ON THE 2-PRIMARY v1-PERIODIC HOMOTOPY GROUPS OF SPACES A.K. BOUSFIELD Abstract.We develop foundations of a general approach for calculating p-p* *rimary v1-periodic homotopy groups of spaces using their p-adic KO-cohomologies and K-cohomo* *logies with par- ticular attention to the case p = 2. As a main application, we derive a m* *ethod for calculating v1-periodic homotopy groups of simply-connected compact Lie groups using * *their complex, real, and quaternionic representation theories. This method has been applied ve* *ry effectively by D.M. Davis in recent work. We rely heavily on the v1-stabilization functor 1f* *rom spaces to spectra. Roughly speaking, we obtain the p-primary v1-periodic homotopy of a space* * X from the p-adic KO-cohomology of 1X, which we obtain from the p-adic KO-cohomology and K* *-cohomology of X by a v1-stabilization process under suitable conditions. Contents 1. Introduction 2 2. The general theory of v1-periodic homotopy groups and the functor 1 * * 4 3. Pontrjagin duality for united p-adic K-theory * * 9 4. Even and odd CR-modules 13 5. Even and odd K=p*-local spectra 19 6. Unstable operations in p-adic K-theory * * 21 7. The v1-stabilization homomorphism 26 8. The v1-stabilizations of odd spheres * * 29 9. The v1-stabilizations of simply-connected compact Lie groups * * 33 References 37 __________ Date: March 24, 2004. Key words and phrases. v1-periodic homotopy groups, v1-stabilizations, p-adic* * K-theory, united K-theory, representation theory. 2000 Mathematics Subject Classification. Primary: 55Q51; Secondary: 55N15 55P* *60, 55S25, 57T20. Research partially supported by the National Science Foundation. 1 2 1.Introduction The p-primary v1-periodic homotopy groups v-11ß*X of a space X, as defined by* * Davis and Mahowald [26], are a localization of the portion of the homotopy groups of X de* *tected by p-adic K-theory. In [19], we showed that the groups v-11ß*X are naturally isomorphic t* *o stable homotopy groups ß*øp 1X where øp 1X is the p-torsion part of the spectrum 1X obtained u* *sing the v1- stabilization functor 1 constructed in [15], [20], [27], and [31]. Moreover, i* *n [19], we developed an approach for calculating v-11ß*X ~=ß*øp 1X from K*(X; ^Zp) via K*( 1X; ^Zp) * *when p is an odd prime and X is a suitable space such as a simply-connected finite H-space. * * This approach has been applied very successfully by Don Davis to simply-connected compact Lie* * groups in [24]. After considerable effort, we recently found that most of the results of [19] c* *an be extended to the case p = 2 with some modifications and restrictions, and these new results * *have already been applied by Don Davis to complete his 13-year project of computing the v1-period* *ic homotopy groups of all simply-connected compact Lie groups [25]. In this paper and its sequel, * *we shall develop the promised 2-primary extensions of results of [19], giving a general approach for* * calculating v1-periodic homotopy groups of suitable spaces. When possible, we work at an arbitrary prim* *e p, although our main concern is with p = 2. We begin in Section 2 by discussing the general theory of v1-periodic homotop* *y groups and of the functor 1. This functor carries a pointed space X to a K=p*-local spectrum* * 1X such that v-11ß*X ~=ß*øp 1X and, in fact, such that v-11ß*(X; W) ~=[W, 1X]* for each fin* *ite p-torsion spectrum W. Thus, the study of v1-periodic homotopy groups may be centered arou* *nd the spectra 1X. These spectra are especially well-behaved when X is a simply-connected fin* *ite H-space or is spherically resolved or, more generally, when 1X has an exponent. In such * *cases, we show that 1X is periodic (see Theorem 2.6); we show that X becomes v1-periodically * *equivalent to the infinite loop space 1 1X after finite looping (see Theorem 2.10); and we show* * that the ordinary homotopy groups of X eventually map splittably onto the v1-periodic homotopy gr* *oups of X (see Theorem 2.11). Since the spectra 1X are K=p*-local, they may be studied by the methods of u* *nited K-theory [16]. After establishing a Pontrjagin duality between the united p-adic K-coho* *mology and K- homology theories of spectra (see Theorem 3.1), we show that the groups v-11ß*X* * ~=ß*øp 1X are determined up to extension by KO*( 1X; ^Zp) together with its Adams operations * *(see Theorem 3.2). 3 For many interesting spaces X, including simply-connected compact Lie groups,* * the spectra 1X have cohomologies K*( 1X; ^Zp) concentrated in even or odd degrees. We show tha* *t such K=p*-local spectra 1X are completely classified by their united p-adic K-cohomologies K*CR( 1X; ^Zp) ~= {K*( 1X; ^Zp), KO*( 1X; ^Zp)} equipped with the complexification, realification, conjugation, and Adams opera* *tions (see Theorem 5.3). Moreover, when Kn-1( 1X; ^Zp) = 0, we find that K*CR( 1X; ^Zp) is largely* * determined by a small part Kn( 1X; ^Zp) = {Kn( 1X; ^Zp), KOn( 1X; ^Zp), KOn-4( 1X; ^Zp)} which we call a -module (see Theorems 4.3 and 4.11). We also show that Kn( 1X;* * ^Zp) has a crucial exactness property which facilitates comparisons with other -modules (* *see Theorem 4.4). This work allows us to approach the v1-periodic homotopy groups v-11ß*X of suit* *able spaces X by seeking to calculate the associated -modules Kn( 1X; ^Zp). For a space X and integer n, we approach the stable -module Kn( 1X; ^Zp) by * *starting with the corresponding unstable -module eKn(X; ^Zp) = {Ken(X; ^Zp), gKOn(X; ^Zp), gKOn-4(X; ^Zp)} and dividing out by its " 1-trivial" part. In preparation, we develop an array * *of unstable operations in the p-adic KO-cohomologies and K-cohomologies of spaces using Atiyah's Real * *K-theory (see Theorems 6.4 and 6.5). We then show that the v1-stabilization homomorphism 1:K* *en(X; ^Zp) ! Kn( 1X; ^Zp) must annihilate or linearize these unstable operations, allowing u* *s to calculate Kn( 1X; ^Zp) from eKn(X; ^Zp) under suitable conditions (see Theorem 7.2). When combined wit* *h the preceding work, this gives a v1-stabilization method for calculating v-11ß*X from eK*CR(X* *; ^Zp) for suitable spaces X (see 7.6). To illustrate the v1-stabilization method, we calculate the groups v-11ß*S2n+* *1at p = 2 and recover results of Mahowald and Davis [23] (see Theorem 8.9). As a main application of * *the method, we obtain an approach for calculating 2-primary v1-periodic homotopy groups of sim* *ply-connected com- pact Lie groups using their complex, real, and quaternionic representation theo* *ries. This approach grew, in part, from our extensive correspondence with Bendersky and Davis, who * *were develop- ing another 2-primary approach using complex representation theory together wit* *h the Bendersky- Thompson spectral sequence (see [11]). Our approach eliminates major difficulti* *es with differentials and has now been applied very effectively by Davis in [25] as previously noted.* * For a simply- connected compact Lie group G, our main result (Therem 9.3) expresses KO*( 1G; * *^Z2) in terms of 4 the representation theory of G. This result confirms a general conjecture cited* * in [25, Conjecture 2.2]. This paper provides foundations for a sequel in which we shall continue to de* *velop 2-primary extensions of results of [19]. In particular, we shall obtain explicit 2-primar* *y constructions of the v1-stabilization 1X and of the localization XK=2for various spaces X, includin* *g many simply- connected compact Lie groups. Throughout this paper, we generally follow the terminology of [21], so that "* *space" means "sim- plicial set,ä nd we let Ho* (resp. Hos) denote the homotopy category of point* *ed spaces (resp. spectra). Although we have long been interested in v1-periodic homotopy theory, we were* * prompted to develop the present body of work by Martin Bendersky and Don Davis, and we than* *k them for their questions and comments. 2.The general theory of v1-periodic homotopy groups and the functor 1 Working at an arbitrary prime p, we first recall the v1-periodic homotopy gro* *ups v-11ß*X of a space X and explain how they are captured by the spectrum 1X. We then obtain s* *tronger results on v-11ß*X and 1Xwhen X is a simply-connected finite H-space or is spherically* * resolved or, more generally, when 1X has an exponent. In such cases, we show that the spectrum * *1X is periodic; we show that X becomes v1-periodically equivalent to the infinite loop space 1* * 1X after finite looping; and we show that the ordinary homotopy groups of X eventually map spli* *ttably onto the v1-periodic homotopy groups of X. These results build on insights of Davis and * *Mahowald in [23], [26], and [27]. 2.1. The v1-periodic homotopy groups. For a finite p-torsion spectrum W 2 Hos, * *a v1-map is a K(1)*-equivalence ( = K=p*-equivalence) !: dW ! W with d > 0 such that K(n)** *! = 0 for n > 1, where K(n)*is the n-th Morava K-theory at p. The Hopkins-Smith Periodici* *ty Theorem (see [30] or [36]) ensures that each finite p-torsion spectrum W has a v1-map, which* * becomes unique after sufficient iteration and in fact becomes natural. Since the sequence W -! dW -* *! 