Primary decomposition of the J -groups of complex projective and lens spaces

Transcription

1 Topology and s Applcaons wwwelsevercom/locae/opol Prmary decomposon of he J -groups of complex projecve and lens spaces I Dbag Deparmen of Mahemacs, Blken Unversy, Blken, Ankara, Turkey eceved 3 Ocober 2005 Absrac We deermne he decomposon of J -groups of complex projecve and lens spaces as a drec-sum of cyclc groups 2005 Elsever BV All rghs reserved MSC: prmary 5525; secondary 55Q50, 55S25 Keywords: Sphere bundles; Vecor bundles; J -morphsm; K-heory operaons 1 Inroducon Ths paper s a connuaon of 1 whose resuls we brefly summarze For a fne-dmensonal CW-complex X, le JXdenoe he fne Abelan group of sable fbre homoopy classes of vecor-bundles over X and for a prme p, J p X he p-summand of JXForn, k Z +,lep n C S 2n+1 /U1 and L n p k S 2n+1 /Z p k denoe he complex projecve space of complex-dmenson n and he assocaed lens space respecvely In 1 J p P n C and JL n p k are deermned by means of a se of generaors and a complee se of relaons Le r n be he greaes neger such ha p r n n/p 1 Then for 0 s r n and 0 j r n s we defned a decreasng sequence by j s nps p j 1/p s+j p 1 where for a real number x, x denoes he greaes neger less han or equal o x Pu j j 0Weleω denoe he realfcaon of he reducon of he Hopf bundle over P nc Le ψ k denoe he Adams operaon acng on K P n C and also on JP n C and ρ k he assocaed characersc class akng values n 1 + K P n C Q k where Q k s he sub-rng of raonals whose denomnaors are powers of k m Z s defned o be a sngular s-exponen f and only f he coeffcen of ω m n he power seres ρ p ψpk ωpk s no negral e fraconal for some k Z + Thej-ndex, m j of a sngular s-exponen m s he exponen of p n he denomnaor of he coeffcen of ω m n he expanson of ρ p ψpj ωpj α α 0,α 1,,α rn s an s-admssble sequence f and only f Congruence 1: j Z s sasfed by all sngular s-exponens m p j m We le Φs 0 {s k s k 0 mod p} and Ms se of all sngular s-exponens Then 1, Proposon 444 saes ha here s a bjecon σ s : Φs 0 Ms M M 0 gven by σ s k s 1/2p 1pk k sifs k Φ0 s, s k pν Δ ν 1, E-mal address: dbag@fenblkenedur I Dbag /$ see fron maer 2005 Elsever BV All rghs reserved do:101016/jopol

2 I Dbag / Topology and s Applcaons Δ, p 1 and m σ s k s hen j m p k+νj Δ + k + ν j j s 1, Proposon 627 reduces he queson of relaons n J p P n C o s-admssbly; n parcular, proves ha a relaon: j 0 p s j Ψ ps+j ω 0 0 s r n exss n J p P n C f and only f α { } s an s-admssble sequence In 1, Secon 52 wo dfferen sequences called α- and β- sequences are consruced for each 0 s r n where αj s 1or0andheyareprovedobesadmssble In 1, Proposon 628 and 629 we oban he correspondng se of r n + 1-relaons n J p P n C whch are proved o be complee Hence n 1 J p P n C s deermned by generaors and a complee se of relaons Analogous relaons are hen obaned for he J -groups of lens spaces However, he deermnaon of he srucure of a fne Abelan group s far from beng over unless s prmary decomposon no cyclc groups s uncovered and s he purpose of he presen paper o deermne he prmary decomposons of J p P n C and JL n p k Usng he framework of 1 he prmary decomposon of J p P n C s reduced o he soluon of he followng problem n elemenary number heory For a prme p and a raonal q, le v p q denoe he exponen of p n he prme facorzaon of q Problem Le k,d Z +, {k } and τ are srcly-decreasng sequences such ha 0 k τ k Gven negers { j } 1 d,j τ such ha k, k k +1 and { j } s a srcly- For fxed, j s a srcly-ncreasng sequence n j for j k 1, k 1 decreasng sequence n j for k + 1 j τ For fxed, j 1 for a leas one 0 j k j j >j j for < and j<j Fnd he leas v p α k for a soluon α α 0,,α k α Z of Congruence 1; e α k p k + α k1 + + α 0 p k1 p 0 Z 1 d 1 The man effor of hs paper s concenraed n gvng a soluon o hs problem Le H k {Ω 2 1,2,,k eher Ω ϕ, or,ω { 1,, r },1 1 < 2 < < r d, k+1 k > 0 and k > 0, 1 r} For each Ω H k we defne he assocaed se Φ {{j 1,,j s } 1 j 1 < <j s k, k+j k+j1 >0, 1 j s} Then we observe ha Ω Φ {1, 2,,l}, l d For each elemen Ω 1,, r H k, we pu v p α k+1 v p α k + k+1 k 1 r and α kr 1, and oban a vecor α r1 k+1 α 0,,α k wh α 0 α kr1 0 and α k p The Congruence Theorem eg Theorem 214 whch s orgnal and proved n hs paper for he frs me shows ha he vecor α so defned s a soluon of he sysem of Congruences 1 wh respec o Ω l + 1,l + 2,,k We hen requre all erms of Congruence 1 wh respec o j 1,j 2,,j s be negral Ths necessaes ha v p α kjh +h k+h 1 h s; or, equvalenly, ha v p α k h 1 k+1 k uω max r 1 + k+h Hence f we defne k+1 k ; max 1 h s k +h j >0 h jh h 1 k k+1 k kj + h +h 2 We oban a soluon α α 0,,α k wh v p α k uω We hen defne a unque elemen Ω 0 H k and defne eg Defnon 25 u k uω 0 Hence a soluon o he sysem of Congruences 1 exss wh v p α k u k Acually, α k p u k We hen show eg Proposon 26 ha u k s mnmal among uω as Ω vares over H k The observaon ha when all he erms of Congruence 1 are no negral here are a leas wo erms wh hghes denomnaor wh posve p-exponen leads o Lemmas 28, 29, Corollary 210 and Lemma 211 and Lemma 211 ogeher wh Proposon 26 yeld Proposon 212 whch saes ha for any soluon α α 0,,α k of he sysem of Congruence 1, v p α k u k Hence he problem saed s solved n full Ths complees he elemenary number heory The proof of Theorem 221 whch s he man resul of he paper s sraghforward algebra I combnes 1, Proposon 6214 wh he soluon of he above problem o deduce he prmary decomposon of J p P n C

