v1-PERIODIC HOMOTOPY GROUPS OF THE DWYER-WILKERSON SPACE MARTIN BENDERSKY AND DONALD M. DAVIS Abstract.The Dwyer-Wilkerson space DI(4) is the only exotic 2-compact group. We compute its v1-periodic homotopy groups v-11ss*(DI(4)). 1.Introduction In [13], Dwyer and Wilkerson constructed a 2-complete space BDI(4), so named because its F2-cohomology groups form an algebra isomorphic to the ring of rank* *-4 mod-2 Dickson invariants. Its loop space, called DI(4), has H*(DI(4); F2) finit* *e. In [14], they then defined a p-compact group to be a pair (X, BX), such that X = * *BX (hence X is redundant), BX is connected and p-complete, and H*(X; Fp) is finite* *. In [1], Andersen and Grodal proved that (DI(4), BDI(4)) is the only simple 2-compa* *ct group not arising as the 2-completion of a compact connected Lie group. The p-primary v1-periodic homotopy groups of a topological space X, defined in [12] and denoted v-11ss*(X)(p)or just v-11ss*(X) if the prime is clear, are a f* *irst ap- proximation to the p-primary homotopy groups. Roughly, they are a localization * *of the portion of the actual homotopy groups detected by p-local K-theory. In [11* *], the second author completed a 13-year project, often in collaboration with the * *first author, of determining v-11ss*(X)(p)for all compact simple Lie groups and all p* *rimes p In this paper, we determine the 2-primary groups v-11ss*(DI(4)). Here and thr* *ough- out, (-) denotes the exponent of 2 in the prime factorization of an integer. __________ Date: June 7, 2007. 2000 Mathematics Subject Classification. 55Q52, 57T20, 55N15. Key words and phrases. v1-periodic homotopy groups, p-compact groups, Adams operations, K-theory. 1 2 MARTIN BENDERSKY AND DONALD M. DAVIS Theorem 1.1. For any integer i, let ei= min(21, 4 + (i - 90627)). Then 8 >>>Z=2ei Z=2 d = 1 >>> e >>>Z=2 i d = 2 >< 0 d = 3, 4 v-11ss8i+d(DI(4)) > >>>Z=8 d = 5 >>> >>>Z=8 Z=2 d = 6 :Z=2 Z=2 Z=2 d = 7, 8. Since every v1-periodic homotopy group is a subgroup of some actual homotopy group, this result implies that exp2(DI(4)) 21, i.e., some homotopy group of * *DI(4) has an element of order 221. It would be interesting to know whether this bound* * is sharp. Our proof involves studying the spectrum 1DI(4) which satisfies ss*( 1DI(4)) v-11ss*(DI(4)). We will relate 1DI(4)) to the 2-completed K-theoretic pseudosp* *here TK=2 discussed in [8, 8.6]. We will prove the following surprising result, whic* *h was pointed out by Pete Bousfield. Theorem 1.2. There is an equivalence of spectra 1DI(4) ' 725019TK=2^ M(221), where M(221) is a mod 221Moore spectrum. In Section 3, we will give the easy deduction of Theorem 1.1 from Theorem 1.2. As an immediate corollary of 1.2, we deduce that the 221bound on ss*( 1DI(4)) is induced from a bound on the spectrum itself. Corollary 1.3. The exponent of the spectrum 1DI(4) is 221; i.e., 2e1 1DI(4)is * *null if and only if e 21. In [5], Bousfield presented a framework that enables determination of the v1- periodic homotopy groups of many simply-connected H-spaces X from their united K-theory groups and Adams operations. The intermediate step is KO*( 1X). (All of our K*(-) and KO*(-)-groups have coefficients in the 2-adic integers ^Z2, wh* *ich we omit from our notation.) Our first proof of Theorem 1.1 used Bousfield's exa* *ct sequence [5, 9.2] which relates v-11ss*(X) with KO*( 1X), but the approach via * *the pseudosphere, which we present