2dW -! . . . in Hoseventually desuspends uniquely in Ho*, we may define the v1-periodic homo* *topy groups of a space X 2 Ho*relative to W by v-11ß*(X; W) ~= colimm[ mdW, X]* 5 with naturality in both X 2 Ho*and W 2 Hos. Following Davis and Mahowald [26], * *we may also define the (absolute) v1-periodic homotopy groups of X 2 Ho*by v-11ß*X = colimkv-11ß*+1(X; Z=pk) = colimkv-11ß*(X; SZ=pk) using the Moore spectra SZ=pk = S0[pke1 with the canonical maps SZ=pk+1! SZ=pk. 2.2. The functor 1. By [20] or by earlier work in [15], [27], or [31], there i* *s a v1-stabilization functor or v1-periodic spectrum functor 1: Ho*! Hossuch that: (i)for a space X and finite p-torsion spectrum W, there is a natural isomorph* *ism v-11ß*(X; W) ~= [W, 1X]*; (ii) 1X is K=p*-local for each space X; (iii)for a spectrum E, there is a natural equivalence 1( 1 E) ~= EK=p; (iv) 1 preserves homotopy fiber squares; (v)for a space X and finite complex A in Ho*, there is a natural equivalence 1(XA) ~= ( 1X)A. 2.3. p-torsion parts and p-completions of spectra. A spectrum E has a natural p* *-torsion part øpE ! E given by the homotopy fiber of the localization E ! E[1=p] away fr* *om p with øpE ' E ^ øpS ' E ^ S-1Zp1. It also has a natural p-completion E ! ^Epgive* *n by the SZ=p*-localization [13] with ^Ep' F(S-1Zp1, E). The functors øp: Hosø Hos: (^-)* *pare adjoint and restrict to equivalences between the homotopy categories of p-complete spec* *tra and of p-torsion spectra, with the K=p*-local spectra corresponding to the p-torsion K*-local sp* *ectra. Thus 1X corresponds to the p-torsion K*-local spectrum øp 1X, and by [19] we have: Theorem 2.4.For a space X, there is a natural isomorphism v-11ß*X ~= ß*øp 1X. In some important cases, the spectrum 1X has an exponent and a periodicity, * *which are auto- matically inherited by v-11ß*X. 2.5. Spaces with 1-exponents. A space X is said to have a 1-exponent pr if pr* *' 0: 1X ! 1X. Note that whenever two of the spaces in a fiber seqence have 1-exponents* * (at p), then so does the third by 2.2(iv). A space is also said to have an eventual H-space* * exponent pr if pr ' 0: ( N0X)(p)! ( N0X)(p)for sufficiently large N. This easily implies that* * X has a 1- exponent prand holds (for suitable values of r) whenever X is a sphere or a sim* *ply connected finite H-space (see [35] and [37]). Thus the following theorem will apply to spherical* *ly resolved spaces and to simply-connected finite H-spaces. 6 Theorem 2.6.If a space X has a 1-exponent pr, then the spectrum 1X is periodi* *c with prqr 1X ' 1X where qr= max{8, 2r-1} for p = 2 and qr= 2(p - 1)pr-1for p odd. Proof.We rely on the KCRT*-Adams spectral sequence of [16] and use the associat* *ed notation. Since the united K-homology KCRT* 1X 2 ACRT has exponent pr, it has a canonical* * periodicity ': qrKCRT* 1X ~= KCRT* 1X, and the element d2' 2 Ext2,qr+1ACRT(KCRT*X, KCRT*X) is the obstruction to realizing ' by a map qr 1X ! 1X. Since d2 acts as a de* *rivation on compositions and since pr(d2') = 0, we find that d2('pr) = 0, and hence 'prmay * *be realized by a map prqr 1X ! 1X. This gives the required periodicity. * * |__| Before developing further properties of v1-periodic homotopy groups, we need: 2.7. v1-periodic equivalences. As in [19], we say that a map f :X ! Y in Ho*is * *a v1-periodic equivalence when it satisfies the following equivalent conditions: (i) 1f : 1X ' 1Y ; (ii)f*:v-11ß*X ~= v-11ß*Y ; (iii)f*:v-11ß*(X; Z=p) ~= v-11ß*(Y ; Z=p); (iv))f*:v-11ß*(X; W) ~= v-11ß*(Y ; W) for each finite p-torsion spectrum W. A v1-periodic equivalence may also be characterized homologically. We let X * *! X denote the m-connected cover of a space X, and we call K=p*X the generic K=p-homology o* *f X when m 3, since it does not depend on the choice of such m. By [18, Sections 11.5 * *and 11.11], we have: Theorem 2.8.A map f in Ho*is a v1-periodic equivalence if and only if f is a g* *eneric K=p*- equivalence. In this case, f is also a generic K=p*-equivalence. 2.9. Examples of v1-periodic equivalences. By [18, Corollary 11.2], each K=p*-e* *quivalence of H-spaces is a v1-periodic equivalence. Thus, each K=p*-equivalence f of spaces * *suspends to a v1- periodic equivalence f because f is a K=p*-equivalence of H-spaces. For exam* *ple, whenever a map ff: dA ! A of spaces represents a v1-map 1 ff of spectra, then ff suspend* *s to a v1-periodic equivalence ff, although ff itself need not be a v1-periodic equivalence by [3* *2]. Perhaps the most striking example of a v1-periodic equivalence is the Snaith map s: 2n+10S2n+1 -! 1 1 RP2n 7 for n 0 at p = 2, discovered by Mahowald [33] and extended to odd primes by T* *hompson [39]. This gives 1S2n+1 ' 2n+1( 1 RP2n)K=2 by 2.2 and leads to a computation of the groups v-11ß*S2n+1= ß* 1S2n+1at p = 2 * *(see [23] and Theorem 8.9). Proceeding more generally, we now show that each space with a* * 1-exponent (including each spherically resolved space and each simply-connected finite H-s* *pace) becomes v1- periodically equivalent to an infinite loop space after finite looping. Theorem 2.10.For a space X with a 1-exponent pr, there is a natural v1-periodi* *c equivalence ~: mX ! 1 m 1X for sufficiently large m (depending on pr). This will be proved later in 2.14. We now consider the induced natural homomo* *rphism ~*:ßiX -! v-11ßiX defined for sufficiently large i when X has a 1-exponent. To show that ~*is sp* *littably epic for large i, we must assume that v-11ß*(X; Z=p), or equivalently v-11ß*X, is of finite ty* *pe. This condition holds whenever X is spherically resolved [23], or X is a simply-connected compa* *ct Lie group (see 9.1 and Theorem 9.3), or X is a rationally associative finite H-space for p odd [19* *]. Slightly generalizing a result of [26] or [23], we have: Theorem 2.11.If X is a space with a 1-exponent pr and with v-11ß*(X; Z=p) of f* *inite type, then ~*:ßiX ! v-11ß1X is splittably epic for sufficiently large i. Proof.Whenever an abelian sequence G0! G1! G2. .h.as a finitely generated colim* *it G1 , the colimit map Gm ! G1 must be splittably epic for sufficienly large m. Thus, sinc* *e v-11ß*(X; Z=pr) is of finite type, the colimit map ßi(X; Z=pr) ! v-11ßi(X; Z=pr) must be splittabl* *y epic for sufficiently large i. In the commutative diagram ßi(X; Z=pr)-@*---!ßi-1X ?? ? y~* ?y~* ßi( 1X; Z=pr)@*----!ßi-1( X) for such i, the left ~* is splittably epic since it is equivalent to the colimi* *t map, and the bottom @* is splittably epic since pr ' 0: 1X ! 1X. Hence, the right ~* is also splitta* *bly epic, and the lemma follows. |__| We now prepare to prove Theorem 2.10, relying on work in [20]. 8 2.12. The category W. Let d1 be the integer defined in [20] with d1= 3 for p od* *d and d1= 3, 4, or 5 (but not known) for p = 2. Let W be the category of which an object is a d* *1-connected finite p-torsion complex W 2 Ho*equipped with a v1-map !: mW ! W in Ho*for some m > 0* *, such that ! is a v1-periodic equivalence, and of which a map f :(W, !) ! (W0, !0) is* * a map f :W ! W0 in Ho*with f!i= (!0)j( kf) for some i, j, k > 0. Note that any finite p-torsion* * complex with a v1-map in Ho*may be suspended to give an object of W by 2.9, and that the categ* *ory W is closed under the suspension functor. Also, for objects (W, !) and (W0, !0) of W, any m* *ap W ! W0in Ho* may be finitely suspended to give a map in W. Lemma 2.13. For an object (W, !) 2 W and space X 2 Ho*, there is a natural v1-p* *eriodic equiva- lence h: XW ! 1 ( 1X)W which respects the suspension in W. Proof.Since the map !: mW ! W is an Lf1-equivalence by [20, Corollary 4.8], th* *ere is a natural isomorphism [W, Lf1X] ~=v-11ß0(Lf1X; W). This composes with the natural isomorp* *hisms v-11ß0(Lf1X; W) ~= [W, 1Lf1X] ~= [ 1 1 W, Lf1X] of [20, Corollary 5.9] to induce a natural equivalence 1 1 W ' Lf1W, and hence* * (Lf1X)W ' 1 ( 1X)W by [20, Theorem 5.4]. This combines with the v1-periodic equivalence * *XW ! (Lf1X)W to give the desired h. |_* *_| 2.14. Proof of Theorem 2.10. Choose a sequence of objects (Wi, !i) and maps ffi: mi+1-mi(Wi, !i) -! (Wi+1, !i+1) in W for i 1 using Moore spaces Wi = Mmi(pi) = Smi-1[piemi with canonical maps ffi:Mmi+1(pi) ! Mmi+1(pi+1) in Ho*. Let q: Wi! Smi be the p* *inching map. Using a nullhomotopy pr ' 0: 1X ! 1X, choose a left inverse fl to the map q*:* * mr 1X ! ( 1X)Wr, and let fl :( 1X)Wi ! mi 1X be the induced map for i r. Note that w* *hen i 2r, the map fl does not depend on the choice of left inverse. We claim that the com* *position * h 1fl miX q-!XWi -! 1 ( 1X)Wi ---! 1 mi 1X is a v1-periodic equivalence for i r. For this it suffices to show that the m* *ap of constant towers {v-11ß* miX}i -! {v-11ß* 1 mi 1X}i is a pro-isomorphism and hence an isomorphism. This follows since {v-11ß*(-)}ic* *arries h to an isomorphism by Lemma 2.13 and also carries q* and 1 fl to pro-isomorphisms bec* *ause {ß*(-)}i 9 carries q*: mi 1X ! ( 1X)Wi to a pro-isomorphism. The required map ~ of Theore* *m 2.10 is now given by miX ! 