3 2486 I Dbag / Topology and s Applcaons Thus, relaons, β 0 ω + + β k ψ pk ω 0 wh mnmal v pβ k k + u k acually, β k p k+u k exs I follows from elemenary algebra n a sraghforward way ha J p P n C has a prmary decomposon wh nvarans p k+u k for 0 k r n The frs summand n hs decomposon s generaed by w and s proved ha has order p 0+u 0 M n+1,p p-componen of he Ayah Todd number M n+1 We hen exend hs o J -groups of lens spaces Le Gp, n, r be he sub-group of JL n p r generaed by powers of w The ndex funcons m j are replaced by a ceran reducon mr j wh respec o r The whole heory goes over wh u r k defned analogously wh u k n erms of mr j and Gp, n, r has a prmary decomposon wh nvarans p k+u r k for 1 k r 1 From hs we recover he decomposon of JL n p r no a drec-sum of cyclc groups The frs summand generaed by w has order p 0+u r 0 M n+1 p r where M n+1 p r s defned n 1, Defnon 734 We also recover he prmary decomposons of J p P n C and JL n p r when n p 1p k + s0 s p2 In 1 as a demonsraon we wroe down he α- and β-relaons n J 2 P 164 C In hs paper we wre down, very quckly, he prmary decomposon of J 2 P 164 C These prmary decomposons are exsenal n he sense ha hey only gve he nvarans of J p P n C and JL n p r e he orders of he cyclc groups n hs decomposon whou an explc expresson for her generaors excep ha of he frs summand whch s generaed by w For hs reason, he prevous paper 1 s essenal o hose who seek explc relaons n hese J -groups Wh 1 and he presen paper he algebrac srucure of he J -groups of complex projecve and lens spaces s compleely deermned and here s nohng more o do on he algebrac sde However, J -groups have wo dfferen srucures compable wh each oher; he algebrac srucure and he degree-funcon on hem defned by sable codegrees of vecor bundles jus as vecor spaces wh a norm The algebrac srucure hus deermned, he way s opened o he deermnaon of he degree-funcon and s hoped ha he nfrasrucure of hs problem s lad down n hese wo papers The degree-funcon q on negave mulples of he complex Hopf bundle s he complex sable James number whch s he order of he obsrucon o cross-seconng a ceran Sefel fbraon Le 1+ 1 an x xp 1 n Then a folklore conjecure saes ha he p-prmary componen q p p nη k1 of qnη k1 s equal o LCD{a n :1 k1} for eher p odd or p 2 and n even 2 Prmary decomposon of J p P n C 21 Background maeral from 1 p s a fxed prme hroughou We defne a decreasng sequence k 0 k rn by k npk +1 p k p1 LeΦ0 { k : k 0modp} m s a sngular exponen e m M f he coeffcen of ω m n he expanson of ρ p ψpk ωpk s no negral for some 0 k r n 1, Proposon 444 saes ha here s a bjecon, σ : Φ 0 M gven by σ k 1 2 p 1pk k ;ef k p ν Δ ν 1, Δ, p 1 hen m 1 2 p 1p Δ where k + ν j p j Δ 1, k + 1 j and f we le T m jk en for consecuve elemens, m, m M m >m, T m T m s eher empy or equal o { }{ k } and he laer s always he case f p 2 The j-ndex, m j of m, defned for j, s he p-exponen of he denomnaor of he coeffcen of ω m n he expanson of ρ p ψpj ωpj when ha coeffcen s no negral and s gven by he formula, m j p j Δ + j j {m} j k j ννν1ν2 21 and m j j k1 s a srcly-ncreasng sequence bounded above by ν m k1 ν ff k1 p ν+1 Δ Lemma Le m, m M, m >m, m σ k σp ν Δ 1 2 p 1p Δ, m σ k σp ν Δ 1 2 p 1p Δ Δ, p Δ,p 1, k + ν, k + ν Ifj <j hen m j m> j j m j m Proof j m j j m m j m p j Δ + j j p j Δ + j j p j Δ + j j p j Δ + j j

