbe the 1-connected cover of Lm* * . By Theorems 6.3 and 6.4, h*"Lm~= ^M is a 2-profinite exterior Hopf algebra on primitives M * * h*"Lmwhich are concentrated in a single degree modulo 4. Since the differential @ of 5.8 m* *ust carry primitives to primitives, it must vanish on M and hence vanish on h*Lm ~= ^M by Theorem 5.* *9. Thus, K*CR("Lm; ^Z2) is CR-exact by Proposition 5.10. Moreover, K*CR(K(ss1Lm , 1); ^Z* *2) is CR-exact since ss1Lm is ^Z2, Z=2, or 0. Finally, since Lm ' "Lx K(ss1Lm , 1), we conclude tha* *t K*CR(Lm ; ^Z2) is CR-exact by Theorems 5.4, 5.9, and 5.10. To prove Theorem 6.2, we shall use: 26 6.6. The functor ~j*. For a spectrum or space X, we let ~j*X be the graded prof* *inite Z=2-module ~j*X = j*X=jj*+1X. We may view ~j*X as a module over the graded commutative Z=2-algebra ~j*= ~j*S * *with ~j-8k= Z=2 = for k 2 Z and ~jn= 0 for n 6 0 mod 8. For spaces X and Y , there * *is a product homomorphism ~: ~j*X ^~j*~j*Y -! ~j*(X x Y ) which is an isomorphism by Theorem 5.2 when K*CR(X; ^Z2) is CR-exact with K*(X;* * ^Z2) torsion- free. By Theorem 6.1, this applies when X = Lm where Lm denotes the infinite lo* *op space dKO_m or bK_m. Thus, ~j*Lm is a 2-profinite Hopf algebra over ~j*, and there is a sho* *rt exact sequence 0 -! ~j*+1Lm -j-!j*Lm -q-!~j*Lm -! 0 where q is the quotient map. Furthermore, we have: Lemma 6.7. For each m, there is an exact sequence 0 -! P~j*+1Lm -j-!Pj*Lm -q-!P~j*Lm Proof.This follows by comparing the preceding short exact sequence with the sho* *rt exact sequence 0 -! ~j*+1(Lm x Lm ) -j-!j*(Lm x Lm ) -q-!~j*(Lm x Lm ) -! 0 whose terms are expanded using the Kunneth theorems for j* and ~j*(5.2 and 6.6). Lemma 6.8. The following results hold for each m and n: (i)if P~jn+1Lm = 0 and P~jnLm = 0, then PjnLm = 0; (ii)if P~jn+1Lm = 0, then j = 0 on Pjn+1Lm and j :PjnLm ! Pjn-1Lm is monic; (iii)if oe :~j*B1 Lm ~=P~j*Lm , then oe :j*B1 Lm ~=Pj*Lm . Proof.These results are obtained from Lemma 6.7 using the factorization j = jq * *:Pj*Lm ! Pj*-1Lm and using the short exact sequence 0 -! ~j*+1B1 Lm -j-!j*B1 Lm -q-!~j*B1 Lm -! 0 To obtain Theorem 6.2, we may now use: Theorem 6.9. We have: 27 (i)P~jidKO_0= 0 for i 6 0 mod 8; (ii)P~jidKO_1= 0 for i 6 1 mod 8; (iii)P~jidKO_2= 0 for i 6 2 mod 8 and oe :~j*B1 dKO_2~=P~j*dKO_2; (iv)P~jidKO_3= 0 for i 6 3, -1 mod 8; (v)P~jidKO_-4= 0 for i 6 0, -4 mod 8; (vi)P~jidKO_-3= 0 for i 6 1, -3 mod 8; (vii)P~jidKO_-2= 0 for i 6 -2 mod 8 and oe :~j*B1 dKO_-2= P~j*dKO_-2; (viii)P~jidKO_-1= 0 for i 6 -1 mod 8; (ix)P~jibK_-1= 0 for i 6 1, 0 mod 8 and ff*: P~jibK_-1~=P~jiS1 for i 1 mod 8 where ff: S1 ! bK_-1is obtained from the inclusion U(1) U; (x)P~jibK_0= 0 for i 6 0 mod 8. 6.10. Proof of Theorem 6.2. This theorem follows easily from Lemma 6.8 and Theo* *rem 6.9. We devote the rest of this section to proving Theorem 6.9. We may approach P~* *j*Lm via Ph*Lm using: Lemma 6.11. Let X be a connected loop space such that K*CR(X; ^Z2) is CR-exact * *and K*(X; ^Z2) torsion-free. Then there is an isomorphism c + B-2c: P~j*X P~j*-4X ~=Ph*X. Proof.This follows since there is a Hopf algebra isomorphism c + B-2c: ~j*X ~j*-4X ~=h*X. by 5.5 and Proposition 5.11. The components of