1 mi 1X for i = 2r. |_* *_| 3.Pontrjagin duality for united p-adic K-theory Working at a prime p, we wish to study the spectra 1X using the methods of u* *nited K-theory [16]. In preparation, we now establish a Pontrjagin duality (Theorem 3.1) relat* *ing the united p-adic K-cohomology * * K*CRT(E; ^Zp) = {K*(E; ^Zp), KO*(E; ^Zp), KT*(E; ^Zp)} def={Kb*E, dKOE, dKTE} of a spectrum E to the united p-adic K-homology __ ___ ___ KCRT*(E; Zp1) = {K*(E; Zp1), KO*(E; Zp1), KT*(E; Zp1)} def={K *E, KO*E, KT*E} ~={K*-1(øpE), KO*-1(øpE), KT*-1(øpE)} = KCRT*-1(øpE) thereby extending previous work of [5], [19, Corollary 2.3], and [40]. We also * *give a basic application of this duality showing that the v1-periodic homotopy groups of a space X can b* *e extracted from the p-adic KO-cohomology of 1X (Theorem 3.2). For a locally compact Hausdorff abel* *ian group G, the Pontrjagin dual G# is given by Homcont(G, R=Z). This restricts to a duality bet* *ween the categories of discrete abelian groups and compact Hausdorff abelian groups, with the p-tor* *sion groups corre- __ * * ___ ___ sponding to the p-profinite groups. We consider the p-torsion homologies {K *E,* * KO*E, KT*E} * * and the p-profinite cohomologies {Kb*E, dKOE, dKTE} equipped with stable Adams * *operations _k for p-local units k 2 Zx(p)(or more generally for p-adic units k 2 ^Zxp). __ * * *-4 Theorem 3.1.For a spectrum E, there are natural dualities eC :bK*E ~=(K *E)#, e* *R :dKO E ~= ___ *-3 ___ (KO *E)#, and eT: dKT E ~=(KT *E)# such that __ __ ___ ___ (i)the stable Adams operations _k: K*E ~=K*E, _k: KO*E ~=KO *E, and ___ ___ *-4 _k: KT*E ~=KT*E respectively dualize to _1=k:bK*E ~=bK*E, k2_1=k:dKO E ~= KdO*-4E, and k_1=k:dKT*-3E ~=dKT*-3E for each k 2 Zx(p); __ __ ___ ___ ___ (ii)the periodicities B :K*E ~=K*+2E, BR :KO *E ~=KO*+8E, and BT: KT*E ~= ___ *+4 *-4 KT *+4E respectively dualize to B :bK*+2E ~=bK*E, BR :dKO E ~=dKO E, *+1 *-3 and BT: dKT E ~=dKT E; ___ ___ ___ ___ (iii)the Hopf operations j: KO *E ! KO *+1E and j: KT *E ! KT *+1E re- *-3 *-4 *-2 *-3 spectively dualize to j: dKO E ! dKO E and j: dKT E ! dKT E; ___ __ __ ___ (iv)the complexification c: KO *E ! K*E and the realification r: K*E ! KO *E *-4 *-4 respectively dualize to rB2: bK*E ! dKO E and B-2c: dKO E ! bK*E; 10 ___ ___ ___ ___ ___ (v)the operations ffl: KO *E ! KT *E, ø :KT *E ! KO *+1E, i :KT *E ! __ __ ___ *-3 *-4 K *E, and fl :K*E ! KT*-1E respectively dualize to ø :dKT E ! dKO E, *-3 *-3 *-3 *-4 ffl: dKO E ! dKT E, flB2: bK*E ! dKT E, and B-2i :dKT E ! Kb*E. Before proving this theorem, we apply part (i) to derive the promised result * *on the v1-periodic homotopy groups of a space. Let r 2 Zx(p)be a unit which generates (Z=p)x when * *p is odd and such that r 3 mod 8 when p = 2. Theorem 3.2.For a space X 2 Ho*and for a spectrum E 2 Hos, there are natural lo* *ng exact sequences n _r-r2 n -1 # n+1 _r-r2 . .-.! dKO 1X ----!dKO 1X -! (v1 ßn+3X) -! dKO 1X ----!. . . n _r-r2 n # n+1 _r-r2 . .-.! dKOE ----!dKO E -! (ßn+3øpEK=p) -! dKO E ----!. . . Proof.For a p-torsion K*-local spectrum F, there is a natural long exact sequen* *ce r-1 _r-1 . .-.! KOnF _---!KOnF -! ßn-1F -! KOn-1F ---! . . . obtained by [13, Corollary 4.4] or by using the KCRT*-Adams spectral sequence o* *f [16]. After r is ___ replaced by 1=r and KO*F is expressed as KO *+1F, this sequence dualizes to giv* *e a natural long exact sequence n r-2_r-1 n # n+1 r-2_r-1 . .-.! dKOF ------!dKO F -! (ßn+3F) -! dKO F ------!. . . by Theorem 3.1. The result now follows by taking F = øp 1X or F = øpE and using* * Theorem 2.4. |__| Note that the homotopy groups ß*EK=pmay be determined from ß*øpEK=pas in [13,* * Proposition 2.5]. The proof of Theorem 3.1 will be based on: 3.3. Brown-Comenetz duality. As in [22], for a spectrum E 2 Hos, the Brown-Come* *netz dual ^cE is the function spectrum F(E, ^cS) where ^cS is determined by the natural e* *quivalence [Y, ^cS] = (ß0Y )# for Y 2 Hos. The associated cohomology theory has universal coefficient* * isomorphisms (^cE)nY ~=(EnY )# for all Y 2 Hosand n 2 Z, and thus ßn(^cE) ~= (ß-nE)#. 11 The Brown-Comenetz functor ^crestricts to a contravariant equivalence from the * *homotopy category of spectra whose homotopy groups are finite direct sums of Z=pj's and Zp1's to * *the homotopy cate- gory of spectra whose homotopy groups are finite direct sums of Z=pj's and ^Zp'* *s. For a commutative ring spectrum R and an R-module spectrum E, note that the Brown-Comenetz dual ^* *cE inherits an R-module spectrum structure from E. We may now view ^c(KZp1), ^c(KOZp1), ^c(KTZp1) as module spectra over the com* *mutative ring spectra bKp, dKOp, and dKTp, since they are obtained from the module spect* *ra KZp1 ' bKpZp1, KOZp1 ' dKOpZp1, and KTZp1 ' dKTpZp1. Let fflC 2 ß0^c(KZp1), fflR 2 ß-4^c(KOZp1* *), and fflT 2 ß-3^c(KTZp1) be the elements corresponding to 1 2 ^Zpunder the isomorphi* *sms ^Zp~=(ß0KZp1)# ~= (ß4KOZp1)# ~= (ß3KTZp1)# induced by rB2: ß0KZp1 ~=ß4KOZp1 and ø :ß3KTZp1 ~=ß4KOZp1. Then let eC :bKp! ^c(KZp1) be the bKp-module map with eC(1) = fflC; let eR : -4dKOp! ^c(KOZp1) be* * the dKOp- module map with eR(1) = fflR; and let eT: -3dKTp! ^c(KTZp1) be the dKTp-module* * map with eT(1) = fflT. Lemma 3.4. The maps eC :bKp! ^c(KZp1), eR : -4dKOp! ^c(KOZp1), and eT: -3dKTp! ^c(KTZp1) are equivalences. Proof.Using [16, Section 2.5], we see that: ß*^c(KZp1) is a free ß*bKp-module o* *n fflC; ß*^c(KOZp1) is a free ß*dKOp-module on fflR; and ß*^c(KTZp1) is a free ß*dKTp-module on ffl* *T. Thus the given maps are ß*-equivalences. |_* *_| This lemma combines with 3.3 to give natural duality isomorphisms eC :K*(E; ^Zp) ~= K*(E; Zp1)# eR :KO*-4(E; ^Zp) ~= KO*(E; Zp1)# eT: KT*-3(E; ^Zp) ~= KT*(E; Zp1)# for a spectrum E. Lemma 3.5. The parts (ii)-(v) of Theorem 3.1 hold for the above dualities eC, e* *R, and eT. Proof.Each of the homology operations in (ii)-(v) is represented by some dKOp-m* *odule map OE 2 [EZp1, FZp1]* for spectra E, F 2 {K, KO, KT}. The map OE dualizes via Lemma 3.4* * to a dKOp- module map ^cOE 2 [Fbp, bEp]* which must be shown equal to some specified dKOp-* *module map OE02 [Fbp, bEp]*. In each case, we check that ^cOE and OE0induce the same ß*-homomo* *rphisms, and we 12 conclude that ^cOE = OE0since all dKOp-module maps in [Fbp, bEp]*, except for s* *ome irrelevant ones in [dKTp, dKTp]4n+2, are detected by ß* (see [16, Section 1.9]). * * |__| We must deal separately with part (i) of Theorem 3.1 since it involves p-adic* * Adams operations which are not represented by dKOp-module maps. We first show: Lemma 3.6. For a spectrum E 2 {K, KO, KT}, the homology E*E is a free E*-module* * on gener- ators in degree 0 for E = K, in degree 0 for E = KO, and in degrees 0 and 3 for* * E = KT. Proof.This was shown for E = K by Adams-Clarke [4] and is presumably known for * *E = KO. In general, we may rely on the formula KCRT*E = ~U(ßCRT*E) of [16, Theorem 8.2], u* *sing the exact functor ~U:Inv ! A of [16, Proposition 6.6]. The result of [4] now shows that ~* *U(Z _-1Z) is free abelian, and hence ~U(M) is also free abelian for M = Z with _-1 = 1 sinc* *e M Z _-1Z. Thus K*E ~=~U(ßC*E) is free abelian for each E 2 {K, KO, KT}, and KO*E ~=~U(ßR** *E) is the underlying KO*-module of a free E*-module on 0-dimensional generators. Hence, K* *CRT*E is a free CRT-module by [16, Theorem 3.2] on generators in K0E for E = K, in KO0E for E =* * KO, and in KT3E for E = KT by the structural results of [16, Section 2.4]. The lemma now f* *ollows from these same structural results. |_* *_| Lemma 3.7. The maps in [Kbp, bKp]i, [dKOp, dKOp]i, and [dKTp, dKTp]iare all det* *ected by rational homotopy groups Q ß* with the exception of the maps in [dKO2, dKO2]ifor i 1* *, 2 mod 8 and in [dKT2, dKT2]ifor i 1, 2 mod 4. Proof.Let E denote K, KO, or KT. Since [Ebp, bEp]i~=[E, -ibEp], it suffices to* * show that the maps in [E, N] are detected by Q ß* whenever N is an E-module spectrum with ß0N to* *rsion-free and also with ß3N torsion-free when E = KT. There are natural universal coefficient* * isomorphisms [E, N] ~= HomCRT(KCRT*E, ßCRT*N) ~= HomE*(E*E, ß*N) obtained from [16, Section 9.6] (or [3] when E = K or E = KO) using the freenes* *s results of Lemma 3.6. Thus the rationalization N ! NQ induces a monomorphism [E, N] ! [E, NQ], a* *nd hence the maps in [E, N] are detected by ß* Q as required. * * |__| 3.8. Proof of Theorem 3.1. Using the dualities eC, eR, and eT of Lemma 3.4, we * *have proved parts (ii)-(v) of the theorem in Lemma 3.5, and we easily deduce part (i) from * *Lemma 3.7. |__| 13 4.Even and odd CR-modules For many interesting spaces X, including simply-connected compact Lie groups,* * the complex p- adic K-theory K*( 1X; ^Zp) is concentrated in even or odd degrees. In such case* *s, the united p-adic K-theory of 1X is completely captured by K*CR( 1X; ^Zp) = {K*( 1X; ^Zp), KO*( 1X; ^Zp)} without KT*( 1X; ^Zp), and in fact is largely captured by a small part of K*CR(* * 1X; ^Zp) which we call a -module. In this section, we first recall the underlying theory of * *CR-modules from [16, Section 4.7] using cohomological indexing and working over an arbitrary ab* *elian category. We then develop some crucial special properties of even and odd CR-modules, and fi* *nally introduce the theory of -modules. 4.1. CR-modules. A CR-module over an abelian category M consists of a pair M = * *{M*C, M*R} of Z-graded objects in M 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 t2= 1 tB = -Bt rt = r tc = c cBR = B4c rB4 = BRr cr = 1 + t rc = 2 rBc = j2 rB-1c = 0 We may sometimes write _-1z or z* in place of tz for z 2 M*C. We let CRM deno* *te the abelian category of CR-modules over M. For a CR-module M, we call -1 j . .-.! M*+1Rj--!M*R-c-!M*CrB---!M*+2R--!. . . the Bott sequence and call M Bott exact when this chain complex is exact. We al* *so call M CR-exact [16, Section 4.7] when it is Bott exact and the chain complex . .-.! M*+1R=r -j-!M*R=r -j-!M*-1R=r -j-!. . . is exact. Some examples of Bott exact CR-modules are K*CR(E; G) = {K*(E; G), KO*(E; G)} KCR*(E; G) = {K*(E; G), KO*(E; G)} for arbitrary spectra E and coefficients G, where the operations come from stan* *dard maps for the spectra K and KO [16, Section 1.9] with Bott exactness shown by [16, Section 1.* *11]. Whenever 14 K*CR(E; G) or KCR*(E; G) is CR-exact, it prolongs canonically to give the group* *s KT*(E; G) or KT*(E; G), which become superfluous [16, Theorem 4.15]. 