4 p Δ p Δ p j p j 2 1 p 1 m m I Dbag / Topology and s Applcaons p j 1 p j > 0 23 Defnon For 0 k r n, M k s he se m M such ha he coeffcen of ω m n he expanson of ρ p ψpj ωpj s no negral for some 0 j klem k {m 1,,m d }, m 1 <m 2 < <m d 24 Observaon Le m, m M,m >mifm 1 m s eher undefned, or, m 1 m 0 1 hen eher 1 m 2 m n undefned, or, 1 m 2 m 0 Proof I follows from 21 ha f m σ k and m σ k Then k k 1 and k and hence k 1 Thus, 1 m 2 m s eher undefned, or, 1 m 2 m 0 25 Defnon Le H k {Ω 2 M k eher Ω φ, or, Ω m 1,,m r, 1 < 1 < 2 < < r d, k+1 k >0 and k > 0, 1 r} Defne he assocaed se Φ {Ω M k Ω eher m<m r, or, m>m r and kr m kr1 m >0}: Le m Φ, m <m<m +1 Then k+1 m s eher undefned, or, k+1 k m k 0 by Observaon 24 whch s a conradcon >0 snce f we assume he conrary hen k+1 Le m l supω Φ I follows from he Observaon 24 ha Ω Φ {m 1,m 2,,m l }, l d and m kr s eher undefned, or kr kr1 m m kr1 m 0form>m l k For Ω H k { } we wre down he correspondng se of nequales IΩ for he ndex funcons j Le Ω m 1,,m r, Φ m j1,,m js r+ s l For 1 r, k > r u+1 ku+1 u ku u and k > r u+1 ku+1 u ku u + k+h for h, k+h > 0 There s a 1 1 correspondence beween Ω H k {φ} and he se of nequales IΩ The se of nequales {IΩ; Ω H k {φ}} are dsjon Two of whch canno hold smulaneously Hence a mos one se of nequales s sasfed We le Ω 0 Ω f IΩs sasfed for some Ω H k and Ω 0 φ f none of he se of nequales IΩs sasfed For Ω {m 1,,m r } H k and Φ {m j1,,m js } we defne r uω max 1 k+1 k ; max 1 h s k +h >0 jh h 1 k+1 k kj + h +h 3 We noe ha uφ maxm k : m M k,m k k1 m >0 We hen defne u k uω 0 where Ω 0 s he unque elemen of H k defned above 26 Proposon Le Ω H k Then v p uω u k Proof Le Ω H k and Φ be he assocaed se Le Ω 0 H k be he unque elemen of H k defned n 25 such ha u k uω 0 LeΩ 0 {m 1,,m r }, Φ 0 {m j1,,m js }, Ω 0 Φ 0 {m 1,,m l } Then by Defnon 25, r jh k+1 u k max k h k+1 ; max 1 h s k kj + h +h 4 1 k +h 1 j >0 h Le k+h > 0forsome1 h s Suppose m 1,,m p Ω and m p+1 / Φ1 p h

