v1-periodic homotopy groups of the Dwyer-Wilkerson space

v 1 -PERIODIC HOMOTOPY GR OUPS OF THE D WYER -WILKERSON SP A CE MAR TIN BENDERSKY AND DONALD M. DA VIS Abstract. The Dwyer-Wilkerson space D I (4) is the only exotic 2-compact group. W e compute its v 1 -p erio dic homotopy groups v − 1 1 π ∗ ( D I (4)). 1. Introduction In [13], D wy er and Wilke rson constructed a 2 -complete space B D I (4), so named b ecause its F 2 -cohomology groups form an algebra isomorphic to the ring of rank-4 mo d-2 Dic kson inv aria n ts. Its lo op space, called D I (4), has H ∗ ( D I (4); F 2 ) finite. In [14], they then defined a p -compact g roup to b e a pair ( X , B X ), such that X = Ω B X (hence X is redundan t), B X is connected and p - complete, a nd H ∗ ( X ; F p ) is finite. In [1], Andersen and Gro dal prov ed that ( D I (4 ) , B D I ( 4 )) is the only simple 2-compact group not arising a s the 2-completion of a compact connected Lie g roup. The p -primary v 1 -p erio dic homotop y groups of a top ological space X , defined in [12] and denoted v − 1 1 π ∗ ( X ) ( p ) or just v − 1 1 π ∗ ( X ) if the prime is clear, ar e a first ap- pro ximation to the p -prima r y homotopy groups. Roughly , they are a lo calization of the p ortion of the actual homotop y groups detected by p -lo cal K -theory . In [11], the second author completed a 13-y ear pro ject, often in collab o r a tion with the first author, of determining v − 1 1 π ∗ ( X ) ( p ) for all compact simple Lie gro ups a nd all primes p In this pap er, we determine the 2-primary groups v − 1 1 π ∗ ( D I (4)). Here and thro ugh- out, ν ( − ) denotes the exp onen t of 2 in the prime factorization o f an in teger. Date : June 7, 20 07. 2000 Mathematics Subje ct Classific ation. 5 5Q52, 5 7T20, 55N15 . Key wor ds and ph r ases. v 1 -p erio dic ho motopy groups, p - compact groups, Adams op erations, K -theory . 1 2 MAR T IN BENDERSKY AND DONALD M. DA VIS Theorem 1.1. F or any inte ger i , le t e i = min(2 1 , 4 + ν ( i − 90627)) . Then v − 1 1 π 8 i + d ( D I (4)) ≈                        Z / 2 e i ⊕ Z / 2 d = 1 Z / 2 e i d = 2 0 d = 3 , 4 Z / 8 d = 5 Z / 8 ⊕ Z / 2 d = 6 Z / 2 ⊕ Z / 2 ⊕ Z / 2 d = 7 , 8 . Since ev ery v 1 -p erio dic homotop y group is a subgroup of some actual homotop y group, this result implies that exp 2 ( D I (4)) ≥ 2 1, i.e., some homoto p y group of D I (4) has an elemen t of order 2 21 . It w ould b e interes ting to kno w whether this b ound is sharp. Our pro of inv olv es studying the sp ectrum Φ 1 D I (4) whic h satisfies π ∗ (Φ 1 D I (4)) ≈ v − 1 1 π ∗ ( D I (4)). W e will relate Φ 1 D I (4)) to the 2-completed K -theoretic pseudosphere T K/ 2 discusse d in [8, 8.6]. W e will pro v e the f ollo wing surprising result, whic h was p oin ted out by P ete Bousfield. Theorem 1.2. Ther e is an e quivalenc e of sp e ctr a Φ 1 D I (4) ≃ Σ 725019 T K/ 2 ∧ M (2 21 ) , wher e M ( 2 21 ) is a mo d 2 21 Mo or e sp e ctru m. In Section 3, w e will give the easy deduction of Theorem 1.1 from Theorem 1.2. As an immediate coro llary of 1.2, w e deduce tha t the 2 21 b ound on π ∗ (Φ 1 D I (4)) is induced fro m a b ound on the sp ectrum itself. Corollary 1.3. T he e xp onent of the sp e ctrum Φ 1 D I (4) is 2 21 ; i.e., 2 e 1 Φ 1 D I (4) is nul l if and only if e ≥ 21 . In [5 ], Bousfield pres en ted a framew ork t ha t e nables determination of the v 1 - p erio dic homotop y gro ups of man y simply-connected H -spaces X from their united K - theory groups and Adams op erat ions. The in termediate step is K O ∗ (Φ 1 X ). (All of our K ∗ ( − ) and K O ∗ ( − )-groups ha v e co efficien ts in the 2-adic in tegers ˆ Z 2 , whic h w e omit from our notatio n.) Our first pro o f of Theorem 1.1 used Bousfield’s exact sequence [5, 9.2] whic h relates v − 1 1 π ∗ ( X ) with K O ∗ (Φ 1 X ), but the approach via the