4.2. Even and odd Bott exact CR-modules. Let M be a Bott exact CR-module over an abelian category M. We call M even or odd when M*Cvanishes in the opposite degr* *ees, allowing us to determine M*Cperiodically from a single term MnCwhere Mn-1C= 0. This impl* *ies that M is CR-exact since j: MiR=r ~=Mi-1R=r for i n mod 2. Moreover, by the following t* *heorem, we can largely determine M*Rfrom the single triad of terms {MnC, MnR, Mn-4R} with oper* *ations t: MnC! MnC c: MnR! MnC r: MnC! MnR c0:Mn-4R! MnC q: MnC! Mn-4R where c0= B-2c and q = rB2. Theorem 4.3.Suppose M is a Bott exact CR-module over M with Mn-1C= 0 for some n* *. Then there are natural isomorphisms 8 >>>MnR for i 0 mod 8 >>> n >> MRn-\c4 >>>MR for i 4 mod 8 >>>Mn-4=q for i 5 mod 8 >: R MnR\c for i 7 mod 8 Moreover, the terms Mn+8k-2R~=Mn-2Rand Mn+8k-6R~=Mn-6Rbelong to natural extensi* *ons 2 B-1c 0 ----! MnR=r--j--!Mn-2R----! MnC\r ----! 0 ?? ? ? yc ?y1 ?yq 2 0 ----!MnC=c0--rB--!Mn-2R-j---!Mn-4R\c0----!0 2 B-3c 0 ----! Mn-4R=q-j---!Mn-6R----!MnC\q----! 0 ?? ? ? yc0 ?y1 ?yr 3 B-1Rj2 0 ----! MnC=c -rB---!Mn-6R----!MnR\c----!0 where the horizontal maps satisfy (rB)(B-1c) = 2, (B-1c)(rB) = 1 - t, (rB3)(B-3* *c) = 2, and (B-3c)(rB3) = 1 - t. Proof.This follows easily from Bott exactness and the CR-module relations. * * |__| We shall see later (Theorem 4.11) that the "difficult" terms Mn-2Rand Mn-6Rma* *y actually be recovered up to isomorphism (though not functorially) from the given triad {MnC* *, MnR, Mn-4R}, at 15 least when M is the category of abelian groups or p-profinite abelian groups. W* *e now obtain a crucial exactness result for {MnC, MnR, Mn-4R}. Theorem 4.4.Suppose M is a Bott exact CR-module over M with Mn-1C= 0 for some n* *. Then there is an exact sequence 0 1-t (r,q) . .-.! MnC-(r,q)---!MnR Mn-4R-c-c---!MnC---! MnC----!MnR Mn-4R-! . . . Proof.This follows from the exact sequences 0 -! MnR=r -c-!MnC=c0-1-t--!MnC\r -q-!Mn-4R\c0-! 0 0 1-t r 0 -! Mn-4R=q -c-!MnC=c ---! MnC\q --!MnR\c -! 0 associated with the ladders of extensions in Theorem 4.3. * * |__| In view of the preceding theorems, we introduce: 4.5. -modules. A -module over an abelian category M consists of a triad A = {* *AC, AR, AH} of objects in M with operations t: AC ! AC c: AR ! AC r: AC ! AR c0:AH ! AC q: AC ! AH satisfying the relations t2 = 1 cr = 1 + t rc = 2 tc = c rt = r c0q = 1 + t qc0 = 2 tc0 = c0 qt = q We may sometimes write _-1z or z* in place of tz for z 2 AC. We let M denote * *the abelian category of -modules over M, and we say that a -module A is exact when the ch* *ain complex 0 1-t (r,q) . .-.! AC -(r,q)---!AR AH -c-c---!AC ---! AC ----!AR AH -! . . . is exact. Hence, if two -modules in a short exact sequence are exact, then so * *is the third. For a CR-module M 2 CRM and integer n, we obtain a -module nM = {MnC, MnR, Mn-4R} 2 M which is exact whenever M is Bott exact with Mn-1C= 0. 16 4.6. Other examples of exact -modules. For a compact Lie group G, there is a * *-module R (G) = {R(G), RR(G), RH(G)} consisting of the complex representation ring R(G)* * with its real and quaternionic parts RR(G), RH(G) R(G) linked by the standard operations. T* *his -module is always exact since it is freely generated by irreducible representations of * *complex, real,and quaternionic types (see [2] and [12]). For an abelian group N with involution * *t: N ~=N, we let N+ = N\(1 - t) and N+ = N=(1 - t). Then there are exact -modules {N, N+ ,* * N+} and {N, N+, N+ } with obvious operations. When t = 1: N ~=N and the map N\2 ! N=2 i* *s trivial (e.g. when t = 1: Z=2k ~=Z=2k for k > 1), there are also exact -modules {2N, * *N, 4N} and {2N, 4N, N} with obvious operations and there are generally many more. For an object N 2 M with involution t: N ~=N, we write h+N = ker(1 - t)= im(1* * + t) and h-N = ker(1 + t)= im(1 - t) for the associated cohomologies. It is straightforw* *ard to show: Lemma 4.7. For an exact -module M 2 M , there are isomorphisms c + c0:MR=r MH=q ~= h+MC (r, q): h-MC ~= MR\c MH\c0 We suspect that exact -modules are determined up to isomorphism by their com* *plex parts together with their direct sum splittings of h+ and h- terms, but we shall not * *pursue this here. Instead, we use these terms to give comparison lemmas for exact -modules and C* *R-modules. Lemma 4.8. Suppose f :L ! M is a map of exact -modules over M. Then (i)f is an isomorphism if and only if fC :LC ! MC is an isomorphism; (ii)f is epic if and only if fC :LC ! MC and f*:h+LC ! h+MC are both epic; (iii)f is monic if and only if fC :LC ! MC and f*:h-LC ! h-MC are both monic. Proof.In view of Lemma 4.7, part (ii) follows by comparing the sequences LC ! L* *R ! LR=r ! 0 and MC ! MR ! MR=q ! 0, as well as their quaternionic counterparts. The other p* *arts follow similarly. |__| Lemma 4.9. Suppose f :L ! M is a map of Bott exact CR-modules over M with Ln-1C* *= 0 and Mn-1C= 0 for some n. Then (i)f is an isomorphism if and only if fC :LnC! MnCis an isomorphism; (ii)f is epic (resp. monic) if and only if f :LnC! MnC, f*:h+LnC! h+MnC, and f*:h-LnC! h-MnCare all epic (resp. monic). 17 Proof.This follows by combining Lemma 4.8 with Theorems 4.3 and 4.4. * * |__| In the remainder of this section, we shall establish a very close corresponde* *nce between exact -modules and even or odd Bott exact CR-modules. We first consider 4.10. Adjoints of n. For any n, the functor n: CRM ! M has a left adjoint CR* *(-, n): M ! CR M where CR(M, n)n-iC~=MC for i even, CR(M, n)n-iC~=0 for i odd, and 8 >>>MR for i 0 mod 8 >>>M =r for i 1 mod 8 >>> R >0 for i 3, 7 mod 8 >>> >>>MH for i 4 mod 8 >>>MH=q for i 5 mod 8 : MC=c for i 6 mod 8 Hence n(CR(M, n)) = M, and if M is an exact -module with h-MC = 0 (i.e. with * *c and c0 monic), then CR(M, n) is a Bott exact CR-module. The functor n: CRM ! M also * *has a right adjoint CR0(-, n): M ! CRM where CR0(M, n)n-iC~=MC for i even, CR0(M, n)n-iC~=* *0 for i odd, and 8 >>>MR for i 0 mod 8 >>>0 for i 1, 5 mod 8 >>> >MH\c0 for i 3 mod 8 >>> >>>MH for i 4 mod 8 >>>MC\q for i 6 mod 8 : MR\c for i 7 mod 8 Hence n(CR0(M, n)) = M, and if M is an exact -module with h+MC = 0 (i.e. with* * r and q epic), then CR0(M, n) is a Bott exact CR-module. We now address the general problem of prolonging an exact -module M 2 M to * *give a Bott __ __ __n-1 exact CR-module M 2 CRM with nM = M and M C = 0 for some integer n. When h-MC* * = 0 __ or h+MC = 0, such an M will be given by the above CR(M, n) or CR0(M, n). Howeve* *r, when * * __ M is the exact -module {Z=2, Z=2, Z=2} with c = 1 and q = 1, such a prolongati* *on M cannot exist in the category of Z=2-modules. Fortunately, this trouble disappears in o* *ur preferred abelian categories. Theorem 4.11.Suppose M is the category of abelian groups or p-profinite abelian* * groups. For __ an exact -module M 2 M and integer n, there exists a Bott exact CR-module M * *2 CRM with __ __n-1 __ nM = M and M C = 0. Moreover, M is unique up to (noncanonical) isomorphism. 18 Thus, the isomorphism classes of even or odd Bott exact CR-modules over M cor* *respond to the isomorphism classes of exact -modules over M via the functor n for an even or* * odd n. This theorem will be proved below in 4.13 using: 4.12. Special resolutions. In the category of abelian groups, we consider the f* *ollowing elementary exact -modules: F(C) = {Z tZ, Z, Z} with c = 1 + t and c0= 1 + t; F(R) = {Z,* * Z, Z} with t = 1, c = 1, and q = 1; F(H) = {Z, Z, Z} with t = 1, c0= 1, and r = 1; F0(R) = {Z, Z=* *2, 0} with t = -1 and r onto; and F0(H) = {Z, 0, Z=2} with t = -1 and q onto. A -module is calle* *d free when it is a direct sum on copies of F(C), F(R), and F(H); it is called parafree when i* *t is a direct sum of copies of F0(R) and F0(H). For an abelian exact -module M, we use Lemma 4.7 to* * construct a parafree -module F0 and map F0! M inducing an isomorphism h-F0C~=h-MC. We then* * find a free -module F and map F ! M such that F F0 ! M is onto. This determines a* * short exact sequence of -modules 0 ! eF! F F0! M ! 0 called a special resolution o* *f M. The -module eFis exact with eFCfree abelian and with h-FeC= 0. Hence eFis a free * *-module by [16, Proposition 4.8] applied to CR(Fe, 0). 4.13. Proof of Theorem 4.11. It suffices to prove the theorem for abelian group* *s, since it then follows for p-profinite abelian groups by Pontrjagin dualization. Let 0 ! eF! F* * F0! M ! 0 be a special resolution for the exact -module M. Then the induced map of CR-modul* *es CR(Fe, n) ! CR(F, n) CR0(F0, n) is monic by Lemma 4.9, and one easily checks that its coker* *nel is the required __ __ M . The desired uniqueness follows from a more general property of this M . Nam* *ely, for any Bott exact CR-module N with Nn-1C= 0, we claim that each -module map M ! nN prolon* *gs __ (nonuniquely) to a CR-module map M ! N. For this, it is fairly straightforward * *to check that the induced map F F0 ! nN prolongs (nonuniquely) to a map CR(F, n) CR0(F0,* * n) ! N, __ which must be trivial on CR(Fe, n). The desired map M ! N is now obtained by di* *viding out CR(F~, n). |__| We conclude with a technical lemma for later use. Lemma 4.14. Suppose 0 ! M0! M ! M00! 0 is a short exact sequence of exact -m* *odules over an abelian category M with h-M0C= 0 and h-MC = 0. Then there are natural e* *xact sequences 0 -! M00R\c -! M0C=c -! MC=c -! M00C=c -! 0 0 -! M00R\c -! M0H=q -! MH=q -! M00H=q -! 0 0 -! M00H\c0-! M0C=c0-! MC=c0-! M00C=c0-! 0 19 0 -! M00H\c0-! M0R=r -! MR=r -! M00R=r -! 0 Proof.Since M0R\c = 0 and MR\c = 0, we obtain the first exact sequence by the s* *erpent lemma. We then obtain the second exact sequence from the first by using the isomorphis* *m of the left and middle kernels in the ladder of exact sequences 0 0 ----! M0H=q--c--!M0C=c----!M0C=M0+C----!0 ?? ? ? y ?y ?y 0 0 ----! MH=q --c--!MC=c ----!MC=M+C ----!0 where the right kernel is trivial. We obtain the third and fourth exact sequenc* *es similarly. |__| Roughly speaking, the exact sequences in this lemma combine to give a long ex* *act sequence for the real parts of the associated Bott exact CR-modules. 