5 2488 I Dbag / Topology and s Applcaons h 1 < k+1 p 1 p 1 p 1 k + k +h k+1 k+1 k+1 jhh k + p+1 k + kp1 p+1 snce by defnon of Ω 0 we have kp1 p+1 r 1 k+1 k k+1 h p+2 k kp p+1 k+1 k kj h +h k kp + p+1 <uω 5 p < 1 p 1 k+1 k+1 < h p+2 k+1 k kp + p+1 k kp1 p+1 + k+h Also, r p+2 k+1 k k kp + p+1 < uω 6 27 Defnon Movaed by Proposon 26, we can gve an alernave defnon of u k LeH k be defned n Defnon 25 If Ω H k defne uω as n 25 Then defne u k mn uω Proposon 26, shows he equvalence of Defnons 25 and 27 If we use Defnon 25, we have o check ou he se of nequales IΩ and hence pck he unque Ω 0 and wh Defnon 27 we have o check ou uω for Ω H k and ake s mnmum and he wo requre equal labour However, Defnon 25 gves more nsgh 28 Lemma If all he erms n Congruence 1; e k j0 1, 2,,kconanng a leas wo elemens such ha p j m Z are no negral here exss a subse U of v p α < m U; v p α v p < m j m, j U; e v p α m U s a consan k U ; v p α v p < m j m for 0 j< k, / U,j U; e v p α m >k U / U Proof Assume he conrary Then eher a There s no par, j, 0<j< k such ha v p α v p m j m, or, b For each subse U of 0, 1,,k wh he propery ha v p α v p <m j m, j U hen here exss s/ U such ha v p α s v p m s j m j U α0 In eher case, le 0 0 k be such ha v p p 0 vp α 0 0 m mnv p α m Then we have src m nequaly; e v p α 0 0 m <v p α m 0Ifv p α 0 0 m 0 hen v p α m 0 0 k and all he erms of Congruence 1 would be negral whch s a conradcon Hence v p α 0 0 m < 0 Thus, Congruence 1 does no hold whch s a conradcon 29 Lemma Le α α 0,,α k be an admssble sequence such ha all he erms of Congruence 1 wh respec o m j1,,m js be negral and m 1,,m r be he remanng elemens of M k wh respec o whch no all he erms of Congruence 1 are negral Le U be he se defned n Lemma 28 for each m 1 r and le V 0, 1,,k U be s complemen Le k sup U and l nf U Then {0 k, > l } V +1 and U +1 {; 0 k, l } Hence eher l k +1,or,l V +1 1 r 1

6 I Dbag / Topology and s Applcaons Proof Le U +1 for >l Then eher U and v p α v p α l l,or, V and v p α v p α l > l, and hence n eher case, v p α v p α l l By Lemma 22, l > +1 l +1 Thus, v p α +1 >v p α l l +1 By of Lemma 28, mn 0 j k v p j +1 k U+1 and hence / U +1,e V Corollary Le α α 0,,α k be an admssble sequence such ha all he erms of Congruence 1 wh respec o m j1,,m js be negral, j 1 <j 2 < <j s e v p α, 0 k, 1 h s and m 1,,m r are he remanng elemens 1 < 2 < < r, wh respec o whch no all he erms of Congruence 1 are negral Then here exss a srcly-decreasng sequence l 0 r n he nerval 0,k wh l 0 k and l k such ha v p α l < l l 1 l > 0 and l > 0 1 r If we defne r jh h K α max hen v p α k K α 1 l 1 l ; max 1 h s l h j >0 h 1 l 1 l l jh h +, 7 Proof Le U be he se defned n Lemma 28 and le l nf U 1 r Pu l 0 k I follows from Lemma 29 ha {l } forms a srcly-decreasng sequence such ha l 1 l >0 and l > 0 Eher k sup U 1 and v p α k v p α l1 k 1 l 1 1,or,k/ U 1 and v p α k v p α l1 > k 1 l 1 1 In eher case, 1 v p α k v p α l1 k 1 l 1 1 Smlarly, we have 2 v p α l1 v p α l2 l 1 2 l 2 2 v p α l1 v p α l l 1 h l v p α k 1 1 l Summng up hese nequales for 1 r we oban l For 1 h s, summng up he frs h nequales above ogeher wh he nequaly; v p α ljh h l h we oban j h h l v p α k 1 l l jh h Lemma Le α α 0,,α k be an admssble sequence and K α be as defned n Corollary 210 Then K α uω for some Ω H k ; 8 Proof Le α α 0,,α k be an admssble sequence Le he elemens m j1,,m js and m 1,,m r of M k and he number K α be defned as n Corollary 210 Defne Ω {m k+1 k >0 and k > 0} Then Ω H k LeΦ be he assocaed se o Ω as defned n 25 Then Φ m j1,,m js LeΩ {m 1,,m r } and Φ m j1,,m js where r r and s s Then by repeaed applcaon of Lemma 22 and for h 1 h s we oban K α k 1 l l 1 2 l 2 l l + l k 1 k1 1 + k1 2 k k+1 k 1 k1 1 + k1 2 k k+1 Smlarly, K α r 1 k+1 k k k Hence K α uω + k ++ l l + l