here, is stronger and more elegant. The insight * *for v1-PERIODIC HOMOTOPY GROUPS OF DI(4) 3 Theorem 1.2 is the observation that the two spectra have isomorphic Adams modul* *es KO*(-). In several earlier e-mails, Bousfield explained to the authors how the result* *s of [5] should enable us to determine KO*( 1DI(4)). In Section 4, we present our account of these ideas of Bousfield. We thank him profusely for sharing his insights wi* *th us. The other main input is the Adams operations in K*(BDI(4)). In [18], Osse and Suter showed that K*(BDI(4)) is a power series algebra on three specific genera* *tors, and gave some information toward the determination of the Adams operations. In private communication in 2005, Suter expanded on this to give explicit formulas* * for _k in K*(BDI(4)). We are very grateful to him for sharing this information. In Sec* *tion 2, we will explain these calculations and also how they then lead to the determ* *ination of KO*( 1DI(4)). 2.Adams operations In this section, we present Suter's determination of _k in K*(BDI(4)) and sta* *te a result, proved in Section 4, that allows us to determine KO*( 1DI(4)) from th* *ese Adams operations. Our first result, communicated by Suter, is the following determination of Ad* *ams operations in K*(BDI(4)). An element of K*(X) is called real if it is in the im* *age of KO*(X) c-!K*(X). Theorem 2.1. (Suter) There is an isomorphism of algebras K*(BDI(4)) ^Z2[[,8, ,12, ,24]] (2.2) such that the generators are in K0(-) and are real, _-1 = 1, and the matrices o* *f _2 and _3 on the three generators, mod decomposables, are 0 1 0 1 24 0 0 34 0 0 2 B@-2 26 0 CA, 3 B@-33 36 0CA. 0 -2 214 36=527 -35. 41=17 314 Proof.The subscripts of the generators indicates their "filtration," meaning th* *e di- mension of the smallest skeleton on which they are nontrivial. A standard prope* *rty of Adams operations is that if , has filtration 2r, then _k(,) equals kr, plus * *elements of higher filtration. 4 MARTIN BENDERSKY AND DONALD M. DAVIS The isomorphism (2.2) is derived in [18, p.184] along with the additional inf* *orma- tion that 4,24- ,212has filtration 28, and ,12 = ~2(,8) + 8,8 (2.3) ,24 = ~2(,12) + 32,12+ c1,28+ c2,38+ c3,8,12, for certain explicit even coefficients ci. The Atiyah-Hirzebruch spectral sequence easily shows that ,8 is real, since t* *he 11- skeleton of BDI(4) equals S8. Since ~2(c(`)) = c(~2(`)), and products of real b* *undles are real, we deduce from (2.3) that ,12and ,24are also real. Since tc = c, wher* *e t denotes conjugation, which corresponds to _-1, we obtain that the generators are invariant under _-1, and hence so is all of K*(BDI(4)). We compute Adams operations mod decomposables, writing for equivalence mod decomposables. Because 4,24- ,212has filtration 28, we obtain _k(,24) k14,24. (2.4) Here we use, from [18, p.183], that all elements of K*(BDI(4)) of filtration gr* *eater than 28 are decomposable. Equation (2.4) may seem surprising, since ,24has filt* *ration 24, but there is a class ,28 such that 4,24- ,212= ,28, and we can have _k(,24) k12,24+ ffk,28consistently with (2.4). Using (2.3) and that _2 -2~2 mod decomposables, we obtain _2(,8) 24,8- 2,12 (2.5) _2(,12) 26,12- 2,24, yielding the matrix 2 in the theorem. Applying _2_3 = _3_2 to _3(,12) 36,12+fl,24yields -2.36+214fl = 26fl -2.314, from which we obtain fl = -35 . 41=17. Applying the same relation to _3(,8) = 34,8 + ff,12+ fi,24, coefficients of ,12yield -2 . 