5. Even and odd K=p*-local spectra Using results of [16], we now obtain an algebraic classification of the K=p*-* *local spectra E whose complex K-cohomologies K*(E; ^Zp) are concentrated in even or odd degrees. This* * classification will depend only on the CR-modules K*CR(E; ^Zp) together with their stable Adams ope* *rations and will apply to most of the spectra E = 1X of interest on this paper. 5.1. Stable p-adic Adams modules. By a finite stable p-adic Adams module we mea* *n a finite abelian p-group G with automorphisms _k: G ~=G for k 2 Zx(p)such that: (i)_1= 1 and _j_k = _jkfor all j, k 2 Zx(p); (ii)for a sufficiently large integer n, the condition j k mod pn implies _j=* * _k on G. By a stable p-adic Adams module we mean the topological inverse limit of an inv* *erse system of finite stable p-adic Adams modules. Such a module G has an underlying p-profini* *te abelian group structure with continuous automorphisms _k: G ~=G for k 2 Zx(p). In fact, a sta* *ble p-adic Adams module is just the Pontrjagin dual of a stable p-torsion Adams module in the se* *nse of [14, Section 1] or [16, Section 5.1]. We let ^Abe the abelian category of stable p-adic Adam* *s modules, and we let ~Si:^A! ^A, for i 2 Z, be the functor with ~SiG equal to G as a group but w* *ith _k: ~SiG ~=~SiG equal to ki_k: G ~=G for k 2 Zx(p). 20 5.2. A^CR-modules. By an ^ACR-module we mean a CR-module M of stable p-adic Ada* *ms modules with _-1 = t in M*Cand with _-1 = 1 in M*R, where the homomorphisms B :~SM*C~=M*-2C BR :~S4M*R~=M*-8R j: M*R! M*-1R c: M*R! M*C r: M*C! M*R are all maps in ^A. Equivalently, an ^ACR-module is just the Pontrjagin dual of* * a p-torsion ACR- module in the sense of [16, Section 5.5], where the duality is taken with respe* *ct to the componentwise rules of Theorem 3.1. We let ^ACRbe the abelian category of ^ACR-modules. The m* *ain examples of ^ACR-modules are the cohomologies K*CR(E; ^Zp) = {K*(E; ^Zp), KO*(E; ^Zp)} for arbitrary spectra E, which are Pontrjagin dual to the p-torsion ACR-modules KCR*(E; Zp1) = {K*(E; Zp1), KO*(E; Zp1)} ~={K*-1(øpE), KO*-1(øpE)} = KCR*-1(øpE) by Theorem 3.1. We can now give our main classification theorem for even or od* *d K=p*-local spectra. Theorem 5.3.Suppose M is an even or odd Bott exact ^ACR-module. Then there exis* *ts a K=p*- local spectrum E with K*CR(E; ^Zp) ~=M, and E is unique up to (noncanonical) eq* *uivalence. Proof.The Pontrjagin dual M# is a p-torsion Bott exact ACR-module which prolong* *s canonically to a p-torsion CRT-exact ACRT-module by [16, Lemma 4.14]. Hence, there exists a* * p-torsion K*- local spectrum X with KCR*(X; Zp1) ~=KCR*-1X ~=M# by [16, Theorem 10.1], and th* *e spectrum E = ^Xphas the desired properties. It is unique by Theorem 5.4 below. * * |__| Theorem 5.3 shows that the homotopy types of even or odd K=p*-local spectra c* *orrespond to the isomorphism classes of even or odd Bott exact ^ACR-modules. We have used: Theorem 5.4.Suppose E and F are K=p*-local spectra with Kn-1(E; ^Zp) = 0 and Kn* *-1(F; ^Zp) = 0 for some n. Then, for each ^ACR-module homomorphism OE: K*CR(F; ^Zp) ! K*CR(E* *; ^Zp), there exists a map f :E ! F with OE = f*. Proof.It suffices to prove the corresponding result for the p-torsion K-local s* *pectra øpE and øpF, and that result follows from [16, Section 9.8] by dualization as in the proof o* *f Theorem 5.3. |__| We remark that the map f in this theorem is generally not unique. For instanc* *e, there is a map f 6= 0: SK=2! SK=2of order 2 with f* = 0 on K*CR(SK=2; ^Z2). 21 6.Unstable operations in p-adic K-theory For a space X and integer n, we may approach the stable -module Kn( 1X; ^Zp) = {Kn( 1X; ^Zp), KOn( 1X; ^Zp), KOn-4( 1X; ^Zp)} in favorable cases by starting with the corresponding unstable -module eKn(X; ^Zp) = {Ken(X; ^Zp), gKOn(X; ^Zp), gKOn-4(X; ^Zp)} and dividing out by its " 1-trivial" part. In preparation, we now discuss vario* *us unstable operations in p-adic K-theory. Although some of these operations are well-known, others ma* *y not be, and we shall explain how they may be constructed using Atiyah's Real K-theory [7]. We * *start by recalling: 6.1. ~-rings without identity. A ~-ring without identity consists of a commutat* *ive ring A without identity together with functions ~m :A ! A for m = 1, 2, 3, . .w.here ~1(a) = a* * and where the usual expressions for ~m(a + b), ~m(ab), and ~m~n(a) hold when a, b 2 A and m, * *n 1 (see Atiyah- Tall [10]). We note that such an A may be viewed as the augmentation ideal of a* * ~-ring Z A with identity 1 2 Z. We call such an A semigraded when it has the form A = A0 * *A1 with the Z=2-gradation properties that for all elements x, y 2 A0, u, v 2 A1, and m, k * * 1, the following conditions hold: xy 2 A0, xu 2 A1, uv 2 A0, ~mx 2 A0, ~2ku 2 A0, and ~2k-* *1u 2 A1. Of course, uv = vu instead of uv = -vu for u, v 2 A1. For a compact Lie group G, w* *e note that the augmentation ideals eR(G) and eRR(G) eRH(G) are ~-rings without identity,* * where the latter is semigraded. We may view Re (G) = {Re(G), eRR(G), eRH(G)} as a prototype for: 6.2. ~-rings. A ~-ring A consists of a ~-ring AC without identity and a semig* *raded ~-ring AR AH without identity together with a -module structure on {AC, AR, AH} suc* *h that the following conditions hold for all elements z, w 2 AC, x, y 2 AR, u, v 2 AH, and* * n, k 1: (i)c(xy) = (cx)(cy), c(uv) = (c0u)(c0v), c0(xu) = (cx)(c0u), and (zw)* = z*w* where -* denotes t-; (ii)(rz)x = r(z(cx)), (rz)u = q(z(c0u)), (qz)x = q(z(cx)), and (qz)u = r(z(c0u* *)); (iii)c(~mx) = ~m(cx), c(~2ku) = ~2k(c0u), c0(~2k-1u) = ~2k-1(c0u), and (~mz)*= ~m(z*); (iv)the operation ~OE:AC ! AR given by ~OE(z) = ~2(rz) - r(~2z) has the proper- ties c~OE(z) = z*z, O~E(zw) = (~OEz)(~OEw), and O~E(z + w) = ~OEz + ~OEw + * *r(z*w); 22 (v)using the operation ~OE:AC ! AR, we have k-1X ~2k(qz) = ~2k(rz) = r(~2kz) + ~OE(~kz) + r (~iz)(~2k-iz*) i=1 k-1X ~2k-1(rz) = r(~2k-1z) + r (~iz)(~2k-1-iz*) i=1 k-1X ~2k-1(qz) = q(~2k-1z) + q (~iz)(~2k-1-iz*). i=1 6.3. p-adic ~-rings. We say that a ~-ring A = {AC, AR, AH} is of finite type * *when AC, AR, AH are finitely generated as abelian groups, and we say that A is fl-nilpotent * *when AC and AR AH are fl-nilpotent, i.e., when they are nilpotent with vanishing operations flm f* *or sufficiently large m (see [17, Section 4]). For a ~-ring A of finite type and a fixed prime p, the * *tensor product with the ~-ring ^Zpgives a ~-ring A ^Zpwhose underlying -module is p-profinite and w* *hose operations are all continuous. By a weak (resp. strong) p-adic ~-ring, we mean the topolo* *gical inverse limit limff(Aff ^Zp) of an inverse system of ~-rings Aff ^Zpwhere each Affis of fi* *nite type (resp. of finite type and fl-nilpotent). For a strong p-adic ~-ring B, we note that ^Zp * *BC and ^Zp BR BH are p-adic ~-rings in the sense of [17, Section 5]. Our main topological examples of p-adic ~-rings will be given by eKn(X; ^Zp) = {Ken(X; ^Zp), gKOn(X; ^Zp), gKOn-4(X; ^Zp)} for a space X and integer n. We define an internal multiplication * on eKn(X; ^* *Zp) by z * w = zw when n = 0 and z * w = 0 when n 6= 0 for elements z, w 2 Ken(X; ^Zp). We also * *define an n n-4 internal multiplication * on gKO (X; ^Zp) gKO (X; ^Zp) by the following form* *ulae for elements n n-4 x, y 2 gKO (X; ^Zp) and u, v 2 gKO (X; ^Zp): (i) x * y = jnxy when n 0 and x* * * y = 0 when n < 0; (ii) x * u = jnxu when n 0 and x * u = 0 when n < 0; and (iii) u * v =* * B-1Rjnuv when * n 0 and u * v = 0 when n < 0. Note that jn = 0 in gKO(X; ^Zp) unless p = 2 an* *d n 2. Theorem 6.4.For a space X 2 Ho* and integer n, eKn(X; ^Zp) has a natural weak p* *-adic ~- ring structure with the above internal multiplication, where the structure is s* *trong whenever X is connected or n 6= 0. Moreover, the ~-ring eKn(X; ^Zp) is isomorphic to eK0( |n* *|X; ^Zp) for n 0, while the ~-ring eKn(X; ^Zp) is isomorphic to eK0( |n|X; ^Zp) for all n. Fi* *nally, the operation ~OE:eKn(X; ^Zp) ! gKOn(X; ^Zp) is trivial for n > 0. 23 This will be proved in 6.10, and it provides a wide array of exterior power o* *perations in eK*(X; ^Zp) * * * n and gKO(X; ^Zp). The following theorem expresses the operation ~OE:eKn(X; ^Zp) * *! gKO(X; ^Zp) for 0 n 0 in terms of a more basic operation OE: eKn(X; ^Zp) ! gKO(X; ^Zp). Theorem 6.5.For a space X 2 Ho* and n 0, there is a natural operation OE: eKn* *(X; ^Zp) ! gKO0(X; ^Zp) with the following properties for elements z, w 2 eKn(X; ^Zp): (i)cOE(z) = Bn(z*z); (ii)OE(zw) = (OEz)(OEw); (iii)OE(z + w) = OEz + OEw + rBn(z*w); (iv)OE(Bz) = OE(z); (v)O~E(z) = j|n|OE(z). This extends a result of Seymour [38] and will be proved in 6.11 using: 6.6. Atiyah's Real K-theory. In [7], Atiyah introduced a common generalization * *KR(-) of real and complex K-theory. It applies to a compact Real space Y , which consists of * *a compact Hausdorff space Y equipped with a map ø :Y ! Y such that ø2 = 1. A Real vector bundle ove* *r Y consists of a complex vector bundle q: E ! Y equipped with a map ø :E ! E such that ø2 =* * 1 and øq = qø with ø :Ey! Eøyantilinear for each y 2 Y . The ring KR(Y ) is then obta* *ined by applying the Grothendieck construction to the semiring of Real vector bundles on Y . Ati* *yah-Segal [9] and Dupont [28] extended the ring KR(Y ) to a semigraded ring KM(Y ) = KR(Y ) KH(* *Y ) using the same definitions, but with the condition ø4 = 1 on vector bundles in place of ø* *2 = 1, where the summand KH(Y ) is generated by the vector bundles with ø2 = -1, which are calle* *d Symplectic. For a compact Real space Y , we now obtain a -module K (Y ) = {K(Y ), KR(Y )* *, KH(Y )} with operations as follows: (i)t: K(Y ) ! K(Y ) is defined on vector bundles by t(E) = ø*E~; (ii)c: KR(Y ) ! K(Y ) and c0:KH(Y ) ! K(Y ) are defined on vector bundles by forgetting the ø-actions; (iii)r: K(Y ) ! KR(Y ) and q: K(Y ) ! KH(Y ) are defined on vector bundles by sending E to E ø*E~with the natural Real or Symplectic ø-action. When Y is connected or has a specified base component (closed under ø), we defi* *ne augmentations ffl: K(Y ) ! Z, ffl: KR(Y ) ! Z, and ffl: KH(Y ) ! Z sending vector bundles to * *their complex dimensions over the base component, and we let eK (Y ) = {Ke(Y ), gKR(Y ), gKH(* *Y )} be the - module of augmentation kernels. We now obtain a precursor to Theorem 6.4. 