pseudosphere, whic h w e presen t here, is stronger and more elegan t. The insigh t for v 1 -PERIODIC HOMOTOPY GROUPS OF DI (4) 3 Theorem 1 .2 is the observ ation that the tw o sp ectra hav e isomorphic Adams mo dules K O ∗ ( − ). In sev eral earlier e-mails, Bousfield explaine d to t he authors how the results of [5] should enable us to determine K O ∗ (Φ 1 D I (4)). In Section 4 , we presen t our account of these ideas o f Bousfield. W e thank him profusely for sharing his insigh ts with us. The other main input is the Adams op erations in K ∗ ( B D I (4)). In [18], Osse a nd Suter sho w ed that K ∗ ( B D I (4)) is a p ow e r series algebra on t hree sp ecific generator s, and ga v e some information to w ard the determination of the Adams op erations. In priv ate c ommunic atio n in 2005, Suter expanded on this to giv e explicit form ulas for ψ k in K ∗ ( B D I (4)). W e a re v ery grateful t o him for sharing this information. In Section 2, w e will explain these calculations and also how they then lead to the determination of K O ∗ (Φ 1 D I (4)). 2. Ad ams opera tions In this section, w e pres ent Suter’s determin atio n of ψ k in K ∗ ( B D I (4)) a nd state a result, prov ed in Section 4, that allows us to determine K O ∗ (Φ 1 D I (4)) from these Adams op era t io ns. Our first result, communicated b y Suter, is the f o llo wing determination of Adams op erations in K ∗ ( B D I (4)). An elemen t of K ∗ ( X ) is called r e al if it is in the image of K O ∗ ( X ) c − → K ∗ ( X ). Theorem 2.1. (Suter) The r e is an isomorph ism of algebr as K ∗ ( B D I (4)) ≈ ˆ Z 2 [ [ ξ 8 , ξ 12 , ξ 24 ] ] (2.2) such that the gener ators ar e in K 0 ( − ) and ar e r e al, ψ − 1 = 1 , and the matric es of ψ 2 and ψ 3 on the thr e e gener ators, mo d de c omp osables, a r e Ψ 2 ≡    2 4 0 0 − 2 2 6 0 0 − 2 2 14    , Ψ 3 ≡    3 4 0 0 − 3 3 3 6 0 36 / 527 − 3 5 · 41 / 17 3 14    . Pr o of . The subscripts of the generators indicates their “filtrat ion,” meaning the di- mension of the smallest sk eleton on whic h they are non tr ivial. A standard prop ert y of Adams op erations is t ha t if ξ has filtration 2 r , then ψ k ( ξ ) equals k r ξ plus elemen ts of higher filtration. 4 MAR TIN BENDER SKY A ND DONALD M. DA VIS The isomorphism (2 .2 ) is deriv ed in [18 , p.18 4] along with the a dditional informa- tion that 4 ξ 24 − ξ 2 12 has filtrat io n 28 , and ξ 12 = λ 2 ( ξ 8 ) + 8 ξ 8 (2.3) ξ 24 = λ 2 ( ξ 12 ) + 32 ξ 12 + c 1 ξ 2 8 + c 2 ξ 3 8 + c 3 ξ 8 ξ 12 , for certain explicit ev en co efficien ts c i . The A tiy ah-Hirzebruc h sp ectral sequence easily show s that ξ 8 is real, since the 11- sk eleton of B D I (4) equals S 8 . Since λ 2 ( c ( θ )) = c ( λ 2 ( θ )), and pro ducts of real bundles are real, w e deduce from (2.3) that ξ 12 and ξ 24 are also real. Since tc = c , where t denotes conjug a tion, whic h corresponds to ψ − 1 , w e obtain that the generato r s a r e in v ariant under ψ − 1 , and hence so is all of K ∗ ( B D I (4)). W e compute Adams o p erations mo d decomp o sables, writing ≡ for equiv alence mo d decomp osables. Because 4 ξ 24 − ξ 2 12 has filtrat io n 28, we obtain ψ k ( ξ 24 ) ≡ k 14 ξ 24 . (2.4) Here w e use, from [18, p.183], that all elemen ts of K ∗ ( B D I (4)) of filtration greater than 28 are decomp osable. Equation (2.4 ) may seem surprising, since ξ 24 has filtration 24, but there is a class ξ 28 suc h that 4 ξ 24 − ξ 2 12 = ξ 28 , and w e can hav e ψ k ( ξ 24 ) ≡ k 12 ξ 24 + α k ξ 28 consisten tly with (2 .4). Using (2.3) and that ψ 2 ≡ − 2 λ 2 mo d decomp osables, w e obta in ψ 2 ( ξ 8 ) ≡ 2 4 ξ 8 − 2 ξ 12 (2.5) ψ 2 ( ξ 12 ) ≡ 2 6 ξ 12 − 2 ξ 24 , yielding the matrix Ψ 2 in the theorem. Applying ψ 2 ψ 3 = ψ 3 ψ 2 to ψ 3 ( ξ 12 ) ≡ 3 6 ξ 12 + γ ξ 24 yields − 2 · 3 6 + 2 14 γ = 2 6 γ − 2 · 3 14 , from whic h w e obtain γ = − 3 5 · 41 / 17. Applying the