Ph*X are now given by P~j*X ~=Ph*X \ c~j*X and P~j*-4X ~=Ph* **X \ B-2c~j*-4X. This lemma may be combined with: Lemma 6.12. We have Ph*K(Z=2r, 1) = 0 for r 1, Ph*K(^Z2, 2) = 0, Ph1K(^Z2, 1)* * = Z=2, and PhiK(^Z2, 1) = 0 for i 6 1 mod 4. Moreover, h0K(^Z2, 2) = Z=2 and hiK(^Z2, 2)* * = 0 for i 6 0 mod 4. Proof.By Theorem 6.3, for a finite 2-torsion group G, we have K1(K(G, 1); ^Z2) * *= 0 and K0(K(G, 1); ^Z2) ~= J(G]) ~= ^Z2(G]) where ^Z2(G]) is the group ring on G] with _-1 inverting elements of G]. Thus, * *Ph*K(G, 1) = 0 since h0K(G, 1) ~=Z=2(G=2)] and since hiK(G, 1) = 0 for i 6 0 mod 8. Now the * *results for 28 K(Z=2r, 1) follow immediately; the results for K(^Z2, 2) follow by limit argume* *nts since h*K(^Z2, 2) ~= h*K(Z21, 1); and the results for K(^Z2, 1) follow since h*K(^Z2, 1) ~=h*S1. 6.13. Proof of Theorem 6.9 except for parts (iii) and (vii). Let Lm denote dKO_* *mor bK_m, and let "Lmdenote the 1-connected cover of Lm . For m odd, we have Ph*Lm ~=Ph*"Lm P* *h*K(ss1Lm , 1) by Theorem 6.3, and we easily determine these summands using Lemma 6.12 and The* *orem 6.4. We then extract the groups P~j*Lm from Ph*Lm using Lemma 6.11 together with the op* *erations ` of 3.5 (when Lm = dKO_m) and OE of 3.3 (when Lm = bK_m). This gives parts (ii), (i* *v), (vi), (viii), and (ix) of Theorem 6.9. Next, let Lm = dKO_mwith m = 0 or m = 4. Then K1(Lm ; ^Z* *2) = 0 and K0(Lm ; ^Z2) is torsion-free with _-1 = 1 by Theorems 6.3 and 6.4. Hence, oe :^QK*+1("Lm+1; Z=2) ~= PK*(Lm ; Z=2) ~= Ph*Lm by [8, Theorem 10.2]. We may now determine Ph*Lm by Theorems 6.3 and 6.4, and w* *e then extract the groups P~j*Lm from Ph*Lm using Lemma 6.11 together with the operator ` of 3* *.5. This gives parts (i) and (v) of Theorem 6.9. Finally, let L0 = bK_0and let "L0be the 2-co* *nnected cover of L0. Using the splitting L0 ' "L0x K(^Z2, 2) of [13], we obtain Ph*L0 ~=Ph*"L0by* * Theorem 5.4 and Lemma 6.12. We may now determine Ph*L0 and extract P~j*L0 as above to give * *part (x) of Theorem 6.9. Before proving parts (iii) and (vii) of Theorem 6.9, we must make some algebr* *aic preparations. For a module M over F2(=Z=2) with involution t: M ~=M, we let h(M) be the homol* *ogy of M with respect of the differential d = 1 + t. For such modules M and N, there is a pai* *ring h(M) h(N) ! h(M N) since d(x y) = dx y +x dy +dx dy, and we obtain h(M) h(N) ~=h(M N) sin* *ce such modules decompose as direct sums of components F2 tF2 and F2. Thus, if A * *is an abelian F2-Hopf algebra, then h(A) inherits this structure, where t: A ~=A is the invol* *ution given by the antipode (see [17]). As in [8, Appendix A], we let W1 = F2[x0, x1, . .].be the * *infinite Witt Hopf algebra. Then: Lemma 6.14. h(W1 ) is an exterior Hopf algebra on the generator x0. Proof.We give W1 the grading with |xi| = 2ifor i 0, and we consider the short* * exact sequence of graded abelian Hopf algebras W0 ae W1 i W01of [8, Appendix A], where W0 = F2[x0* *] and where W01= F2[x00, x01, . .].is the infinite Witt Hopf algebra with |x0i| = 2i+1for i* * 0. Then W1 is a free graded W0-module on any graded set of elements in W1 which project to a graded * *basis for W01, as shown by an argument using the