24 Theorem 6.7.For a compact Real space Y with a specified base component, the -m* *odules K (Y ) and eK(Y ) have natural ~-ring structures. Moreover, if Y is a finite complex,* * then K (Y ) and Ke (Y ) are of finite type, and if Y is a connected finite complex, then eK(Y )* * is fl-nilpotent. Proof.In [28, Theorem 2], Dupont showed that the classical Splitting Principle * *[8, Corollary 2.7.11] generalizes to Real and Symplectic vector bundles over compact Real spaces, usi* *ng Real and Sym- plectic "line bundlesö f complex dimension 1. Thus, we obtain ~-ring structur* *es on K(Y ) and KR(Y ) KH(Y ) by the usual constructions for vector bundles, and these give * *~-ring structures on K (Y ) and eK(Y ) by straightforward arguments. When Y is a finite complex, * *K*(Y ) is of finite type, and hence KR(Y ) is finitely generated abelian by Segal's spectral sequen* *ce (see [38, Theorem 3.1]). More generally, KR(Y ) KH(Y ) is finitely generated abelian since it is * *additively isomorphic to KR(Y x S3,0) by [28, Theorem 1]. When Y is a connected finite complex, the i* *deal eK(Y ) and the kernel of c: gKR(Y ) ! eK(Y ) are nilpotent by [38, Theorem 3.1]. Hence, gK* *R(Y ) gKH(Y ) is also nilpotent. Moreover, the elements of gKR(Y ) and gKH(Y ) must have finite * *fl-dimension by the Splitting Principle, since a Real or Symplectic line bundle ! has has flm (! - * *1) = 0 for m > 1 and has flm (1 - !) = (1 - !)m = 0 for sufficiently large m. Hence, eK(Y ) is fl-ni* *lpotent. |__| We also obtain a precursor to Theorem 6.5. This applies to a compact Real sp* *ace Y with basepoint (fixed under ø), and it involves the compact Real spaces Y "xY and Y * *"^Y given by Y x Y with ø(y1, y2) = (øy2, øy1) and by Y ^ Y with ø(y1^ y2) = øy2^ øy1. Theorem 6.8.For a compact Real space Y with basepoint, there are natural operat* *ions ÖE:K(Y ) ! KR(Y "xY ) and ÖE:eK(Y ) ! gKR(Y "^Y ) with the following properties on element* *s of K(Y ) or eK(Y ): (i)cÖE(z) = z*x z; (ii)ÖE(zw) = (ÖEz)(ÖEw); (iii)ÖE(z + w) = ÖEz + ÖEw + r(z*x w); (iv) *ÖE(z) = ~OE(z) where is the diagonal Y ! Y "xY or Y ! Y "^Y . Proof.The operation ÖE:K(Y ) ! KR(Y "xY ) is defined on vector bundles by sendi* *ng E to ø*E~ E with the twisting ø-action, and this induces an operation ÖE:eK(Y ) ! gKR(Y "^Y* * ). The properties (i)-(iv) are easily verified on vector bundles. * * |__| As a final preparation for our main proofs, we use Real K-theory to approach: 6.9. The p-adic K-cohomology of spaces. For m, n 0, let m,nbe the pointed Re* *al (m + n)- sphere obtained as the 1-point compactification of Rm+n = Rm x Rn with ø(x, y) * *= (-x, y). If X 25 is a pointed finite complex then the results of Atiyah [7], Atiyah-Segal [9], D* *upont [28], or Seymour [38] give a natural isomorphism of -modules ( 0,|n| eKn(X; Z) ~= eK( ^ X) for n 0 eK( n,0^ X) for n 0. Moreover, the internal multiplication in eKn(X; Z) (defined by the formulae pre* *ceding Theorem 6.4) agrees with the Real K-theoretic multiplication in eK( 0,|n|^ X) or eK( n,0^ X)* *. This follows easily for n 0 and follows for n > 0 since the diagonal n,0! n,0^ n,0= 2n,* *0is equivariantly homotopic to the standard inclusion and hence induces the operator 2n 2n-4 n n-4 jn: gKO (X) gKO (X) -! gKO(X) gKO (X) by [7, Proposition 3.2]. More generally, if X is a pointed complex (possibly in* *finite), then there is a natural isomorphism of p-adic -modules ( 0,|n| ^Zfor n 0 eKn(X; ^Zp) ~= limffeK( ^ Xff) p limffeK( n,0^ Xff) ^Zpfor n 0. where Xffranges over the finite pointed subcomplexes of X. Moreover, the intern* *al multiplication in eKn(X; ^Zp) agrees with the limit of Real K-theoretic multiplications. 6.10. Proof of Theorem 6.4. By Theorem 6.7 and 6.9, eKn(X; ^Zp) has a natural w* *eak p-adic ~- ring structure obtained as an inverse limit of the structures of eK( 0,|n|^ Xff* *) ^Zpor eK( n,0^ Xff) Z^pfor the finite pointed Xff X. This structure is strong when X is conne* *cted (with a single vertex for simplicity) or n 6= 0, since the complexes 0,|n|^ Xffand n,0^ Xffa* *re then connected. n Moreover, the operation ~OE:eKn(X; ^Zp) ! gKO (X; ^Zp) is trivial for n > 0 by * *Theorem 6.8, since the diagonals : n,0-! n,0"^ n,0~= n,n : n,0^ Xff-! ( n,0^ Xff) "^( n,0^ Xff) are equivariantly nullhomotopic. * *|__| 6.11. Proof of Theorem 6.5. For a pointed finite complex W and n 0, there is * *a natural 0 operation OE: eKn(W; Z) ! gKO(W; Z) given by the composition * ~ Ke( 0,|n|^W) -ÖE-!gKR(( 0,|n|"^ 0,|n|) ^ (W^"W)) (h^-)----!gKR( |n|,|n|^W) ~=gK* *R(W) where h: |n|,|n|~= 0,|n|"^ 0,|n|is the standard equivariant homeomorphism and * *where ~ comes * * 0 from Atiyah's (1, 1)-Periodicity Theorem [7]. Moreover, the operations OE: eKn(* *W; Z) ! gKO(W; Z) satisfy the conditions of Theorem 6.5(i)-(v) by Theorem 6.8 and by [7, Proposit* *ion 3.2], since the diagonal : 0,|n|! 0,|n|"^ 0,|n|is equivalent to the standard inclusion 0,|n* *| |n|,|n|. The 26 desired operations OE: eK(X; ^Zp) ! gKO0(X; ^Zp) are now obtained by tensoring * *with ^Zpand passing to inverse limits. |__| 7.The v1-stabilization homomorphism For a space X and prime p, we now introduce the v1-stabilization homomorphism* * 1:Ke*CR(X; ^Zp) ! K*CR( 1X; ^Zp) and explain how it may be used to determine K*CR( 1X; ^Zp) under* * suitable condi- tions (see 7.6). This will be applied to odd spheres and to simply-connected co* *mpact Lie groups in Sections 8 and 9. 7.1. The v1-stabilization homomorphism. For a K=p*-local spectrum E (such as bK* *por dKOp) and an integer n, the cohomology eEn(X) of a pointed space X is represented by * *the space E_n= 1 ( nE), which has 1E_n' nE by 2.2. Thus there is a natural v1-stabilization* * homomorphism 1:Een(X) -! En( 1X) sending each f :X ! E_nto 1f : 1X ! nE. Equivalently, this is obtained by app* *lying En to the map 1X ! 1 1 1 X ' ( 1 X)K=pinduced by the adjunction unit X ! 1 1 X. * *The homomorphism 1 respects the cohomology suspension oe, so that the diagram eEn(X) ---1-! En( 1X) ?? ? yoe ~=?yoe eEn-1( X)--1--!En-1( 1 X) commutes, and hence an element x 2 eEn(X) has 1x = 0 whenever oeix = 0 for som* *e i > 0. Any element x 2 eEn(X) with 1x = 0 is called 1-trivial. For K=p*-local spectra * *D, E 2 Hosand integers m, n 2 Z, let !: eDm(X) ! eEn(X) be a natural cohomology operation wit* *h representing map !: D_m! E_n. Then ! induces a commutative diagram eDm(X)---1-!Dm ( 1X) ?? ? y! ?y 1! Een(X)---1-!En( 1X) and we note that 1 preserves stable cohomology operations, that is, it gives * *1! = whenever ! = 1 . For a pointed space X, the homomorphisms 1:Ke*(X; ^Zp) -! K*( 1X; ^Zp) 27 1:KgO*(X; ^Zp) -! KO*( 1X; ^Zp) now combine to give the v1-stabilization homomorphism 1:Ke*CR(X; ^Zp) -! K*CR( 1X; ^Zp) of p-profinite CR-modules. For each integer n, this restricts to a homomorphism 1:Ken(X; ^Zp) -! Kn( 1X; ^Zp) of p-profinite -modules, which we may use to determine Kn( 1X; ^Zp) and eventu* *ally K*CR( 1X; ^Zp) under favorable conditions. Theorem 7.2.For a pointed space X and integer n, suppose that Kn-1( 1X; ^Zp) = * *0 and suppose that M eKn(X; ^Zp) is a 1-trivial p-profinite -submodule such that eKn(X; ^* *Zp)=M is an exact -module with 1:Ken(X; ^Zp)=MC ~=Kn( 1X; ^Zp). Then 1:Ken(X; ^Zp)=M ~=Kn( 1X;* * ^Zp). Proof.This follows by Lemma 4.8 since the -module Kn( 1X; ^Zp) is exact by The* *orem 4.4. |__| To construct the needed 1-trivial elements in eKn(X; ^Zp), we use operations* * obtained from Theorems 6.4 and 6.5. Lemma 7.3. For a pointed space X and integers n and k with k > 0, the internal * *~-ring operations n n-4 ~pk, _pk, and `p in eKn(X; ^Zp) and in gKO (X; ^Zp) gKO (X; ^Zp) are all ann* *ihilated by 1. 0 Moreover, when n 0, the operation OE: eKn(X; ^Zp) ! gKO(X; ^Zp) is annihilate* *d by 1. Here, `p is the natural ~-ring operation with _px = xp+ p`px as in [17]. This* * lemma will be proved in 7.8. 7.4. Examples of 1-trivial elements. For a pointed space X, we obtain from 7.1* * and Lemma * 7.3 the following useful examples of 1-trivial elements in eK*(X; ^Zp) and gKO* *(X; ^Zp): all graded * decomposables in eK*(X; ^Zp) and their realifications in gKO(X; ^Zp); all grade* *d decomposables in gKO*(X; ^Zp); all image elements of ~p or `p in eKn(X; ^Zp) and in gKOn(X; ^Zp)* * gKOn-4(X; ^Zp) for 0 n = -1, 0, 1; all image elements of OE: eKn(X; ^Zp) ! gKO(X; ^Zp) for n = -1, 0* *; and all elements generated by the preceding ones using the CR-module operations. We may obtain o* *ther examples by relaxing the conditions on n, but these are generally superfluous. 