same relation to ψ 3 ( ξ 8 ) = 3 4 ξ 8 + αξ 12 + β ξ 24 , co efficien ts of ξ 12 yield − 2 · 3 4 + α · 2 6 = 2 4 α − 2 · 3 6 and hence α = − 3 3 . No w co efficien ts o f ξ 24 yield − 2 α + 2 14 β = 2 4 β − 2 γ and hence β = 36 / 5 2 7. Let Φ 1 ( − ) denote the functor from spaces to K/ 2 ∗ -lo cal sp ectra describ ed in [5, 9.1], whic h satisfies v − 1 1 π ∗ X ≈ π ∗ τ 2 Φ 1 X , where τ 2 Φ 1 X is the 2-torsion part o f Φ 1 X . In Section 4, w e will use results of Bousfield in [5 ] to pro v e the following result. As p ects v 1 -PERIODIC HOMOTOPY GROUPS OF DI (4) 5 of Theorem 2.1, suc h as K ∗ ( B D I (4)) b eing a p o we r series algebra on real generators, are also used in prov ing this theorem. Recall that K O ∗ ( − ) has p erio d 8. Theorem 2.6. Th e gr oups K O i (Φ 1 D I (4)) ar e 0 if i ≡ 0 , 1 , 2 mo d 8 , and K 0 (Φ 1 D I (4)) = 0 . L et M denote a fr e e ˆ Z 2 -mo dule on thr e e gener ators, acte d on b y ψ 2 and ψ 3 by the matric es of The or em 2.1 , with ψ − 1 = 1 . L et θ = 1 2 ψ 2 act on M . T hen ther e a r e exa c t se quenc es 0 → 2 M θ − → 2 M → K O 3 (Φ 1 D I (4)) → 0 → 0 → K O 4 (Φ 1 D I (4)) → M / 2 θ − → M / 2 → K O 5 (Φ 1 D I (4)) → M / 2 θ − → M / 2 → K O 6 (Φ 1 D I (4)) → M θ − → M → K O 7 (Φ 1 D I (4)) → 0 and 0 → M θ − → M → K 1 (Φ D I (4)) → 0 . F or k = − 1 and 3 , the action of ψ k in K O 2 j − 1 (Φ 1 D I (4)) , K O 2 j − 2 (Φ 1 D I (4)) , and K 2 j − 1 (Φ D I (4)) agr e es w ith k − j ψ k in a d jac ent M -terms. In the remainder of this section, we use 2.1 and 2 .6 to g iv e explicit formulas for the Adams mo dule K O i (Φ 1 D I (4)). A similar argumen t w orks for K ∗ (Φ 1 D I (4)). If g 1 , g 2 , and g 3 denote the three g enerato r s of M , then t he action of θ is given b y θ ( g 1 ) = 8 g 1 − g 2 θ ( g 2 ) = 2 5 g 2 − g 3 θ ( g 3 ) = 2 13 g 3 . Clearly θ is injectiv e on M and 2 M . W e hav e K O 7 (Φ 1 D I (4)) ≈ coker( θ | M ) ≈ Z / 2 21 with g enerator g 1 ; note that g 2 = 2 3 g 1 in this coke rnel, and then g 3 = 2 8 g 1 . Similarly K O 3 (Φ 1 D I (4)) ≈ coke r ( θ | 2 M ) ≈ Z / 2 21 . Also K O 4 (Φ 1 D I (4)) ≈ k er ( θ | M / 2) = Z / 2 with g enerator g 3 , while K O 6 (Φ 1 D I (4)) ≈ coke r ( θ | M / 2) = Z / 2 with generator g 1 . There is a short exact sequence 0 → cok er( θ | M / 2) → K O 5 (Φ 1 D I (4)) → ke r ( θ | M / 2) → 0 , with the groups at either end b eing Z / 2 as b efo r e. T o see that this short exact sequence is split, w e use the map S 7 f − → D I (4) whic h is inclus ion o f the b ottom cell. The morphism f ∗ sends the first summand o f K O 5 (Φ 1 D I (4)) to one of the t w o 6 MAR TIN BENDER SKY A ND DONALD M. DA VIS Z / 2-summands of K O 5 (Φ 1 S 7 ), pro viding a splitting homomorphism. Th us w e hav e pro v ed the first part of the fo llo wing result. Theorem 2.7. We hav e K i (Φ 1 D I (4)) ≈    0 i = 0 Z / 2 21 i = 1 , K O i (Φ 1 D I (4)) ≈              0 i = 0 , 1 , 2 Z / 2 21 i = 3 , 7 Z / 2 i = 4 , 6 Z / 2 ⊕ Z / 2 i = 5 . F or k = − 1 and 3 , we have ψ k = 1 on the Z / 2 ’s, and on K O 2 j − 1 (Φ 1 D I (4)) with j even and K 2 j − 1 (Φ 1 D I (4)) , ψ − 1 = ( − 1) j and ψ 3 = 3 − j (3 4 − 3 3 · 2 3 + 36 527 2 8 ) . Completion of pr o of. T o obtain ψ 3 on the Z / 2 ’s, w e use the last part of Theorem 2.6 and t he matrix Ψ 3 of Theorem 2 .1. If ψ 3 is as in Ψ 3 , then, mo d 2, ψ 3 − 1 sends g 1 7→ g 2 , g 2 7→ g 3 , and g 3 7→ 0 . T hus ψ 3 − 1 equals 0 on K O 4 (Φ 1 D I (4)) and K O 6 (Φ 1 D I (4)). Clearly ψ − 1 = 1 on these groups. T o see that ψ k − 1 is 0 on K O 5 (Φ 1 D I (4)), w e use the comm utative diag ram 0 − − − → Z / 2 i − − − → K O 5 (Φ 1 D I (4)) ρ − − − → Z / 2 − − − → 0 ≈    y f ∗    y 0    y 0 − − − → Z / 2 − − − → K O 5 (Φ 1 S 7 ) − − − → Z / 2 − − − → 0 . W e can c ho ose generators G 1 and G 2 of K O 5 (Φ 1 D I (4)) ≈ Z / 2 ⊕ Z / 2 so that G 1 ∈ im( i ), ρ ( G 2 ) 6 = 0, and