faithfulness of F2 W0 -. We may now obtain an* * F2-basis for W1 consisting of 1 and x0 followed by elements of the form {gff, tgff}ffin high* *er dimensions. This 29 is accomplished by copying partial bases of W1 to W01, then lifting these copie* *s back to W1 , and then multiplying these lifts by powers of x0, with special arguments for the lo* *west lifts. Next, recall that the abelian F2-Hopf algebras form an abelian category [18, * *Corollary 4.16], and let B ae A i C be a short exact sequence of such Hopf algebras. Suppose A is ir* *reducible (see e.g. [7, Section 10.5]); suppose B ~=W1 ; and suppose C ~= (JZ^2=2)]. Then: Lemma 6.15. The map h(B) ! h(A) is an isomorphism. Proof.Following Radford [14], we let B = B(0) B(1) B(2) . .b.e the natural f* *iltration of A by "wedges" of B, and we recall that there is a natural isomorphism of left B-modu* *les B Cs=Cs-1~= B(s)=B(s-1)for each s, where F2 = C0 C1 C2 . .i.s the coradical filtratio* *n of C. Since the above filtrations all respect the action of t, we obtain a spectral sequence {E* *srA} of left h(B)-modules converging to h(A) with Es1A = h(B(s)=B(s-1)) for s 0. We find that the spect* *ral sequence {EsrC} collapses to h(C) = F2 with Cs=Cs-1= F2 for s 0 and with d1: E2m1C ~=E2m-11C * *for m 1. Using the natural chain isomorphism F2 h(B)E*1A ~=E*1C together with our knowl* *edge of h(B) from Lemma 6.14, we deduce that d1: E2m1A ~=E2m-11A for m 1. Hence, the spect* *ral sequence {EsrA} collapses to give h(A) ~=h(B). For a 2-profinite abelian group A with involution t: A ~=A, we let h A = {h+A* *, h-A} be the Z=2-graded profinite F2-module with h+A = ker(1-t)= im(1+t) and with h-A = ker(* *1+t)= im(1- t). When A is torsion-free, we obtain a natural isomorphism h+A h-A ~=h(A=2)* * using the decomposition of A described in the proof of Theorem 5.4. Moreover, for such to* *rsion-free groups A and B, we obtain a natural isomorphism h A^ h B ~=h (A^ B) which agrees with * *the natural isomorphism h(A=2)^ h(B=2) ~=h(A=2^ B=2). We now consider the torsion-free 2-pr* *ofinite Hopf algebra W(f) of [7, Sections 7.6 and 7.9], where f :M ! H is a map of torsion-f* *ree 2-profinite abelian groups. Using the involutions induced by t = -1 on M and H, we obtain a* * Z=2-graded profinite F2-Hopf algebra h W(f) with a natural map h M ! Ph W(f). Lemma 6.16. If f :M ! H is a map of torsion-free 2-profinite abelian groups wit* *h H = 0 or H = ^Z2, then the natural map h M ! Ph W(f) is an isomorphism. Proof.Since M and W(f) are torsion-free, it suffices to show that h(M=2) ! Ph(W* *(f)=2) is an isomorphism. When f :^Z2! 0, this follows by [7, Section 7.6] and Lemma 6.14. W* *hen f :^Z2! ^Z2, it follows by [7, Section 7.7] and Lemma 6.15. When f :0 ! ^Z2, it follows by r* *etraction from the preceding case with f = 0. The general result now follows by limit arguments si* *nce the given map 30 f :M ! 0 or f :M ! ^Z2may be decomposed as (possibly infinite) products of maps* * ^Z2! 0, ^Z2! ^Z2, and 0 ! ^Z2. 6.17. Proof of parts (iii) and (vii) of Theorem 6.9. For Lm = dKO_mwith m = 2,* * we have oe :h*B1 Lm ~=Ph*Lm by Theorem 6.3 and Lemma 6.16, and hence oe :~j*B1 Lm ~=P~j* **Lm by Lemma 6.11. The desired results now follow since ~j*B1 Lm = 0 for i 6 m mod 8* * by Theorem 6.4. 