34 + ff . 26 = 24ff - 2 . 36* * and hence ff = -33. Now coefficients of ,24yield -2ff + 214fi = 24fi - 2fl and hence fi =* * 36=527. || Let 1(-) denote the functor from spaces to K=2*-local spectra described in [* *5, 9.1], which satisfies v-11ss*X ss*o2 1X, where o2 1X is the 2-torsion part of* * 1X. In Section 4, we will use results of Bousfield in [5] to prove the following resul* *t. Aspects v1-PERIODIC HOMOTOPY GROUPS OF DI(4) 5 of Theorem 2.1, such as K*(BDI(4)) being a power series algebra on real generat* *ors, are also used in proving this theorem. Recall that KO*(-) has period 8. Theorem 2.6. The groups KOi( 1DI(4)) are 0 if i 0, 1, 2 mod 8, and K0( 1DI(4* *)) = 0. Let M denote a free ^Z2-module on three generators, acted on by _2 and _3 by* * the matrices of Theorem 2.1, with _-1 = 1. Let ` = 1_2_2 act on M. Then there are e* *xact sequences 0 ! 2M -`!2M ! KO3( 1DI(4)) ! 0 ! 0 ! KO4( 1DI(4)) ! M=2 `-!M=2 ! KO5( ` 6 1DI(4)) ! M=2 -!M=2 ! KO ( 1DI(4)) ! M `-!M ! KO7( 1DI(4)) ! 0 and 0 ! M -`!M ! K1( DI(4)) ! 0. For k = -1 and 3, the action of _k in KO2j-1( 1DI(4)), KO2j-2( 1DI(4)), and K2j-1( DI(4)) agrees with k-j_k in adjacent M-terms. In the remainder of this section, we use 2.1 and 2.6 to give explicit formula* *s for the Adams module KOi( 1DI(4)). A similar argument works for K*( 1DI(4)). If g1, g2, and g3 denote the three generators of M, then the action of ` is given by `(g1) = 8g1- g2 `(g2) = 25g2- g3 `(g3) = 213g3. Clearly ` is injective on M and 2M. We have KO7( 1DI(4)) coker(`|M) Z=221 with generator g1; note that g2 = 23g1 in this cokernel, and then g3 = 28g1. Si* *milarly KO3( 1DI(4)) coker(`|2M) Z=221. Also KO4( 1DI(4)) ker(`|M=2) = Z=2 with generator g3, while KO6( 1DI(4)) coker(`|M=2) = Z=2 with generator g1. There is a short exact sequence 0 ! coker(`|M=2) ! KO5( 1DI(4)) ! ker(`|M=2) ! 0, with the groups at either end being Z=2 as before. To see that this short exact f sequence is split, we use the map S7 -! DI(4) which is inclusion of the bottom cell. The morphism f* sends the first summand of KO5( 1DI(4)) to one of the two 6 MARTIN BENDERSKY AND DONALD M. DAVIS Z=2-summands of KO5( 1S7), providing a splitting homomorphism. Thus we have proved the first part of the following result. Theorem 2.7. We have 8 < 0 i = 0 Ki( 1DI(4)) : Z=221 i = 1, 8 >>>0 i = 0, 1, 2 >>< Z=221 i = 3, 7 KOi( 1DI(4)) > >>>Z=2 i = 4, 6 >: Z=2 Z=2 i = 5. For k = -1 and 3, we have _k = 1 on the Z=2's, and on KO2j-1( 1DI(4)) with j even and K2j-1( 1DI(4)), _-1 = (-1)j and _3 = 3-j(34- 33. 23+ 36_52728). Completion of proof.To obtain _3 on the Z=2's, we use the last part of Theorem * *2.6 and the matrix 3of Theorem 2.1. If _3 is as in 3, then, mod 2, _3-1 sends g1 * *7! g2, g2 7! g3, and g3 7! 0. Thus _3 - 1 equals 0 on KO4( 1DI(4)) and KO6( 1DI(4)). Clearly _-1 = 1 on these groups. To see that _k - 1 is 0 on KO5( 1DI(4)), we use the commutative diagram ae 0 ---! Z=2 --i-! KO5( 1DI(4)) ---! Z=2 ---! 0 ?? ?? ?? ?y f*?y 0?y 0 ---! Z=2 ---! KO5( 1S7) ---! Z=2 ---! 0. We can choose generators G1 and G2 of KO5( 1DI(4)) Z=2 Z=2 so that G1 2 im(i), ae(G2) 6= 0, and f*(G2) = 0. Since _k - 1 = 0 on the Z=2's on either sid* *e of KO5( 1DI(4)), the only way that _k - 1 could be nonzero on KO5( 1DI(4)) is if (_k - 1)(G2) = G1. However this yields the contradiction 0 = (_k - 1)f*G2 = f*(_k - 1)G2 = f*(G1) 6= 0. On KO2j-1( 1DI(4)) with j even and K2j-1( 1DI(4)), _3 sends the generator g1 to 3-j(34g1- 33g2+ 36_527g3) = 3-j(34- 33. 