f ∗ ( G 2 ) = 0. Since ψ k − 1 = 0 on t he Z / 2 ’s on either side of K O 5 (Φ 1 D I (4)), the o nly w a y tha t ψ k − 1 could b e nonzero o n K O 5 (Φ 1 D I (4)) is if ( ψ k − 1)( G 2 ) = G 1 . Ho w ev er this yields the contradiction 0 = ( ψ k − 1) f ∗ G 2 = f ∗ ( ψ k − 1) G 2 = f ∗ ( G 1 ) 6 = 0 . On K O 2 j − 1 (Φ 1 D I (4)) with j ev en and K 2 j − 1 (Φ 1 D I (4)), ψ 3 sends t he generator g 1 to 3 − j (3 4 g 1 − 3 3 g 2 + 36 527 g 3 ) = 3 − j (3 4 − 3 3 · 2 3 + 36 527 2 8 ) g 1 , and ψ − 1 ( g 1 ) = ( − 1) j g 1 b y Theorem 2.6. v 1 -PERIODIC HOMOTOPY GROUPS OF DI (4) 7 3. Rela tionship with pseudosphere In this section, w e prov e Theorems 1.2 and 1.1. F ollow ing [8 , 8.6], w e let T = S 0 ∪ η e 2 ∪ 2 e 3 , and consider its K / 2-lo calization T K/ 2 . The groups π ∗ ( T K/ 2 ) are giv en in [8 , 8.8 ], while the Adams mo dule is giv en b y K i ( T K/ 2 ) =    ˆ Z 2 i ev en, with ψ k = k − i/ 2 0 i o dd; K O i ( T K/ 2 ) =        ˆ Z 2 i ≡ 0 mo d 4 , with ψ k = k − i/ 2 Z / 2 i = 2 , 3 , with ψ k = 1 0 i = 1 , 5 , 6 , 7 . Bousfield calls this the 2- completed K -theoretic pseudosphere. Closely related sp ectra ha v e b een also considered in [15] and [4]. Let M ( n ) = S − 1 ∪ n e 0 denote the mo d n Moo r e spectrum. Then, for e > 1 a nd k o dd, K i ( T K/ 2 ∧ M (2 e )) =    Z / 2 e i ev en, with ψ k = k − i/ 2 0 i o dd; K O i ( T K/ 2 ∧ M (2 e )) =              Z / 2 e i ≡ 0 mo d 4 , with ψ k = k − i/ 2 Z / 2 i = 1 , 3 , with ψ k = 1 Z / 2 ⊕ Z / 2 i = 2 , with ψ k = 1 0 i = 5 , 6 , 7 . (3.1) Pr o of . Let Y = T K/ 2 ∧ M (2 e ). Most of (3 .1) is immediate from the exact sequence 2 e − → K O i ( T K/ 2 ) − → K O i ( Y ) − → K O i +1 ( T K/ 2 ) 2 e − → . T o see that K O 2 ( Y ) = Z / 2 ⊕ Z / 2 and not Z / 4, one can first note that M ( 2 e ) ∧ M (2) ≃ Σ − 1 M ( 2) ∨ M (2) . (3.2) The exact sequence K O 2 ( Y ) 2 − → K O 2 ( Y ) − → K O 2 ( Y ∧ M (2)) − → K O 3 ( Y ) 2 − → (3.3) implies that if K O 2 ( Y ) = Z / 4, then | K O 2 ( Y ∧ M (2)) | = 4. Ho w ev er, b y (3.2), K O 2 ( Y ∧ M (2)) ≈ K O 2 ( T K/ 2 ∧ M (2)) ⊕ K O 3 ( T K/ 2 ∧ M (2)) . (3.4) 8 MAR TIN BENDER SKY A ND DONALD M. DA VIS Also, there is a cofib er sequence T ∧ M (2) → Σ − 1 A 1 → Σ 5 M ( 2) , (3.5) where H ∗ ( A 1 ; F 2 ) is is omorphic to the subalgebra of the mo d 2 Steenro d a lgebra generated by Sq 1 and Sq 2 , and satisfies K O ∗ ( A 1 ) = 0. Th us K O i ( T K/ 2 ∧ M (2)) ≈ K O i (Σ 4 M ( 2)) ≈    Z / 4 i = 2 Z / 2 i = 3 , so that | K O 2 ( Y ∧ M (2)) | = 8, con tradicting a consequenc e of the hypothesis that K O 2 ( Y ) = Z / 4. W e conclude the pro of by sho wing that, fo r o dd k , ψ k = 1 on K O 2 ( Y ). F irst note that ψ k = 1 on K O ∗ ( M (2)). This follows immediately from the Adams op erations on the sphere, except fo r ψ k on K O − 2 ( M (2)) ≈ Z / 4. This is isomorphic to g K O ( RP 2 ), where ψ k = 1 is well-kno wn. No w use (3.13) to deduce that ψ k = 1 on K O ∗ ( T K/ 2 ∧ M ( 2)), and then (3.4) to see that ψ k = 1 on K O 2 ( Y ∧ M (2)). F inally , use (3.3) to deduce that ψ k = 1 on K O 2 ( Y ). Comparison of 2.7 and (3.1) yields an isomorphism of graded ab elian groups K O ∗ (Σ 8 L +3 T K/ 2 ∧ M (2 21 )) ≈ K O ∗ (Φ 1 D I (4)) (3.6) for an y in teger L . W e will show that if L = 9062 7, then the Adams op erations agree to o. By [9 , 6.4 ], it suffices to prov e they agree f or ψ 3 and ψ − 1 . Note that one w ay of distinguishing a K -theoretic pseu dosphere from a sphere is that in K O ∗ (sphere) (resp. K O ∗ (pseudosphere)) the Z / 2-groups are in dimensions 1 and 2 less t han the dimensions in whic h ψ 3 ≡ 1 mo d 16 (resp. ψ 3 ≡ 9 mo d 16), a nd similarly after smashing with a mo d 2 e Mo ore sp ectrum. Since 3 4 − 6 3 + 36 527 2 8 ≡ 9 mo d 16, the Z / 2- groups in K O ∗ (Φ 1 D I (4)) o ccur in dimensions 1, 2, and 3 less that the dimension in whic h ψ 3 ≡ 9 mo d 