7. Proofs of relations for unstable operations In this section, we prove Propositions 3.7, 3.9, 3.10, 3.11, 3.12, and 3.15, * *giving all of the remaining relations for unstable operations in {K*(X; ^Z2), KO*(X; ^Z2)}. Before focusing* * on specific relations, we describe: 7.1. The basic method of proof. Suppose that we seek to prove a relation ff = f* *i for natural operations ff, fi :KOm (X; ^Z2) ! KOn(X; ^Z2) defined for spaces X. We may assu* *me that X is con- ` * * Q nected since our operations ff and fi will act coordinatewise on KO*( ffXff; ^* *Z2) ~= ffKO*(Xff; ^Z2) ` for a disjoint union of spaces ffXff. We check: (i)ff = fi when X is a point; (ii)ff - fi is additive for all X; (iii)cff = cfi :KOm (X; ^Z2) ! Kn(X; ^Z2) for all X. m n By (i) and (ii), it now suffices to show ff = fi :gKO (X; ^Z2) ! gKO (X; ^Z2) f* *or connected spaces X. Equivalently, using the universal example X = dKO_m, it suffices to show ff' = * *fi' for the canonical m class ' 2 gKO (dKO_m; ^Z2). This stable class ' is primitive in the sense that* * ~*' = p*1' + p*2' in gKOm(dKO_mxKdO_m; ^Z2) where p1, p2, and ~ are the projection and addition maps* * from dKO_mxKdO_m n to dKO_m. Hence, the element ff' - fi' 2 gKO (dKO_m; ^Z2) is also primitive sin* *ce ff - fi is additive. Moreover, ff' - fi' is in the image of j since it is in the kernel of c, and th* *us ff' - fi' corresponds to a unique element ffl 2 Pjn+1dKO_mwith jffl = ff' - fi' by Theorems 5.2 and 6.1. I* *t now suffices to show that this error element ffl 2 Pjn+1dKO_mis trivial, and this may usually be acc* *omplished using our sparseness results on Pjn+1dKO_mfrom Theorem 6.2. In a few cases, we shall rely* * on other methods involving Real K-theory. 7.2. Proof of Proposition 3.11. For a space X and element x 2 KOn(X; ^Z2), we h* *ave x2 = j~2x = j`x for n -1, -5 mod 8 by Crabb [11, p.67], Minami [12, Proposition 2.* *2], and [10, Section 6.9]. We likewise have x2 = 0 for n 1, -3 mod 8 by Crabb [11, p.66] a* *nd [10, Section 31 6.9]. For an element x 2 KOn(X; ^Z2), we must show x2 = -rB-2`Bcx when n = 2 a* *nd show x2 = -rB2`B-1cx when n = -2, using the operation ` on K0(X; ^Z2). We verify 7.* *1(i)-(iii) using the relations `(y + z) = `y + `z - yz and `(y) + `(-y) = -y2 in K0(X; ^Z2* *). Hence, there is an error element ffl 2 Pj2n+1dKO_nwith an isomorphism oe :j2n+1B1 dKO_n~=Pj2* *n+1dKO_nby Theorem 6.2. Since the v1-stabilization homomorphism of [10, Section 7.1] is* * left inverse to oe, it also gives an isomorphism : Pj2n+1dKO_n~=j2n+1B1 dKO_n, and there is a * *monomorphism j :j2n+1B1 dKO_n! KO2n(B1 dKO_n; ^Z2) by 5.1. We now deduce that ffl = 0 since * *j ffl = 0 by [10, Section 7.4], and the result follows. 7.3. Proof of Proposition 3.9. For a (possibly nonconnected) pointed space X an* *d element x 2 gKOn(X; ^Z2) with n = 1 or n = -3, we must show `jx = j`x in gKO0(X; ^Z2). When* * X is a finite complex, this follows as in [10, Section 6.9] since 1 -3 0 -4 j :gKO (X; ^Z2) gKO (X; ^Z2) -! gKO (X; ^Z2) gKO (X; ^Z2) may be obtained by tensoring the ~-homomorphism i*: gKR( 1,0^ X) gKH( 1,0^ X) -! gKR( 0,0^ X) gKH( 0,0^ X) with the ~-ring ^Z2where i: 0,0^ X ! 