7 2490 I Dbag / Topology and s Applcaons Proposon Le α α 0,,α k be an admssble sequence Then v p α k u k Proof v p α k u α by Corollary 210 K α uω for some Ω K by Lemma 211 and uω u k by Proposon 26 We shall prove eg Proposon 218 ha here exss an admssble sequence α α 0,,α k wh α k p u k and we need he Congruence Theorem for hs purpose The Congruence Theorem s an orgnal conrbuon of hs paper 213 emark Consder he sysem of congruences, β k p k + β k1 + + β 0 p k1 p 0 Z 1 d If we defne h max j :0 j k, 1 d 1 hen any smulaneous soluon β β 0,,β k s deermned n β Z p h k+1 e β j Z p h,0 j k 214 Theorem Congruence Theorem Le p be a prme and j Z 0 j k,1 d and k < <k+1 k > k1 > > 0 Le h maxj :0 j k, 1 d and β Z ph Then here exss a smulaneous soluon β β 0,,β k Z p h k+1 of he sysem of congruences: β k p k + β k1 + + β 0 p k1 p 0 If β s a un n Z p h so are β j 0 j k Z 1 d wh β k β Proof Defne d j j j1 1 0 j 1 and d j j j+1 1 j k Le1 r h We shall show by nducon on r ha he followng sysem of congruences have a unque smulaneous soluon n Z p r k+1 wh β k β 0 β + β 1 p d γ + pd1 γ 1 mod p r, 1 γ 1 2 γ 2 + β 2 p d2 γ 2 mod p r, + β 1 p d1 γ 1 mod p r, 1 γ 1 + β 0 0modp r, γ + β +1 p d+1 γ +1 mod p r, + β +2 p d+2 γ +2 mod p r, + 1 γ +1 k 1 γ k1 + β k 0modp r For r 1, β + β 1 p d γ + pdj1 γ 1 0modp, e β β 1 mod p 1 j k 1Leβ k β and hs deermnes β j 1 kj β mod p γ j + β j1 p dj1 γ j1 0modp and hus γ j 1 kj β mod p j 1 and, smlarly, γ j 1kj β mod p j k 1Ler>1and assume he nducon-hypohess for r 1 β + β 1 p d γ + pd1 γ 1 mod p r d,d1 1 where γ, γ 1 are he unque soluons mod p r1 Hence β + β 1 s unquely deermned mod p r 1 k From hs and he fac ha β k β, all he β 0 k are unquely deermned n Z p r From he equaon, γ j deermned mod p r1, d j1 > 1, β j1 s deermned mod p r, he varables γ j + β j1 p dj1 γ j1 mod p r and he fac ha γ j1 s are unquely deermned mod p r 2 j 1 γ 1 s unquely deermned n Z p r from he equaon, γ 1 +β 0 0modp r, γ j s unquely deermned n Z p r j k 1 from he equaon, γ j + β j+1 p dj+1 γ j+1 mod p r and he fac ha he class of γ j+1 s

8 I Dbag / Topology and s Applcaons deermned n Z p r1, d j+1 from he equaon, γ k1 > 1, and ha β j+1 s deermned n Z p r The varable γ k1 s deermned unquely n Z p r + β k 0modp r Hence all he varables are unquely deermned n Z p r Leγ j x j mod p r1 and γ j x j mod p r1 Then γ j x j + k j pr1 The equaons 0, 1,, 2, ī, + 1,,k 1 are deermned wh γ j x j and γ j x j on he HS of he congruence and f we now pu γ j and γ j nsead of x j and x j, he HS of he congruences dffer by elemens of he form kj pdj p r1,or,k j pdj p r1 whch are congruen o 0 mod p r Hence he unquely deermned varables sasfy he gven sysem of congruences n Z p r The varables β 0,,β k of he sysem of congruences for r h s a soluon of he orgnal sysem of congruences wh β k β 215 emark α α 0,,α k s called admssble wh respec o a subse S of M k f and only f Congruence 1 s sasfed for all m S 216 Proposon Le Ω {m 1,,m r } H k 1 1 < 2 < < r d and Φ {m j1,,m js } 1 j 1 <j 2 < <j s, be s assocaed se as n Defnon 25 Then here exss an admssble sequence α α 0,,α k wh r1 respec o he se Ω M k Ω Φ, wh α k p k+1 k Proof Le M k {m 1,,m d }, Ω Φ {m 1,,m l }, l d Suppose krj+1 krj+1 krj s undefned for j>eleα k+1 p l j1 krj krj+1 βkr 1 e and β j 0 k r e<j k Le r+j l + j Z m Ω M k Ω Φ can be wren down as Congru- α kr p 1 j e Then Congruence 1; e k j0 ence 2; e β k p δk We clam he + β k1 p δk1 Saemen δ k <δ k1 Proof For 1 j r, ru u ku+1 k krj 0 1 j e and u β k+1 1 r, α kr β kr, p j m + + β kre Z 1 r + e 10 p δkre < <δ k+1 δ k >δ k+1 > >δ kre 1 r + e δ kj+1 δ kj1+1 e δ kj+1 For 1 r, δ k+1 δ k e δ k+1 δ kj+1 kj+1 kj+2 j1 >δ kj1+1 k+1 k+1 δ k r s k For 1 r, <j r 1, δ kj+2 r sj kj+1 j1 ks+1 s ks+1 s ks s kj+2 ks s k k+1 k 0 kj+2 r ks+1 sj1 kj+1 > 0 by Lemma 22 r ks+1 s+1 s s ks s ks s 11 12