16, and so Φ 1 D I (4) should b e identified with a suspension of T K/ 2 ∧ M (2 21 ) a nd not S K/ 2 ∧ M (2 21 ). In K O 4 t − 1 (Σ 8 L +3 T K/ 2 ∧ M (2 21 )), ψ 3 = 3 − 2( t − 2 L − 1) and ψ − 1 = 1. Thus if L satisfies 3 4 L +2 ≡ 3 4 − 6 3 + 36 527 2 8 mo d 2 21 , (3.7) then K O ∗ (Σ 8 L +3 T K/ 2 ∧ M (2 21 )) and K O ∗ (Φ 1 D I (4)) will b e isomorphic Adams mo d- ules. Maple easily verifie s that (3.7) is satisfied for L = 906 2 7. v 1 -PERIODIC HOMOTOPY GROUPS OF DI (4) 9 A wa y in whic h this num b er L can b e fo und b egins with the mo d 2 18 equation 6 X i =1  2 L − 1 i  8 i − 1 ≡ 3 4 L − 2 − 1 8 ≡ 1 9 ( 2 7 527 − 3) ≡ 192725 , where we use Maple a t the last step. This easily implies L ≡ 3 mo d 8, and so we let L = 8 b + 3. Again using Maple and w orking mo d 2 18 w e compute 6 X i =1  16 b +5 i  8 i − 1 − 192 725 ≡ 2 10 u 0 + 2 4 u 1 b + 2 10 u 2 b 2 + 2 17 b 3 , with u i o dd. Th us w e mus t ha v e b ≡ 64 mo d 12 8, and so L ≡ 515 mo d 2 10 . Sev eral more steps of this type lead t o the desired v alue of L . Th us, in the terminology of 4.3, w e hav e prov ed the following result. Prop osition 3.8. If L = 90627 , then ther e is an isomorphism of A d a ms mo dules K ∗ C R (Σ 8 L +3 T K/ 2 ∧ M (2 21 )) ≈ K ∗ C R (Φ 1 D I (4)) . Theorem 1 .2 follows immediately from this using the remark able [8, 5.3], whic h say s, among other things, t ha t 2- lo cal sp ectra X ha ving some K i ( X ) = 0 are determined up to equiv alence b y their Adams mo dule K ∗ C R ( X ). Theorem 1.1 follows immediately from Theorem 1.2 and the following result. Prop osition 3.9. F or al l inte gers i , π 8 i + d ( T K/ 2 ∧ M (2 21 )) ≈                        Z / 2 ⊕ Z / 2 min(21 ,ν ( i )+4) d = − 2 Z / 2 min(21 ,ν ( i )+4) d = − 1 0 d = 0 , 1 Z / 8 d = 2 Z / 2 ⊕ Z / 8 d = 3 Z / 2 ⊕ Z / 2 ⊕ Z / 2 d = 4 , 5 . Pr o of . F or the most part, these gro ups are immediate from the groups π ∗ ( T K/ 2 ) g iv en in [8, 8.8] a nd the exact sequence 2 21 − → π j +1 ( T K/ 2 ) → π j ( T K/ 2 ∧ M (2 21 )) → π j ( T K/ 2 ) 2 21 − → . (3.10) 10 MAR TIN BENDER SKY A ND DONALD M. DA VIS All that needs to b e done is to sho w that the follo wing short exact sequences, obtained from (3.10 ) , are split. 0 → Z / 2 → π 8 i +3 ( T K/ 2 ∧ M (2 21 )) → Z / 8 → 0 (3.11) 0 → Z / 2 ⊕ Z / 2 → π 8 i +4 ( T K/ 2 ∧ M (2 21 )) → Z / 2 → 0 0 → Z / 2 → π 8 i +5 ( T K/ 2 ∧ M (2 21 )) → Z / 2 ⊕ Z / 2 → 0 0 → Z / 2 min(21 ,ν ( i )+4) → π 8 i − 2 ( T K/ 2 ∧ M (2 21 )) → Z / 2 → 0 . Let Y = T K/ 2 ∧ M (2 21 ). W e consider the exact sequence f or π ∗ ( Y ∧ M (2)), 2 − → π i +1 ( Y ) → π i ( Y ∧ M (2)) → π i ( Y ) 2 − → . (3.12) If the four sequences (3.11) are all split, then by (3.12) the groups π 8 i + d ( Y ∧ M (2)) for d = 2 , 3 , 4 , 5 , − 2 ha ve orders 2 3 , 2 5 , 2 6 , 2 5 , and 2 3 , resp ectiv ely , but if a ny of the sequence s (3.11) fails to split, then some of the orders | π 8 i + d ( Y ∧ M (2)) | will hav e v alues smaller than those listed here. By (3.2) , π i ( Y ∧ M (2)) ≈ π i +1 ( T K/ 2 ∧ M (2)) ⊕ π i ( T K/ 2 ∧ M (2)) . By (3.5), since lo calization pr eserv es cofibra tions and ( A 1 ) K/ 2 = ∗ , t here is an equiv- alence Σ 4 M K/ 2 ≃ T K/ 2 ∧ M (2) , (3.13) and hence π i ( Y ∧ M (2)) ≈ π i − 3 ( M K/ 2 ) ⊕ π i − 4 ( M K/ 2 ) . (3.14) By [10, 4.2], π 8 i + d ( M K/ 2 ) =              0 d = 4 , 5 Z / 2 d = − 2 , 3 Z / 2 ⊕ Z / 2 d = − 1 , 2 Z / 4 ⊕ Z / 2 d = 0 , 1 . This is the sum of tw o “lightning flashes,” one b eginning in 8 d − 2 and the other in 8 d − 1. Substituting this information into (3.14) yields exactly the or ders whic h w ere sho wn in the previous parag raph t o b e true if and only if all the exact sequenc es (3.11) split. v 1 -PERIODIC HOMOTOPY GROUPS OF DI (4) 11 4. Determina tion of K O ∗ (Φ 1 D I (4)) In this section, w e pro v e Theorem 2.6, whic h show s ho w ψ k in K ∗ ( B D I (4)) leads to the determination of K O ∗ (Φ 1 D I (4)). Our presen tatio n here follo ws suggestions in sev eral e-mails from Pete Bousfield. The first result explains ho w K O ∗ ( B D I (4)) follows from K ∗ ( B D I (4)). Theorem 4.1. Ther e ar e cl a sses g 8 , g 12 , an d g 24 in K O 0 ( B D I (4)) such that c ( g i ) = ξ i , with ξ i as in 2.1, and K O ∗ ( B D I (4)) ≈ K O ∗ [ [ g 8 , g 12 , g 24 ] ] . The A dams op er ations ψ 2 and ψ 3 mo d de c omp osables on the b asis of g i ’s is a s in 2. 