1,0^ X is the standard inclusion. The g* *eneral result * * n now follows by an inverse limit argument. For a pointed space X and element x 2* * gKO (X; ^Z2) with n = 0 or n = -4, we must show `jx = j_2x. This follows from the relation * *`j = j` on gKOn+1( X; ^Z2). Finally, for a space X and element x 2 KOn(X; ^Z2) with n = 2* *, we must show `jx = 0. We verify 7.1(i)-(iii) using Propositions 3.6 and 3.13, and we obtain* * an error element ffl 2 PjnKdO_n. As in 7.2, there is a monomorphism j : PjnKdO_n! KOn-1(B1 dKO_n* *; ^Z2), and we deduce that ffl = 0 since j ffl = 0. This shows `jx = 0. 7.4. Proof of Proposition 3.12. For a space X and element x 2 KOn(X; ^Z2), we m* *ust show OEcx = x2 when n = 0 and show OEB-2cx = B-1Rx2 when n = -4. The case n = 0 foll* *ows since OEcx = r`cx - `rcx = 2`x - `(2x) = x2 by Propositions 3.13 and 3.14, and the ca* *se n = -4 follows similarly. For x 2 KOn(X; ^Z2), we must show OEcx = 0 when n = -1 and show OEB-* *2cx = 0 when n = -5. We easily verify 7.1(i)-(iii) and obtain an error element ffl 2 Pj1dKO_* *n. The results now follow since ffl = 0 by Theorem 6.2. For x 2 KOn(X; ^Z2), we must show OEBcx = * *j`x when n = 1 and show OEB-1cx = j`x when n = -3. We easily verify 7.1(i)-(iii), and we also * *obtain jOEBcx = j2`x and jOEB-1cx = j2`x by Propositions 3.9, 3.13, and 3.14. Hence, there is a* *n error element ffl 2 Pj1dKO_nwith jffl = 0 in Pj0dKO_n. The results now follow since ffl = 0 b* *y Theorem 6.2. Finally, 32 for x 2 KOn(X; ^Z2), we must show OEBcx = r`Bcx when n = 2 and show OEB-1cx = r* *`B-1cx when n = -2. The case n = 2 follows since OEBcx = r`Bcx - `rBcx = r`Bcx - `j2x * *= r`Bcx by Propositions 3.9 and 3.14, and the case n = -2 follows similarly. 7.5. Proof of Proposition 3.10. Since , = rB2c, the result now follows easily f* *rom Propositions 3.12 and 3.14. 7.6. Proof of Proposition 3.15. For a space X and element z 2 K0(X; ^Z2), we mu* *st show `OEz = r(z2(`z*)+(`z)(`z*)) in KO0(X; ^Z2). We verify the 7.1(i)-(iii) conditio* *ns using 3.3, 3.4, and 3.13, and we obtain an error element ffl 2 Pj1Kb_0. The result now follows sinc* *e ffl = 0 by Theorem 6.2. For z 2 K-1(X; ^Z2), we must show `OEz = OE`z. We verify the 7.1(i)-(iii) * *conditions using 3.3, 3.4, 3.6, 3.13, and 3.14, and we obtain an error element ffl 2 Pj1Kb_-1. Using * *the map ff: S1 ! bK_-1 of 6.2(ix), we obtain ff*ffl = 0 in Pj1S1 since the relation `OE = OE` holds on* * eK-1(S1; ^Z2). The result now follows since ffl = 0 by Theorem 6.2. 7.7. Proof of Proposition 3.7. It suffices to prove the corresponding results f* *or external co- homology products of spaces X and Y . Thus, for a pair of elements x 2 KOm (X;* * ^Z2) and y 2 KOn(Y ; ^Z2), we must verify the given formula for `(xy) 2 KOq(X x Y ; ^Z2)* * with q = m + n or q = m + n + 4 depending on the dimensions m and n. Using the method of 7.1 w* *ith two vari- ables, we first check the given formula when X or Y is a point; we next check t* *he additivity in each variable of the difference in the sides of the given formula; and we then * *check the complex- ification of the given formula. This is all