28 7.5. Kb 1-good spaces. Dividing eK*(X; ^Zp) by its submodule of known 1-trivia* *l elements, we obtain a v1-stabilization homomorphism 1:Q^K*(X; ^Zp)=`p -! K*( 1X; ^Zp) as in [19, Lemma 7.10], where ^QK*(X; ^Zp) denotes the p-profinite quotient of * *eK*(X; ^Zp) by its graded decomposables, and where ^QK*(X; ^Zp)=`pdenotes the Bott periodic quotie* *nt of ^QK*(X; ^Zp) by `p^QK0(X; ^Zp) and `p^QK-1(X; ^Zp). A pointed space X will be called bK 1-go* *od if 1:Q^K*(X; ^Zp)=`p ~=K*( 1X; ^Zp). Here, we could equivalently replace `p by ~p since each `px is congruent to (-1* *)p+1~px modulo decomposables. We also note that the operation `p in eK-1(X; ^Zp) corresponds t* *o the operation _p of [19] in eK1(X; ^Zp). Working at an odd prime p in [19, Theorem 9.2], we prov* *ed the bK 1-goodness of an arbitrary 1-connected H-space X with H*(X; Q) associative and with H*(X; * *Z(p)) finitely generated over Z(p). As explained below in 8.1 and 9.1, we can also prove the b* *K 1-goodness of an odd sphere and of a simply-connected compact Lie group at the prime p = 2. In t* *hese cases and others, we may apply: 7.6. The general v1-stabilization method. Suppose that X is a bK 1-good pointed* * space with Kn-1( 1X; ^Zp) = 0 for a suitable n. Then, under favorable conditions, we may * *apply a ver- sion of Theorem 7.2 to determine Kn( 1X; ^Zp); then apply Theorems 4.3 and 4.11* * to determine K*CR( 1X; ^Zp); and finally apply Theorem 3.2 or methods of united K-theory to * *determine the v1-periodic homotopy groups v-11ß*X ~=ß*øp 1X. This general method should provi* *de additional information on the spectrum 1X since the ^ACR-module K*CR( 1X; ^Zp) determines* * the homotopy type of 1X by Theorem 5.3. We conclude this section by proving Lemma 7.3 using the following 1-triviali* *ty criterion, which applies to a natural cohomology operation !: eDm(X) ! eEn(X) with representing * *map !: D_m! E_nfor K=p*-local spectra D, E 2 Hosand integers m, n 2 Z. Using the p-torsion * *subgroup func- tor tp, we say that tpß*E_nhas exponent ps if pstpßiE_n= 0 for all i > 0. We a* *lso say that !*:ß*D_m=tp! ß*E_n=tp becomes boundlessly p-divisible if, for each power pj, it* * becomes pjdivisi- ble in sufficiently high dimensions. Lemma 7.7. Suppose that tpß*E_nhas exponent ps for some s > 0; suppose that !*:* *ß*D_m=tp ! ß*E_n=tp becomes boundlessly p-divisible; and suppose that the maps in [D, E]m-* *n are all detected 29 by Q ß*. Then 1! = 0 and the image elements of !: eDm(X) ! eEn(X) are 1-tri* *vial for each space X. Proof.For each k > 0, the image of !*:ß*(D_m; Z=pk) ! ß*(E_n; Z=pk) has exponen* *t p2sin suffi- ciently high dimensions, and thus the image of !*:v-11ß*D_m! v-11ß*E_nhas expon* *ent p2s. Hence, by Theorem 2.4, the image of øp 1!*:ß*øp 1D_m! ß*øp 1E_nhas exponent p2s, and t* *he image of 1!*:ß* 1D_m! ß* 1E_nhas exponent p4s. Since the maps in [D, E]m-n are detected* * by Q ß*, we conclude that 1! = 0, and the lemma follows. * * |__| 7.8. Proof of Lemma 7.3. When X is a sphere, the operations ~pk, _pk, and `p ar* *e easily de- termined modulo torsion, since they are preserved by complexification and are k* *nown in the ~-ring Ken(X; ^Zp) ~=eK0( |n|X; ^Zp). The results that 1~pk= 0, 1_pk = 0, and 1`p * *= 0 now fol- low by Lemma 7.7, using Lemma 3.7 to verify the Q ß* condition. When n 0, t* *he operation cOE: eKn(X; ^Zp) ! eK0(X; ^Zp) is annihilated by 1since the elements cOE(z) = * *Bn(z*z) are 1-trivial by 7.1. Thus, c*( 1OE) = 0 in the Bott exact sequence [KZ^p, KO^Zp]n-1-j*-![KZ^p, KO^Zp]n -c*-![KZ^p, KZ^p]n and 1OE = 0 since j*= 0. |__| 8. The v1-stabilizations of odd spheres We now illustrate the v1-stabilization method of 7.6 by applying it to an odd* * sphere S2n+1at the prime p = 2. In particular, we show that S2n+1is bK 1-good; we determine th* *e 2-adic united K-theory and other homotopical properties of 1S2n+1; and we recover the v1-per* *iodic homotopy groups v-11ß*S2n+1~=ß*ø2 1S2n+1which were originally determined by Mahowald usi* *ng other methods (see [23]). For simplicity, we rely on certain results of Mahowald and * *Thompson to show the bK 1-goodness of S2n+1at p = 2, although we hope to give a more general sel* *f-contained account in a subsequent paper. Theorem 8.1.For n 1, the sphere S2n+1is bK 1-good at an arbitrary prime p. Proof.At p odd, this follows by [19, Theorem 9.2]. At p = 2, we have 1S2n+1' * *2n+1( 1 RP2n)K=2 by 2.9, and hence ( Ki( 1S2n+1; ^Z2) ~= 0 for i = 0 Z=2n for i = -1 30 by [1]. Since these groups agree with ^QKi(S2n+1; ^Z2)=`2, it suffices to show * *that 1:Ke-1(S2n+1; ^Z2) ! K-1( 1S2n+1; ^Z2) is onto, and this follows since the map from 1S2n+1to ( 1 S2* *n+1)K=2has ho- motopy cofiber ( 1 D2S2n+1)K=2with K0( 1 D2S2n+1; ^Z2) = 0 by [34] and 2.2. * * |__| Focusing on the case p = 2, we now apply our v1-stabilization method to deter* *mine KO*( 1S2n+1; ^Z2) from the -module eK-1(S2n+1; ^Z2) or eK1(S2n+1; ^Z2) with its internal operati* *on `2 = -~2. We deal separately with the four possible cases of n modulo 4. 8.2. Determining KO*( 1S2n+1; ^Z2) for n 1 mod 4 . We consider the -module e* *K-1(S2n+1; ^Z2) ~= {^Z2, ^Z2, ^Z2} with its operations t = 1, c = 2, r = 1, c0= 1, q = 2, `2C= 2n,* * `2R= 2n, and `2HR= 2n-1, using the notation `2C, `2R, and `2HRfor the complex, real, and quaternionic-to* *-real components of `2. This -module has a 1-trivial submodule M = {2n^Z2, 2n-1^Z2, 2n+1^Z2} = {im`2C, im`2HR, imq`2C} giving an exact quotient -module {Z=2n, Z=2n-1, Z=2n+1} whose complex componen* *t goes isomor- phically to K-1( 1S2n+1; ^Z2) by Theorem 8.1. Thus, we obtain v1-stabilization * *isomorphisms K-1( 1S2n+1; ^Z2) ~= eK-1(S2n+1; ^Z2)=M ~= {Z=2n, Z=2n-1, Z=2n+1}, and we can now apply Theorems 4.3 and 4.11 to determine K*CR( 1S2n+1; ^Z2). In* * fact, using our knowledge of the stable Adams operations _k in eK-1(S2n+1; ^Z2) for k 2 Zx(* *2), we find that KOi( 1S2n+1; ^Z2) is: Z=2n-1for i = -1 with _k = kn+1; Z=2n+1for i = -5 with * *_k = kn+3; Z=2 for i = -3, -4, -6, -7 with _k = 1; and 0 for i = -2, -8. 8.3. Determining KO*( 1S2n+1; ^Z2) for n 2 mod 4. We consider the -module eK* *1(S2n+1; ^Z2) ~= {^Z2, ^Z2, ^Z2} with its operations t = 1, c = 2, r = 1, c0= 1, q = 2, `2C= 2n,* * `2R= 2n, and `2HR= 2n-1. Then, as in 8.2, we determine K*CR( 1S2n+1; ^Z2) and find that KOi( 1S2n+1; ^Z2* *) is: Z=2n-1for i = 1 with _k = kn; Z=2n+1for i = -3 with _k = kn+2; Z=2 for i = -1, -2, -4, -5* * with _k = 1; and 0 for i = 0, -6. 8.4. Determining KO*( 1S2n+1; ^Z2) for n 3 mod 4. We consider the -module eK* *-1(S2n+1; ^Z2) ~= {^Z2, ^Z2, ^Z2} with its operations t = 1, c = 1, r = 2, c0= 2, q = 1, `2C= 2n,* * `2R= 2n, and `2HR= 2n+1. This -module has a 1-trivial submodule M = {2n^Z2, 2n^Z2, 2n^Z2} = {im`2C, im`2R, imq`2C} 31 giving an exact quotient -module {Z=2n, Z=2n, Z=2n} whose complex component go* *es isomorphi- cally to K-1( 1S2n+1; ^Z2) by Theorem 8.1. Thus, we obtain v1-stabilization iso* *morphisms K-1( 1S2n+1; ^Z2) ~= eK-1(S2n+1; ^Z2)=M ~= {Z=2n, Z=2n, Z=2n}, and we can now apply Theorems 4.3 and 4.11 to determine K*CR( 1S2n+1; ^Z2). In * *fact, using our knowledge of _k in eK-1(S2n+1; ^Z2) for k 2 Zx(2), we find that KOi( 1S2n+1; ^Z* *2) is: Z=2n for i = -1 with _k = kn+1; Z=2n for i = -5 with _k = kn+3; Z=2 for i = -2, -4 with _k = 1;* * Z=2 Z=2 for i = -3 with _k = 1 by [41, Lemma 4.5]; and 0 for i = -6, -7, -8. 8.5. Determining KO*( 1S2n+1; ^Z2) for n 0 mod 4. We consider the -module eK* *1(S2n+1; ^Z2) ~= {^Z2, ^Z2, ^Z2} with its operations t = 1, c = 1, r = 2, c0= 2, q = 1, `2C= 2n,* * `2R= 2n, and `2HR= 2n+1. Then, as in 8.4, we determine K*CR( 1S2n+1; ^Z2) and find that KOi( 1S2n+1; ^Z2* *) is: Z=2n for i = 1 with _k = kn; Z=2n for i = -3 with _k = kn+2; Z=2 for i = 0 or -2 with _k = 1; * *Z=2 Z=2 for i = -1 with _k = 1 by [41, Lemma 4.5]; and 0 for i = -4, -5, -6. We could now apply Theorem 3.2 to determine the v1-periodic homotopy groups v* *-11ß*S2n+1 up to extension. However, to circumvent extension problems, we prefer to treat* * 1S2n+1as a K-theoretic two-cell spectrum built from: 8.6. The K-theoretic sphere and pseudosphere. Suppose that X is a K*-local spec* *trum with K0X ~=Z, K1X ~=0, and XQ ' SQ. Then, by [16, Proposition 10.6], either X ' SK o* *r X ' TK for the spectrum T = S0[je2[2e3. We call TK the K-theoretic pseudosphere and find t* *hat KO*TK is a free KO*-module on a generator of degree 4, instead of the usual degree 0.* * We note that the above classification implies that TK ^TK ' SK , and hence the functor TK ^- act* *s as an equivalence on the homotopy category of K*-local spectra. We let g: SK ! TK be the bottom c* *ell map and find that it gives the doubling homomorphism on K0 using the standard isomorphi* *sms K0SK ~=Z and K0TK ~=Z. Moreover, g generates the group [SK , TK ] ~=ß0TK ~=Z by [13, Cor* *ollary 4.4]. We can now give the promised K-theoretic two-cell model for the homotopy type of * *1S2n+1. Theorem 8.7.The spectrum 2n+1 1S2n+1is equivalent to the homotopy fiber of 2n-* *1g: SK=2! TK=2 for n 1, 2 mod 4 and of 2n: SK=2! SK=2 for n 0, 3 mod 4 . Proof.Using the results of 8.2-8.5, it is straightforward to show that the ^ACR* *-module K*( 2n+1 1S2n+1; ^Z2) is isomorphic to K*(Fn; ^Z2) for the required homotopy fiber Fn, and hence 2n+* *1 1S2n+1' Fn by Theorem 5.3. |__| 32 The above model for 1S2n+1might also be obtained from Yosimura's analysis of* * ( 1 RP2n)K in [41] or from Mahowald and Thompson's work in [34]. To describe the resulting* * homotopy groups, we let 2(j) denote the greatest power of 2 dividing j (where 2(0) = +1). Theorem 8.8.For each i, we have: 8 >>>^Z2 if i = -1 >>>^Z Z=2 if i = 0 >>> 2 > Z=2 Z=2if i 1 mod 8 >>> >>>Z=8 if i 3 mod 8 >>>0 if i 4, 5, 6 mod 8 : (j)+4 Z=2 2 if i = 8j - 1 with j 6= 0 8 >>>^Z2 if i = 0, -1 >>> >><0 if i 0, 1, 2 mod 8 with i 6= 0 ßiTK=2 ' > Z=8 if i 3 mod 8 >>>Z=2 if i 4, 6 mod 8 >>>Z=2 Z=2if i 5 mod 8 >: Z=2 2(j)+4if i = 8j - 1 with j 6= 0 Proof.The results for ßiSK=2follow from [13, Corollary 4.5], while those for ßi* *TK=2follow from Theorem 3.2 except in the case i 5 mod 8 where there is an extension problem.