1 . Pr o of . In [2, 2.1], it is prov ed that if there is a torsion-free subgroup F ∗ ⊂ K O ∗ ( X ) suc h that F ∗ ⊗ K ∗ ( pt ) → K ∗ ( X ) is an isomorphism, then so is F ∗ ⊗ K O ∗ ( pt ) → K O ∗ ( X ). The pro of is a Fiv e Lemma argumen t using exact sequences in [17, p.257 ]. Although the result is stated for ordinary (not 2-completed) K O ∗ ( − ), the same ar- gumen t a pplies in the 2-completed con text. If F ∗ is a m ultiplicativ e subgroup, then the result holds as rings. Our result then follo ws from 2.1, since the generators there are real. A similar pro of can b e deriv ed from [5, 2 .3]. Next we need a similar sort of resu lt ab out K O ∗ ( D I (4)). W e could deriv e m uc h of what w e need b y an argumen t similar to t ha t just used, using the result of [16] ab out K ∗ ( D I (4)) as input. Ho w ev er, as w e will need this in a sp ecific form in order to use it to draw conclusions ab o ut K O ∗ (Φ 1 D I (4)), w e b egin b y in tro ducing m uch terminology fr om [5]. The study of united K -theory b egins with tw o categories, whic h will then be en- do w ed with additiona l structure. W e b egin with a partial definition of eac h, and their relationship. F or complete details, the reader will need to refer to [5] or an earlier pap er of Bousfield. Definition 4.2. ([5 , 2 . 1]) A C R -mo dule M = { M C , M R } c onsists of Z -gr ade d 2 - pr ofinite ab el i a n gr oups M C and M R with c ontinuous addi tive op er ations M ∗ C B − → ≈ M ∗− 2 C , M ∗ R B R − → ≈ M ∗− 8 R , M ∗ C t − → ≈ M ∗ C , M ∗ R η − → M ∗− 1 R , M ∗ R c − → M ∗ C , M ∗ C r − → M ∗ R , satis- fying 15 r elations, wh ich we w il l men tion as ne e de d. 12 MAR TIN BENDER SKY A ND DONALD M. DA VIS W e omit the descriptor “2- a dic,” whic h Bousfield prop erly uses, just as w e omit writing the 2-a dic co efficien ts ˆ Z 2 whic h are presen t in all o ur K - and K O -g roups. Example 4.3. F or a sp e ctrum or sp ac e X , the unite d 2 -adi c K -c ohomo lo gy K ∗ C R ( X ) := { K ∗ ( X ) , K O ∗ ( X ) } is a C R -mo d ule, with c omplex and r e a l Bott p erio dicity, c onjugation, the Hopf map, c omplex i fi c ation, and r e alific ation giving the r esp e ctive op er ation s . Definition 4.4. ( [5 , 4 . 1]) A ∆ -mo dule N = { N C , N R , N H } is a triple of 2 -p r ofinite ab elian gr oups N C , N R , and N H with c ontinuous additive op e r ations N C t − → ≈ N C , N R c − → N C , N C r − → N R , N H c ′ − → N C , and N C q − → N H satisfying nine r elations. Example 4.5. F or a C R - m o dule M and an inte ger n , ther e i s a ∆ -mo dule ∆ n M = { M n C , M n R , M n − 4 R } with c ′ = B − 2 c and q = r B 2 . In p a rticular, for a sp ac e X and inte ger n , ther e is a ∆ -m o dule K n ∆ ( X ) := ∆ n K ∗ C R ( X ) . No w w e add additional structure to these definitions. Definition 4.6. ([5 , 4 . 3 , 6 . 1]) A θ ∆ -mo d ule is a ∆ -mo d ule N to gether with homo- morphisms N C θ − → N C , N R θ − → N R , and N H θ − → N R satisfying c ertain r elations liste d in [5, 4.3] . An A dams ∆ -m o dule is a θ ∆ -mo dule N to gether with A dams op er ations N ψ k − → ≈ N for o dd k satisfying the familiar pr op erties. Example 4.7. In the notation of Exam ple 4.5, K − 1 ∆ ( X ) is an A d ams ∆ -mo dule with θ = − λ 2 . Definition 4.8. ([5 , 2 . 6 , 3 . 1 , 3 . 