9 2492 I Dbag / Topology and s Applcaons δ kj+1 δ kj+1+1 δ kj+1 kj+1 kj+1 e δ kj+1 >δ kj+1+1 v For 1 r, 0 j e 1, δ krj+1 δ krj+1+1 v For 1 r, δ k+1 v For 1 j e, δ klj+1 l+ δ klj1+1 l+ v For 0 e 1, δl+ kr+1 δl+ kr v For 1 <j e, δ kj r sj kj δ krj+1 krj+1 krj+1 ks+1 s ks s kj r ks+1 sj+1 kj+1 j kj j > 0 by Lemma 22, δ krj j1 krs s1 krj krs+1 krj+1 >δ k and he proof s smlar o ha of δ klj+1 l+ klj+1 l+ krj+2 1 kr+1 l+ kr+1 l+ δ krj+1 l+ δ krj+1+1 l+ 1 δ klj+2 l+ j1 krs s1 krj+1 1 krs s1 kr l+ δ krj+1 l+ krj+1 l+ krj+1 l+ krj krs+1 krj+2 krs+1 kr+1 l+ δ krj l+ j1 krs s1 krj l+ kr l+ s krj ks s j krs s1 13 krs+1 > 0 by Lemma krj+2 l+ krj+1 kr l+ krs+1 krj+1 j2 krs s1 krs+1 > 0 by Lemma krs s1 krs krj krj l+ j krs s1 krs+1 > 0 by Lemma Hence Congruence 2 sasfes he hypohess of he Congruence Theorem and we deduce from he Congruence Theorem ha here exss a soluon β β 0,,β k wh β k 1 Thus, Congruence 1 adms a soluon α α 0,,α k r1 wh respec o m Ω M k Ω Φ wh α k p k+1 k whch by defnon s an admssble sequence wh respec o Ω M k Ω Φ 217 Proposon Le Ω m 1,,m r H k, 1 < 2 < < r Then here exss an admssble sequence α α 0,,α k wh α k p uω Proof Le Φ m j1,,m js be he assocaed se o Ω so ha Ω Φ m 1,m 2,,m l By Proposon 216 here exss an admssble sequence α α 0,,α k wh respec o he se Ω {m l+1,,m d } such ha α k p r1 k+1 k By defnon

10 uω max λ + r 1 r 1 I Dbag / Topology and s Applcaons k+1 k ; max 1 h s k h j >0 h k+1 k for λ 0 r 1 k+1 k kj + h h Defne α α 0,,α k by p λ α j k r j k Then snce Congruence 1 s homogeneous wh respec o he varables, follows ha α s also admssble wh respec o he se Ω {m l+1,,m d }, If we subsue α k+1 p ru u ku+1 ku j1 krj u β k+1 1 r, α kr β kr, α kr p krj+1 βkr 1 e 18 and β j 0 k r e<j k as n he proof of Proposon 216, Congruence 1 akes he form 2 kj0 β j Z Then by precsely he same argumens as used n he proof of Proposon 216, we can esablsh p δj m he nequales: δ k <δ k1 δ k+h < <δ k+h k+h λ k+h λ + >δ k+h1 > >δj 0 h 1 h s r k+1 k h+1 r 1 k+h uω + k+1 h 1 k+1 k + h 1 k+1 k k 0 19 by defnon of uω Thus, δj h <δ k+h 0for k + h, by he above nequaly Hence all erms of Congruence 2 and hence of Congruence 1 wh respec o m jh are negral 1 h s I follows ha α s an admssble sequence wh α k p uω 218 Proposon There exss an admssble sequence α α 0,,α k wh α k p uω Proof Take Ω Ω 0 n Proposon Lemma Le m M, m σ k, k p ν Δν 1, Δ,p 1, k +ν Then for j k + 1, p j Δ+ j < j1 Proof j1 p j Δ + j p +j+1 Δ + pkj+1 Δ1 p1 p j Δ + j by 1, Lemma 446 p j+1 Δ + k j + 1 p j Δ + j p 1p j Δ ν + 1 p 1p k1 ν 1 p 1p ν1 Δ ν 1>0 220 Corollary For j<k, k + u k < j Proof Snce j s a srcly-decreasng sequence, suffces o prove ha k + u k < k1 LeΦ {m 1,m 2,,m l } be he assocaed se o he empy-se φ H k Then j k k1 j >0; e f m j σ kj hen k k j Le kj p ν j Δ j ν j 1, Δ j,p 1, j k j + ν j, m j 1 2 p 1p j Δ j 1 j l Then j k p j k Δ j + j k k < k1 k by Lemma 219 Thus, u k uφ max 1 j l j k < k1 k e k + u k < k1 221 Theorem J p P n C r n k0 Z p k +u k The order of he frs summand generaed by ω s he p-componen, M n+1,p of he Ayah Todd number M n+1