23+ 36_52728)g1, and _-1(g1) = (-1)jg1 by Theorem 2.6. || v1-PERIODIC HOMOTOPY GROUPS OF DI(4) 7 3. Relationship with pseudosphere In this section, we prove Theorems 1.2 and 1.1. Following [8, 8.6], we let T = S0[je2[2e3, and consider its K=2-localization * *TK=2. The groups ss*(TK=2) are given in [8, 8.8], while the Adams module is given by 8 < ^Z2 i even, with_k = k-i=2 Ki(TK=2) = : 0 i odd; 8 >><^Z2 i 0 mod 4, with _k = k-i=2 KOi(TK=2) = > Z=2 i = 2, 3, with _k = 1 >: 0 i = 1, 5, 6, 7. Bousfield calls this the 2-completed K-theoretic pseudosphere. Closely related * *spectra have been also considered in [15] and [4]. Let M(n) = S-1 [n e0 denote the mod n Moore spectrum. Then, for e > 1 and k odd, 8 >>Z=2e i 0 mod 4, with _k = k-i=2 >>< Z=2 i = 1, 3, with _k = 1 KOi(TK=2^ M(2e)) = > k (3.1) >>>Z=2 Z=2i = 2, with _ = 1 >: 0 i = 5, 6, 7. Proof.Let Y = TK=2^ M(2e). Most of (3.1) is immediate from the exact sequence 2e-!KOi(T i i+1 2e K=2) -!KO (Y ) -!KO (TK=2) -! . To see that KO2(Y ) = Z=2 Z=2 and not Z=4, one can first note that M(2e) ^ M(2) ' -1M(2) _ M(2). (3.2) The exact sequence KO2(Y ) 2-!KO2(Y ) -!KO2(Y ^ M(2)) -!KO3(Y ) 2-! (3.3) implies that if KO2(Y ) = Z=4, then |KO2(Y ^ M(2))| = 4. However, by (3.2), KO2(Y ^ M(2)) KO2(TK=2^ M(2)) KO3(TK=2^ M(2)). (3.4) 8 MARTIN BENDERSKY AND DONALD M. DAVIS Also, there is a cofiber sequence T ^ M(2) ! -1A1 ! 5M(2), (3.5) where H*(A1; F2) is isomorphic to the subalgebra of the mod 2 Steenrod algebra generated by Sq1and Sq2, and satisfies KO*(A1) = 0. Thus 8 >>Z=2 Z=2min(21, (i)+4)d = -2 >>> min(21, (i)+4) >>>Z=2 d = -1 >< 0 d = 0, 1 ss8i+d(TK=2^ M(221)) > >>>Z=8 d = 2 >>> >>>Z=2 Z=8 d = 3 :Z=2 Z=2 Z=2 d = 4, 5. Proof.For the most part, these groups are immediate from the groups ss*(TK=2) g* *iven in [8, 8.8] and the exact sequence -221!ss 21 221 j+1(TK=2) ! ssj(TK=2^ M(2 )) ! ssj(TK=2) -! . (3.10) 10 MARTIN BENDERSKY AND DONALD M. DAVIS All that needs to be done is to show that the following short exact sequences, * *obtained from (3.10), are split. 0 ! Z=2 ! ss8i+3(TK=2^ M(221))! Z=8 ! 0 (3.11) 0 ! Z=2 Z=2 ! ss8i+4(TK=2^ M(221))! Z=2 ! 0 0 ! Z=2 ! ss8i+5(TK=2^ M(221))! Z=2 Z=2 ! 0 0 ! Z=2min(21, (i)+4)! ss8i-2(TK=2^ M(221))! Z=2 ! 0. Let Y = TK=2^ M(221). We consider the exact sequence for ss*(Y ^ M(2)), 2-!ss 2 i+1(Y ) ! ssi(Y ^ M(2)) ! ssi(Y ) -!.(3.12) If the four sequences (3.11) are all split, then by (3.12) the groups ss8i+d(Y * *^ M(2)) for d = 2, 3, 4, 5, -2 have orders 23, 25, 26, 25, and 23, respectively, but if* * any of the sequences (3.11) fails to split, then some of the orders |ss8i+d(Y ^ M(2))| wil* *l have values smaller than those listed here. By (3.2), ssi(Y ^ M(2)) ssi+1(TK=2^ M(2)) ssi(TK=2^ M(2)). By (3.5), since localization preserves cofibrations and (A1)K=2= *, there is an* * equiv- alence 4MK=2' TK=2^ M(2), (3.13) and hence ssi(Y ^ M(2)) ssi-3(MK=2) ssi-4(MK=2).