2]) A sp e cial φC R -a lgebr a { A C , A R } is a C R -mo dule with bilin e ar A m C × A n C → A m + n C and A m R × A n R → A m + n R and al s o A 0 C φ − → A 0 R and A − 1 C φ − → A 0 R satisfying numer ous pr op erties. Remark 4.9. The o p erations φ , whic h w ere initially defined in [7], are less familiar than the others. Tw o properties are cφa = t ( a ) a and φ ( a + b ) = φa + φ b + r ( t ( a ) b ) for a, b ∈ A 0 C . F or a connected space X , K ∗ C R ( X ) is a sp ecial φ C R -algebra. The f o llo wing imp ortant lemma is tak en from [5]. v 1 -PERIODIC HOMOTOPY GROUPS OF DI (4) 13 Lemma 4.10. ([5 , 4 . 5 , 4 . 6 ]) F or any θ ∆ -m o dule M , ther e is a universal s p e ci a l φC R - algebr a ˆ LM . This me a ns that ther e is a morphism M α − → ˆ LM such that any morphism fr om M into a φ C R -algebr a factors as α fol lowe d by a unique φC R -algebr a mo rphism. Ther e is an algebr a is o morphism ˆ Λ M C → ( ˆ LM ) C , wh e r e ˆ Λ( − ) is the 2-adic exterior algebr a functor. In [5, 2.7], Bousfield defines, for a C R - a lgebra A , the indecomposable quotient ˆ QA . W e apply this to A = K ∗ C R ( B D I (4)), a nd consider the ∆- mo dule ˆ QK 0 ∆ ( B D I (4)), analogous to [5, 4.10 ]. W e need the fo llo wing result, whic h is more delicate than the K − 1 ∆ -case considered in [5, 4.10]. Lemma 4.11. With θ = − λ 2 , the ∆ -mo dule ˆ QK 0 ∆ ( B D I (4)) b e c omes a θ ∆ -mo dule. Pr o of . First w e need that θ is an additive op eration. In [7, 3.6], it is sho wn that θ ( x + y ) = θ ( x ) + θ ( y ) − xy if x, y ∈ K O n ( X ) with n ≡ 0 mod 4. The additivity follo ws since w e are mo dding out the pro duct terms. (In the case n ≡ − 1 mo d 4 considered in [5, 4.10], the additivit y of θ is already presen t b efore mo dding out indecomp o sables.) There are fiv e additional prop erties whic h m ust b e satisfied b y θ . That θ cx = cθ x and θ tz = tθ z are easily obtained from [7, 3.4]. That θ c ′ y = cθ y follow s from [8, 6.2(iii),6.4]. That θ q z = θ r z follows from preceding [7, 3.10] b y c , whic h is surjectiv e for us. Here w e use that r c = 2 and q = r B 2 . Finally , θ r z = r θ z for us, since c is surjectiv e; here w e ha v e used the result ¯ φcx = 0 given in [5, 4.3]. No w w e obtain the follo wing imp ort a n t description of the C R - algebra K ∗ C R ( D I (4)). Theorem 4.12. Ther e is a morphism of θ ∆ -mo dules ˆ QK 0 ∆ ( B D I (4)) → ˜ K − 1 ∆ ( D I (4)) which induc es an isomorphism of sp e cial φC R -algebr as ˆ L ( ˆ QK 0 ∆ ( B D I (4))) → K ∗ C R ( D I (4)) . Pr o of . The map Σ D I (4) = ΣΩ B D I (4) → B D I (4) induces a morphism K 0 ∆ ( B D I (4)) → K − 1 ∆ ( D I (4)) 14 MAR TIN BENDER SKY A ND DONALD M. DA VIS whic h factors through the indecomp osable quotien t ˆ QK 0 ∆ ( B D I (4)). In [16, 1.2 ], a general result is prov ed whic h implies that K ∗ ( D I (4)) is an exterior algebra on ele- men ts of K 1 ( D I (4)) whic h correspond to the generators of the p ow er series algebra K ∗ ( B D I (4)) under the ab o v e morphism follow ed by the Bott map. Th us our result will follow from [5 , 4.9 ], once w e hav e sho wn that the θ ∆-mo dule M := ˆ QK 0 ∆ ( B D I (4)) is robust([5 , 4.7 ]). This requires tha t M is profinite, which fo llows as in the remark follo wing [5, 4.7 ], together with t w o prop erties regarding ¯ φ , where ¯ φz := θ r z − r θ z for z ∈ M C . In our case, c is surjectiv e, and so ¯ φ = 0 a s used in the previous pro of. One prop erty is that M is torsion-free and exact. This follows from the Bott exactness of the C R -mo dule K ∗ C R ( B D I (4)) no ted in [5, 2.2], and [5, 5.4 ], which states that, for an y n , the ∆-mo dule ∆ n N asso ciat ed to a Bott exact C R -mo dule N with N n C torsion-free a nd N n − 1 C = 0 is torsion-free. The other prop ert y is k er ( ¯ φ ) = cM R + c ′ M H + 2 M C . F or us, b oth sides equal M C since c is surjectiv e