* * To solve this problem, and for later use, we note that TK=2^ S=2 ' TK ^ (S=2)K ' 4(S=2)K by * *Theorem 5.3 or [16, Proposition 10.5]. We then deduce the required splitting of ßiTK=2for i * *5 mod 8 from the orders of the groups ß*(TK=2^ S=2) ~=ß*-4(S=2)K calculated using Theorem 3.2. * * |__| We can now recover the result of Mahowald and Davis [23, p.1041] on the v1-pe* *riodic homotopy groups of S2n+1at p = 2. Theorem 8.9 (Mahowald and Davis).If n 1, 2 mod 4, then 8 >>>Z=2 if i 0, 5 mod 8 min(3,n+1) >>:Z=2 Z=2 if i 2, 3 mod 8 Z=2min(n-1, 2(j)+4)if i = 8j - 2 or 8j - 1 33 If n 0, 3 mod 4, then 8 >>>Z=2 Z=2 Z=2 if i 0, 1 mod 8 >>> >>Z=8 if i 3 mod 8 >>>0 if i 4, 5 mod 8 >>>Z=2min(n, 2(j)+4)if i = 8j - 2 >: Z=2 Z=2min(n, 2(j)+4)if i = 8j - 1 Proof.The map g*: KOi(TK=2; ^Z2) ! KOi(SK=2; ^Z2) is given by 1 for i 0 mod 8* *, given by 4 for i 4 mod 8, and given by 0 otherwise. Hence, using Theorems 3.2 and 8.8,* * we find that g*:ßiSK=2! ßiTK=2is given by (1, 0) for i = 0, given by 4 for i 3 mod 8, give* *n by 1 for i 7 mod 8, and given by 0 otherwise. Using the homotopy fiber sequence ( S=2)K=2! S* *K=2-g!TK=2 and the fact that S=2 has exponent 4, we can now easily calculate the homotopy * *groups v-11ß3+iS3 ~=ßi( 3 1S3) ~= ßi( S=2)K=2 to confirm the theorem for n = 1. We can next calculate the homotopy groups v-1* *1ß2n+iS2n+1~= ßi( 2n+1 1S2n+1) for n > 1 up to extension by using the homotopy exact sequence* *s from Theorem 8.7. Finally we conclude that all of these extensions split since the homomorph* *isms (1 ^ 2n-1g)*:ß*(S=2 ^ SK=2) -! ß*(S=2 ^ TK=2) (1 ^ 2n)*:ß*(S=2 ^ SK=2) -! ß*(S=2 ^ SK=2) are trivial for n > 1, as seen from the exponents of the homotopy groups ß*(S=2* *^SK=2) ~=ß*(S=2)K=2 and ß*(S=2 ^ TK=2) ~=ß*( 4S=2)K=2in each dimension. * * |__| 9.The v1-stabilizations of simply-connected compact Lie groups Finally, we apply the v1-stabilization method to simply-connected compact Lie* * groups at the prime p = 2. For such a group G, our main result (Theorem 9.3) will express KO*( 1G; * *^Z2) in terms of the representation theory of G, assuming that G is bK 1-good. Since we can prove th* *e bK 1-goodness of all simply-connected compact Lie groups, this will confirm our general conje* *cture, which Davis presented in [25, Conjecture 2.2] and used so effectively to calculate v1-perio* *dic homotopy groups. We start by discussing: 9.1. The bK 1-goodness of simply-connected compact Lie groups. In [19, Theorem * *9.2], we proved that each simply-connected compact Lie group G is bK 1-good at an odd pr* *ime p; in fact, we gave explicit constructions of GK=pand 1G. After considerable effort, we recen* *tly showed that this work extends to the prime p = 2 provided that G satisfies a certain Technical C* *ondition involving 34 its representation ring. This result is made useful by recent work of Don Davis* * [25, Theorem 1.3] showing that a simply-connected compact simple Lie group satisfies our Technica* *l Condition if and only if it is not E6or Spin(4k+2) with k not a 2-power. Hence, all simply-conne* *cted compact simple Lie groups are bK 1-good, except possibly for E6and Spin(4k+2) with k not a 2-p* *ower. Fortunately, we can prove that these remaining groups are bK 1-good by a careful analysis of* * (E6=F4)K=2and by fibration arguments. We plan to include a detailed account of this work in a* * subsequent paper showing the bK 1-goodness of all simply-connected compact Lie groups. To state * *our main theorem, we use: 9.2. Indecomposables of representation rings. For a simply-connected compact Li* *e group G, we first let Q(G) = eR(G)=Re(G)2 denote the indecomposables of the complex repr* *esentation ring R(G). We then let QR(G) Q(G) and QH(G) Q(G) denote the real and symplectic * *indecompos- ables given by the images of eRR(G) and eRH(G) in Q(G). These indecomposables c* *ombine to give a ~-ring Q (G) = {Q(G), QR(G), QH(G)} whose structure is inherited from the ~-ring {Re(G), eRR(G), eRH(G)} of 6.1 an* *d 6.2. Since Q (G) has trivial multiplication, its operations ~k are additive for k 1. By standa* *rd results presented in [12, Sections II.6 and VI.4] or [25, Theorem 2.3], Q(G) is a finitely generated* * free abelian group on generators ~æ= æ - dimæ where æ ranges over the basic representations of G; mor* *eover, Q (G) is a free -module on the generators ~æ-dimæ where æ ranges over the complex, real, * *and quaternionic basic representations in Q(G), QR(G), and QH(G) respectively. In particular, Q * *(G) is an exact -module such that c: QR(G) Q(G), c0:QH(G) Q(G), QR(G) \ QH(G) = im(1 + t),* * and QR(G) + QH(G) = ker(1 - t)for the conjugation t: Q(G) ! Q(G). We can now state our main theorem in a form derived directly from Davis [25, * *Conjecture 2.2]. * * For brevity, we write Q = Q(G), QR = QR(G), QH = QH(G), and dKO(-) = KO (-; ^Z2* *). Theorem 9.3.If G is a simply connected compact Lie group which is bK 1-good at * *p = 2, then there is a long exact sequence of abelian groups 0 ~2 1 . .-.! 0 -! dKO( 1G) -! Q=(QR + QH) ---!Q=QR -! dKO( 1G) 2 ~2 -! 0 -! QH=(QR \ QH) -! dKO( 1G) -! QR \ QH ---!QH 3 4 ~2 -! dKO( 1G) -! 0 -! 0 -! dKO( 1G) -! Q=(QR \ QH) ---!Q=QH 5 ~2 -! dKO( 1G) -! (QR + QH)=(QR \ QH) ---!QR=(QR \ QH) 35 -! dKO6( 1G) -! QR + QH -~2--!QR -! dKO7( 1G) -! 0 8 ~2 -! 0 -! dKO( 1G) -! Q=(QR + QH) ---!Q=QR -! . . . which continues by Bott periodicity. Moreover, for any integer i and odd intege* *r k 1, the Adams 2i-1 2i-2 operation _k in dKO ( 1G) and dKO ( 1G) corresponds to k-i_k (or equivalent* *ly k-i+1~k) in the Q-terms under the morphisms of the exact sequence. As explained in 7.6 and [25], this leads to calculations of the v1-periodic h* *omotopy groups v-11ß*G. In cases where G satisfies the Technical Condition (see 9.1), we originally obt* *ained the above exact * sequence (tensored with ^Z2) as the dKO-cohomology exact sequence of a stable (* *co)fiber sequence coming from our explicit construction of 1G at p = 2. We now proceed to prove * *Theorem 9.3 in general using: 9.4. The v1-stabilization of Q (G). For a simply-connected compact Lie group G,* * we first obtain a -module homomorphism ff: eR(G) -! K-1( 1G; ^Z2) by composing the canonical ~-ring homomorphism eR(G) ! eK0(BG; ^Z2) with the * *-module homomorphisms eK0(BG; ^Z2)-1---!K0( 1BG; ^Z2) ?? ? yoe ~=?yoe eK-1(G; ^Z2)-1---!K-1( 1G; ^Z2) This ff factors through the quotient homomorphism eR(G) i Q (G) since it vanish* *es on the terms Re(G)2, eRR(G)2, eRH(G)2, eRR(G)ReH(G), rRe(G)2, qRe(G)2, and OERe(G) by 7.5. W* *e let ff: Q (G) -! K-1( 1G; ^Z2) denote the induced v1-stabilization homomorphism for the exact -module Q (G). 9.5. The exact -modules Q0(G) and Q00(G). We now modify the exact -module Q (* *G) = {Q(G), QR(G), QH(G)} to give an exact -module Q0(G) = {Q(G), QR(G) + QH(G), QR(G) \ QH(G)} with the obvious operations c: QR(G)+QH(G) Q(G), c0:QR(G)\QH(G) Q(G), t: Q(* *G) ! Q(G), r = 1 + t: Q(G) ! QR(G) + QH(G), q = 1 + t: Q(G) ! QR(G) \ QH(G). We note* * that Q0(G) has epic q since QR(G) \ QH(G) equals the image of 1 + t: Q(G) ! Q(G). Th* *e monic operation ~2:Q(G) ! Q(G) now induces a -module monomorphism ~2:Q0(G) ! Q (G) s* *ince 36 ~2t = t~2, ~2QR(G) QR(G), ~2QH(G) QR(G), and ~2(QR(G) \ QH(G)) QR(G) \ QH* *(G). This determines an exact -module Q00(G) = coker~2 belonging to a short exact s* *equence 2 00 0 -! Q0(G) -~-!Q (G) -! Q (G) -! 0 where Q00(G) has monic c by Lemma 4.14 since Q0(G) has epic q. The v1-stabiliza* *tion homomor- phism ff: Q (G) ! K-1( 1G; ^Z2) now factors through the quotient homomorphism Q* * (G) i Q00(G) since it vanishes on image elements of ~2 by 7.4, and we let 00 -1 ff: Q (G) -! K ( 1G; ^Z2) denote the induced v1-stabilization homomorphism. Lemma 9.6. If G is a simply-connected compact Lie group which is bK 1-good at p* * = 2, then ff: Q00(G) ~=K-1( 1G; ^Z2) and K0( 1G; ^Z2) ~=0. Moreover, for any odd integer * *k 1, the Adams operations {_k, _k, _k} in K-1( 1G; ^Z2) correspond to {_k, _k, k2_k} in Q00(G). Proof.Since G is bK 1-good, there are isomorphisms 1:QbK*(G; ^Z2)=~2 ~=K*( 1G; ^Z2) which reduce to isomorphisms ff: Q00(G) ~=K-1( 1G; ^Z2) and 0 ~=K0( 1G; ^Z2) by* * [6] or [29]. Thus ff: Q00(G) ~=K-1( 1G; ^Z2) as in the proof of Theorem 7.2, and the lemma f* *ollows easily. |__| We can now essentially determine K*CR( 1G; ^Z2) from the exact -module Q00(G* *) using the notation Q00= Q00(G), Q00R= Q00R(G), and Q00H= Q00H(G). We write A#B for an ab* *elian group belonging to a short exact sequence 0 ! A ! A#B ! B ! 0. Lemma 9.7. If G is a simply-connected compact Lie group which is bK 1-good at p* * = 2, then there are natural isomorphisms 8 00 >>>QR for m 7 mod 8 >>> 00 >>>QR=r for m 6 mod 8 >>>(Q00=c0)#(Q00\c0)for m 5 mod 8 ><00 0 H KOm ( 1G; ^Z2) ~= >QH\c00 for m 4 mod 8 >>>QH for m 3 mod 8 >>>Q00=q for m 2 mod 8 >>> H00 >>>Q =c for m 1 mod 8 :0 for m 0 mod 8 Moreover, for any integer i and odd integer k 1, the Adams operation _k in KO* *2i-1( 1G; ^Z2) and KO2i-2( 1G; ^Z2) corresponds to the operation k-i_k in the Q-terms. 37 Proof.By 4.1 and Lemma 9.6, K*CR( 1G; ^Z2) is a Bott exact CR-module with K0( 1* *G; ^Z2) ~=0 and K-1( 1G; ^Z2) ~=Q00(G) where c is monic in Q00(G). Hence, the result follow* *s immediately from Theorem 4.3. |__| 9.8. Proof of Theorem 9.3. We first obtain exact sequences 2 00 0 -! Q=(QR + QH) -~-!Q=QR -! Q =c -! 0 00 0 -! QH=(QR \ QH) -! QH=q -! 0 2 00 0 -! QR \ QH -~-!QH -! QH -! 0 00 0 ~2 00 0 0 -! QH\c -! Q=(QR \ QH) --! Q=QH -! Q =c -! 0 00 0 ~2 00 0 -! QH\c -! 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