(3.14) By [10, 4.2], 8 >>>0 d = 4, 5 >>< Z=2 d = -2, 3 ss8i+d(MK=2) = > >>>Z=2 Z=2d = -1, 2 >: Z=4 Z=2 d = 0, 1. This is the sum of two "lightning flashes," one beginning in 8d - 2 and the oth* *er in 8d - 1. Substituting this information into (3.14) yields exactly the orders * *which were shown in the previous paragraph to be true if and only if all the exact se* *quences (3.11) split. || v1-PERIODIC HOMOTOPY GROUPS OF DI(4) 11 4.Determination of KO*( 1DI(4)) In this section, we prove Theorem 2.6, which shows how _k in K*(BDI(4)) leads to the determination of KO*( 1DI(4)). Our presentation here follows suggestions* * in several e-mails from Pete Bousfield. The first result explains how KO*(BDI(4)) follows from K*(BDI(4)). Theorem 4.1. There are classes g8, g12, and g24in KO0(BDI(4)) such that c(gi) = ,i, with ,ias in 2.1, and KO*(BDI(4)) KO*[[g8, g12, g24]]. The Adams operations _2 and _3 mod decomposables on the basis of gi's is as in * *2.1. Proof.In [2, 2.1], it is proved that if there is a torsion-free subgroup F * K* *O*(X) such that F * K*(pt) ! K*(X) is an isomorphism, then so is F * KO*(pt) ! KO*(X). The proof is a Five Lemma argument using exact sequences in [17, p.257]. Although the result is stated for ordinary (not 2-completed) KO*(-), the same a* *r- gument applies in the 2-completed context. If F *is a multiplicative subgroup, * *then the result holds as rings. Our result then follows from 2.1, since the generato* *rs there are real. A similar proof can be derived from [5, 2.3]. || Next we need a similar sort of result about KO*(DI(4)). We could derive much of what we need by an argument similar to that just used, using the result of [* *16] about K*(DI(4)) as input. However, as we will need this in a specific form in o* *rder to use it to draw conclusions about KO*( 1DI(4)), we begin by introducing much terminology from [5]. The study of united K-theory begins with two categories, which will then be e* *n- dowed with additional structure. We begin with a partial definition of each, an* *d their relationship. For complete details, the reader will need to refer to [5] or an* * earlier paper of Bousfield. Definition 4.2. ([5, 2.1]) A CR-module M = {MC, MR} consists of Z-graded 2- profinite abelian groups MC and MR with continuous additive operations M*C B-! j *-1 * c * * r * M*-2C, M*R-BR!M*-8R, M*C-t!M*C, M*R-! MR , MR -! MC, MC -! MR, satis- fying 15 relations, which we will mention as needed. 12 MARTIN BENDERSKY AND DONALD M. DAVIS We omit the descriptor "2-adic," which Bousfield properly uses, just as we om* *it writing the 2-adic coefficients ^Z2which are present in all our K- and KO-group* *s. Example 4.3. For a spectrum or space X, the united 2-adic K-cohomology K*CR(X) := {K*(X), KO*(X)} is a CR-module, with complex and real Bott periodicity, conjugation, the Hopf m* *ap, complexification, and realification giving the respective operations. Definition 4.4. ([5, 4.1]) A -module N = {NC, NR, NH } is a triple of 2-profin* *ite abelian groups NC, NR, and NH with continuous additive operations NC -t! NC, 0 q NR -c!NC, NC -r!NR, NH -c!NC, and NC -!NH satisfying nine relations. Example 4.5. For a CR-module M and an integer n, there is a -module nM = {MnC, MnR, Mn-4R} with c0 = B-2c and q = rB2. In particular, for a space X and integer n, there is a -module Kn(X) := nK*CR(X). Now we add additional structure to these definitions. Definition 4.6. ([5, 4.3, 6.1]) A ` -module is a -module N together with homo- morphisms NC -`!NC, NR -`!NR, and NH -`!NR satisfying certain relations listed in [5, 4.3]. An Adams -module is a ` -module N together with Adams operations _k N -! N for odd k satisfying the familiar properties. Example 4.7. In the notation of Example 4.5, K-1(X) is an Adams -module with ` = -~2. Definition 4.8. ([5, 2.6, 3.1,]3.2) A special OECR-algebra {AC, AR} is a CR-mod* *ule OE 0 with bilinear AmCx AnC! Am+nCand