and ¯ φ = 0. Our Theorem 2.6 now fo llo ws from [5, 9.5] once w e hav e sho wn that the Adams ∆-mo dule M := ˆ QK 0 ∆ ( B D I (4)) is “strong.” ([5 , 7.11 ]) This result ([5, 9.5]) requires that the space (here D I (4)) b e an H -space (a ctually K / 2 ∗ -durable, whic h is satisfied b y H -spaces) and t ha t it satisfie s the conclusion of our 4.12. It then deduces that K O ∗ (Φ 1 D I (4)) fits in to a n exact sequence whic h reduces to ours pro vided M R = M C and M H = 2 M C . These equalities are implied b y ˆ QK 0 ∆ ( B D I (4)) b eing exact, as w as noted to b e true in the previous pro of, plus t = 1 and c surjectiv e, as w ere noted to b e true in 2.1. Indeed, the exactne ss prop ert y , ([5, 4.2 ]), includes that cM R + c ′ M H = k er ( 1 − t ) and cM R ∩ c ′ M H = im(1 + t ). Another p erceptible difference is that Bousfield’s exact sequence is in t erms of ¯ M := M / ¯ φ , while ours inv olv es M , but these are equal since, as a lready observ ed, ¯ φ = 0 since c is surjectiv e. Note also that the Adams op erations in ¯ M in t he exact sequence of [5, 9.5], which reduces to that in our 2.6, are those in the Adams ∆-mo dule ˆ QK 0 ∆ ( B D I (4)), whic h are giv en in our 2.1. The morphism θ in [5 , 9 .5 ] or our 2.6 is 1 2 ψ 2 , since this equals − λ 2 mo d decomp osables. Finally , w e sho w t ha t our M is stro ng . One of the three criteria for b eing strong is to b e robust, and w e ha v e already discuss ed and verified this. The second requiremen t for a n Adams ∆-mo dule to b e strong is that it b e “regular.” This ra ther tec hnical v 1 -PERIODIC HOMOTOPY GROUPS OF DI (4) 15 condition is defined in [5 , 7.8]. In [5, 7.9], a result is prov ed whic h immediately implies that ˜ K − 1 ∆ ( D I (4)) is regular. By 4.12, our M injects in to ˜ K − 1 ∆ ( D I (4)), and so by [5, 7.10], whic h states tha t a submo dule of a regula r mo dule is regular, our M is regular. The third requiremen t for M to b e strong is that it b e ψ 3 -splittable ([5, 7.2]), whic h means t ha t the quotien t map M → M / ¯ φ has a righ t in v erse. As w e hav e noted sev eral times, w e hav e ¯ φ = 0, and so the iden tity map serv es as a righ t in ve rse to the iden tit y map. This completes the pro of that our M is strong, and hence that [5, 9.5] applies to D I (4) to yield our Theorem 2.6. Reference s [1] K. K. S. Andersen and J. Gro dal, The classific ation of 2-c omp act gr oups , preprint. [2] D. W. Anderson, Universal c o efficient the or ems for K -the ory , pr eprint (1971). [3] M. Bender sky and D. M. Davis, v 1 -p erio dic homotopy gr ou ps of S O ( n ), Memoir Amer Math So c 815 (2 004). [4] M. Benders ky , D. M. Davis, and M. Mahow ald, Stable ge ometric dimension of ve ctor bund les over even-dimensional r e al pr oje ctive sp ac es , T rans Amer Math So c 358 (20 0 6) 1585 -1603 . [5] A. K. 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Wilkerson, A new finite lo op sp ac e at t he prime 2 , Jour Amer Math So c 6 (19 9 3) 37-6 4. [14] , Homotopy fixe d-p oint metho ds for Lie gr oups and fin ite lo op sp ac es , Annals of Math 139 (1994) 395 -442. [15] M. J. Hopkins, M. Mahow ald, a nd H. Sadofsk y , Construction of elements in Pic ar d gr oups , Contemp Math Amer Math So c 1 58 (19 9 4) 89-1 26. [16] A. Jeanneret and A. Osse, The K -t he ory of p - c omp act gr oups , Comm Math Helv 7 2 (19 97) 556 - 581. 16 MAR TIN BENDER SKY A ND DONALD M. DA VIS [17] M. Ka roubi, Al gebr` es de Cliffor d et K-t h ´ eorie , Ann Sci de l’Ec ole No r m Sup (1968) 162-2 70. [18] A. Osse and U. Suter, Invari ant the ory and the K -the ory of the Dwyer- Wilkerson sp ac e , Co nt emp Math Amer Math So c, 265 (20 00) 17 5-185 . Hunter College, CUNY, NY , N Y 01220 E-mail addr ess : mbende rs@hun ter.cuny.edu Lehigh University, Bethlehem, P A 18015 E-mail addr ess : dmd1@l ehigh. edu