11 2494 I Dbag / Topology and s Applcaons Proof By Proposon 218, for each 0 k r n, here exss an admssble sequence α α 0,,α k wh α k p u k I follows from 1, Proposon 6214 ha here exss n J p P n C a relaon, p 0α 0 ω + p 1α 1 ψ p ω + + p k+u k ψ pk ω 0 By Corollary 220, j > k + u k 0 j k 1 and hence f we le x k p 0 k +u k α 0 ω + p 1 k +u k α 1 ψ p ω + +p k1 k +u k α k1 ψ pk1 p k+u k x k 0 0 k r n Snce {ω,ψ p ω,,ψprn ω + ψ pk ω hen he above relaon can be wren down as: ω} spans J pp n C and ha he coeffcen of ψ pk ω n he expanson of x k s 1, follows ha {x 0,x 1,,x rn } spans J p P n C Suppose β 0 x 0 + +β rn x rn 0 We clam he followng saemen: Saemen If β 0 x 0 + +β k x k 0 1 k r n hen β k x k 0 and β 0 x 0 + +β k1 x k1 0 Proof Subsung for x j n erms of ψ p ω we oban a relaon, α 0ω + α 1 ψ p ω + + α k1ψ pk1 ω + β k ψ pk ω 0 By 1, Proposon 6214, p j α j 0 j k 1 and β k p kα k where α α 0,,α k s an admssble sequence By Proposon 212, p u k α k and hus p k+u k β k Hence β k x k 0 and hus, β 0 x β k1 x k1 0, provng he saemen I follows from he saemen by nducon on k sarng wh k r n ha β 0 x 0 β 1 x 1 β rn x rn 0 Ths proves he desred prmary decomposon As for he second par of he heorem, 0 p1 n, H 0 {φ} Φ M 0 m M 0 s of he form m 1 2 p 1p Δ, Δ, p 1 m 0 p Δ + 0 > 0 Le r m p Δ p1 2m p1 n Then r m + v p r m 0 > 0 n u 0 max rm + v p r m 0 max r + v p r 0 :1 r m M 0 p 1 n n 0 + u 0 max r + v p r: 1 r,r+ v p r p 1 p 1 max r + v p r: 1 r v p M n+1 n p 1 n,r+ v p r, p emark The second par of he heorem s he soluon of he complex analogue of he vecor feld problem and he smples proof so far has been provded As a corollary o Theorem 221, we recover 1, Proposon 6212, e 223 Corollary Le n p k p 1 + r 0 r p 2 Then J p Pn C Z p p k +k Z p p k1 1 Z p p k2 1 Z p p1 Proof m m {m}, m 1 2 p 1pk 1 k m j p kj 1 1 j, em <1 m < <0 m Thus, H {φ} 1 k and he correspondng se Φ o φ s empy Thus, u 0 1 k and hence he h-summand has order p p pk1 1 k The order of he frs summand follows from he defnon of he Ayah Todd number 224 Example As a demonsraon we wroe down n 1, Example 6213 he α- and β-relaons for J 2 P 164 C We now oban he prmary decomposon of J 2 P 164 C j : { } j 82 : j 80 : j 64 :