AmRx AnR! Am+nRand also A0C-! AR and OE 0 A-1C-!AR satisfying numerous properties. Remark 4.9. The operations OE, which were initially defined in [7], are less fa* *miliar than the others. Two properties are cOEa = t(a)a and OE(a + b) = OEa + OEb + r(* *t(a)b) for a, b 2 A0C. For a connected space X, K*CR(X) is a special OECR-algebra. The following important lemma is taken from [5]. Lemma 4.10. ([5, 4.5, 4.6]) For any ` -module M, there is a universal special O* *ECR- algebra ^LM. This means that there is a morphism M -ff!^LM such that any morphi* *sm from M into a OECR-algebra factors as ff followed by a unique OECR-algebra morp* *hism. v1-PERIODIC HOMOTOPY GROUPS OF DI(4) 13 There is an algebra isomorphism ^MC ! (^LM)C, where ^(-) is the 2-adic exterior algebra functor. In [5, 2.7], Bousfield defines, for a CR-algebra A, the indecomposable quotie* *nt ^QA. We apply this to A = K*CR(BDI(4)), and consider the -module Q^K0(BDI(4)), analogous to [5, 4.10]. We need the following result, which is more delicate th* *an the K-1-case considered in [5, 4.10]. Lemma 4.11. With ` = -~2, the -module ^QK0(BDI(4)) becomes a ` -module. Proof.First we need that ` is an additive operation. In [7, 3.6], it is shown * *that `(x + y) = `(x) + `(y) - xy if x, y 2 KOn(X) with n 0 mod 4. The additivity follows since we are modding out the product terms. (In the case n -1 mod 4 considered in [5, 4.10], the additivity of ` is already present before modding * *out indecomposables.) There are five additional properties which must be satisfied by `. That `cx =* * c`x and `tz = t`z are easily obtained from [7, 3.4]. That `c0y = c`y follows from * *[8, 6.2(iii),6.4]. That `qz = `rz follows from preceding [7, 3.10] by c, which is s* *urjective for us. Here we use that rc = 2 and q = rB2. Finally, `rz = r`z for us, since c* * is surjective; here we have used the result ~OEcx = 0 given in [5, 4.3]. || Now we obtain the following important description of the CR-algebra K*CR(DI(4* *)). Theorem 4.12. There is a morphism of ` -modules ^QK0(BDI(4)) ! "K-1(DI(4)) which induces an isomorphism of special OECR-algebras ^L(Q^K0(BDI(4))) ! K*CR(DI(4)). Proof.The map DI(4) = BDI(4) ! BDI(4) induces a morphism K0(BDI(4)) ! K-1(DI(4)) which factors through the indecomposable quotient ^QK0(BDI(4)). In [16, 1.2], a general result is proved which implies that K*(DI(4)) is an exterior algebra on* * ele- ments of K1(DI(4)) which correspond to the generators of the power series algeb* *ra K*(BDI(4)) under the above morphism followed by the Bott map. Thus our result will follow from [5, 4.9], once we have shown that the ` -module M := ^QK0(BDI(* *4)) 14 MARTIN BENDERSKY AND DONALD M. DAVIS is robust([5, 4.7]). This requires that M is profinite, which follows as in the* * remark following [5, 4.7], together with two properties regarding ~OE, where ~OEz := `* *rz - r`z for z 2 MC. In our case, c is surjective, and so ~OE= 0 as used in the previous* * proof. One property is that M is torsion-free and exact. This follows from the Bott exactness of the CR-module K*CR(BDI(4)) noted in [5, 2.2], and [5, 5.4], which states that, for any n, the -module nN associated to a Bott exact CR-module N with NnCtorsion-free and Nn-1C= 0 is torsion-free. The other property is ker(~O* *E) = cMR + c0MH + 2MC. For us, both sides equal MC since c is surjective and ~OE= 0.