straightforward using the known re* *sults in Section 3. We then obtain an error element ffl 2 jq+1(dKO_mx dKO_n) which is primitive wit* *h respect to the multiplication of dKO_mand of dKO_n. Thus, ffl belongs to the intersection of * *Pj*dKO_m^j*j*dKO_n and j*dKO_m^j*Pj*dKO_nin j*dKO_m^j*j*dKO_n. By Theorem 6.2, we may extend Pj*d* *KO_mand Pj*dKO_nto free j*-submodules ~Pj*dKO_m j*dKO_mand ~Pj*dKO_n j*dKO_non genera* *tors in di- mensions congruent to m and n mod 4. Moreover, these j*-submodules are direct s* *ummands since j*dKO_m=P~j*dKO_mand j*dKO_n=P~j*dKO_nare j-acyclic. Hence, the error element f* *fl belongs to the free j*-submodule ~Pj*dKO_m^j*~Pj*dKO_n j*dKO_m^j*j*dKO_non generators in dime* *nsions congru- ent to m + n mod 4. Since ffl is of dimension congruent to m + n + 1 mod 4, thi* *s implies ffl = 0, and the result follows. 33 7.8. Proofs of the relations 3.16(i)-(iv). The relation 3.16(iii) follows since* * Adams operations commute with exterior power operations in p-adic ~-rings. The remaining relatio* *ns follow by veri- fying the 7.1(i)-(iii) conditions and showing the vanishing of the resulting er* *ror elements as in the preceding proofs. References [1]J.F. Adams, Lectures on Lie Groups, W.A. Benjamin, New York-Amsterdam, 1969. [2]M.F. Atiyah, Vector bundles and the Kunneth formula, Topology 1(1962), 245-2* *48. [3]M.F. Atiyah, K-theory and reality, Quart. J. Math. Oxford 17(1966), 367-386. [4]J.L. Boersema, Real C*-algebras, united K-theory, and the Kunneth formula, K* *-Theory 26(2002), 345-402. [5]A.K. Bousfield, A classification of K-local spectra, J. Pure Appl. Algebra 6* *6(1990), 121-163. [6]A.K. Bousfield, On K*-local stable homotopy theory, Adams Memorial Symposium* * on Al- gebraic Topology, Vol.2, London Math. Soc. Lecture Note Ser. 179, Cambridge * *University Press, 1992, pp.23-33. [7]A.K. Bousfield, On ~-rings and the K-theory of infinite loop spaces, K-Theor* *y 10(1996), 1-30. [8]A.K. Bousfield, On p-adic ~-rings and the K-theory of H-spaces, Mathematisch* *es Zeitschrift 223(1996), 483-519. [9]A.K. Bousfield, The K-theory localizations and v1-periodic homotopy groups o* *f H-spaces, Topology 38(1999), 1239-1264. [10]A.K. Bousfield, On the 2-primary v1-periodic homotopy groups of spaces, Top* *ology 44(2005), 381-413. [11]M.C. Crabb, Z=2-homotopy theory, London Math. Soc. Lecture Note Ser. 44, Ca* *mbridge University Press, 1980. [12]H. Minami, The real K-groups of SO(n) for n 3, 4 and 5 mod 8, Osaka J. Ma* *th. 25(1988), 185-211. [13]G. Mislin, Localization with respect to K-theory, J. Pure Appl. Algebra 10(* *1977), 201-213. [14]D.E. Radford, Pointed Hopf algebras are free over Hopf subalgebras, J. Alge* *bra 45(1977), 266-273. [15]L. Ribes and P. Zalesskii, Profinite Groups, Springer-Verlag, Berlin, 2000. [16]R.M. Seymour, The Real K-theory of Lie groups and homogeneous spaces, Quart* *. J. Math. Oxford 24(1973), 7-30. [17]M.E. Sweedler, Hopf Algebras,Benjamin, New York, 1969. [18]M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras Man* *uscripta Math. 7(1972), 251-170. Department of Mathematics, University of Illinois at Chicago, Chicago, Illino* *is 60607 E-mail address: bous@uic.edu