* * || Our Theorem 2.6 now follows from [5, 9.5] once we have shown that the Adams -module M := ^QK0(BDI(4)) is "strong." ([5, 7.11]) This result ([5, 9.5]) requ* *ires that the space (here DI(4)) be an H-space (actually K=2*-durable, which is sati* *sfied by H-spaces) and that it satisfies the conclusion of our 4.12. It then deduces* * that KO*( 1DI(4)) fits into an exact sequence which reduces to ours provided MR = MC and MH = 2MC. These equalities are implied by Q^K0(BDI(4)) being exact, as was noted to be true in the previous proof, plus t = 1 and c surjective, as were noted to be true in 2.1. Indeed, the exactness property, ([5, 4.2]), includes * *that cMR +c0MH = ker(1-t) and cMR \c0MH = im(1+t). Another perceptible difference is that Bousfield's exact sequence is in terms of M~ := M=~OE, while ours invol* *ves M, but these are equal since, as already observed, ~OE= 0 since c is surjective. Note also that the Adams operations in M~ in the exact sequence of [5, 9.5], * *which reduces to that in our 2.6, are those in the Adams -module ^QK0(BDI(4)), which are given in our 2.1. The morphism ` in [5, 9.5] or our 2.6 is 1_2_2, since thi* *s equals -~2 mod decomposables. Finally, we show that our M is strong. One of the three criteria for being st* *rong is to be robust, and we have already discussed and verified this. The second requi* *rement for an Adams -module to be strong is that it be "regular." This rather technic* *al condition is defined in [5, 7.8]. In [5, 7.9], a result is proved which immedia* *tely implies that "K-1(DI(4)) is regular. By 4.12, our M injects into "K-1(DI(4)), and so by* * [5, 7.10], which states that a submodule of a regular module is regular, our M is r* *egular. The third requirement for M to be strong is that it be _3-splittable ([5, 7.2* *]), which means that the quotient map M ! M=~OEhas a right inverse. As we have noted seve* *ral v1-PERIODIC HOMOTOPY GROUPS OF DI(4) 15 times, we have ~OE= 0, and so the identity map serves as a right inverse to the* * identity map. This completes the proof that our M is strong, and hence that [5, 9.5] app* *lies to DI(4) to yield our Theorem 2.6. References [1]K. K. S. Andersen and J. Grodal, The classification of 2-compact groups, preprint. [2]D. W. Anderson, Universal coefficient theorems for K-theory, preprint (1971* *). [3]M. Bendersky and D. M. Davis, v1-periodic homotopy groups of SO(n), Memoir Amer Math Soc 815 (2004). [4]M. Bendersky, D. M. Davis, and M. Mahowald, Stable geometric dimension of vector bundles over even-dimensional real projective spaces, Trans Amer Math Soc 358 (2006) 1585-1603. [5]A. K. Bousfield, On the 2-adic K-localizations of H-spaces, Homology, Homo- topy, and Applications 9 (2007) 331-366. [6]________, The K-theory localization and v1-periodic homotopy groups of H- spaces, Topology 38 (1999) 1239-1264. [7]________, Kunneth theorems and unstable operations in 2-adic KO- cohomology, K-theory, to appear. 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Osse, The K-theory of p-compact groups, Comm Math Helv 72 (1997) 556-581. [17]M. Karoubi, Algebr`es de Clifford et K-th'eorie, Ann Sci de l'Ecole Norm Sup (1968) 162-270. [18]A. Osse and U. Suter, Invariant theory and the K-theory of the Dwyer- Wilkerson space, Contemp Math Amer Math Soc, 265 (2000) 175-185. 16 MARTIN BENDERSKY AND DONALD M. DAVIS Hunter College, CUNY, NY, NY 01220 E-mail address: mbenders@hunter.cuny.edu Lehigh University, Bethlehem, PA 18015 E-mail address: dmd1@lehigh.edu