12 2 2 k 1: M 1 80, 82, τ k 2: M 2 80, 82, τ k 3: M 3 64, 80, 82, τ k 4: M 4 64, 80, 82, τ k 5: M 5 64, 80, 82, τ k 6: M 6 64, 80, 82, τ k 7: M 7 64, 80, 82, τ M 165, Accordng o Theorem 115, I Dbag / Topology and s Applcaons , H 1 { 80, φ },u80 4, uφ 3, u 1 3;, H 2 {φ}, uφ 3, u 2 3; J 2 P164 C Z Z 2 84 Z 2 40 Z 2 20 Z 2 10 Z 2 4 Z 2 1, H 3 { 64 },u64 1, u 3 1; , H 4 { 64, φ },u64 1, uφ 2, u 4 1; , H 5 {φ}, uφ 0, u 5 0; , H 6 {φ}, uφ 0, u 6 0; , H 7 {φ}, uφ 0, u 7 0; Accordng o hs prmary decomposon, J 2 P 164 C We know from 1 ha J 2 P 164 C 7k k I checks 3 Prmary decomposon of JL n p r 31 Defnon Le n Z +, r r n Then J p,n,r ψ pr J pp n C subgroup of J p P n C generaed by ψ pr ω, ψpr+1 ω,,ψ prn ωlegp, n, r be he subgroup of JLn p r generaed by he powers of ω Then follows from 1 ha Gp, n, r s he quoen, Gp, n, r J p P n C/J p,n,r For deals refers o 1, Defnon 518 and Secon 71 We now defne reduced ndex funcons j mr whch wll play he same role for lens spaces as ndex funcons j m play for complex projecve spaces 32 Defnon Le n Z +, r r n and m 1 2 p 1p Δ M Δ, p 1 Then j mr p j Δ + mn, r 1 j j j 33 Lemma Le n Z +, r r n and m σ k σp ν Δ 1 2 p 1p Δ M, Δ, p 1, k + ν Then m j f <r, mr j m j m r f k + 1 <r, r j f k + 1 <r j, non-posve f k + 1 r Proof If <r, follows from s defnon ha mr j m j If k +1 <r, mr j m j m r pj Δ+r 1j j p j Δ+ j j +p r Δ+ r r p r Δ 1 r 0 by 1, Lemma 443 If k + 1 r, mr j p j Δ + r 1 j p j Δ + k j 0 by 1, Lemma 446

13 2496 I Dbag / Topology and s Applcaons We now sae a slgh varaon of 1, Proposon Proposon If m >mare consecuve elemens n M, m σ k σp ν Δ, m σ k σp ν Δ, Δ, p Δ,p 1 Le k + ν s k and α 0,,α s be admssble n M m n he sense of 1, Defnon 513 Then here exs negers j k+1 such ha α 0,,α s, 0,0, α k+1,,α k+ν,,α rn s an admssble sequence Proof Idencal wh ha of 1, Proposon Proposon There exss a relaon, β 0 ω + +β s ψ ps ω 0 n Gp, n, r s r 1 ff β j p j, 0 j s where α α 0,,α s s an admssble sequence wh respec o j mr Proof Suppose β 0 ω + + β s ψ ps ω 0nGp, n, r Then β 0ω + + β s ψ ps ω 0nJ pp n C mod J p,n,r; e here exs negers β r,β r+1,,β rn such ha β 0 ω + + β s ψ ps ω + β rψ pr ω + + β rn ψ prn ω 0nJ pp n C By 1, Proposon 6214, β j p j 0 j s, r j r n where α α 0,,α s, 0,,0,α r,,α rn s an admssble sequence wh respec o he ndex funcons, m j Suppose ha m σ k σp ν Δ M k + ν<r Then by Lemma 33, mr j m j 0 j k + ν and hus mns,k+ν j0 mns,k+ν p j mr j0 p j m Z k + 1 <r k + ν By Lemma 33, mr j m j m r 0 j r and j m r jr j k + ν s k+ν αj + β Z j0 jr p j m Mulplyng by p r m, we oban: s j0 p j mm r p j mr k+ν + jr p j mm r p rj p jr βp r m Z; e r j k + ν s j0 + k+ν Z, p j mr jr p j mr Thus, s j0 Z p j m r r k + 1 By Lemma 33, mr j 0 and hus s j0 Z p j m r Hence α 0,,α s s an admssble sequence wh respec o he reduced ndex funcons mr j Conversely, le α 0,,α s be an admssble sequence wh respec o mr j and β j p j 0 j s Le m σ k σp ν Δ M Δ, p 1 be such ha k + ν<r Then by Lemma 33, mr j m j and mns,k+ν j0 mns,k+ν p j j0 Z Suppose here exss no m M such ha k + 1 <r k + ν Le m p j m r m sup{m σ k σp ν Δ : k + ν <r} Then α 0,,α s s admssble n M m I follows from 1, Proposon 517 f s k + ν and from Proposon 34 f s>k+ ν ha α 0,,α s exends o an admssble sequence α 0,,α s, 0,,0,α r,,α rn If here exss m σ k σp ν Δ M Δ, p 1 such ha k + 1 <r k + ν hen pu α r s Z and 0 p j m r r+1 α k+ν 0, s j0 p j mm r + α r 0, or, s j0 p j m + α r 0, e p r m

14 I Dbag / Topology and s Applcaons s j0 + k+ν jr Z By 1, Proposon 517, α 0,,α s, 0,,0,α r,, α k+ν exends o an admssble p j m sequence, α 0,,α s, 0,,0,α r,,α rn wh respec o m j Pu β j p j r j r n and we oban from 1, Proposon 6214, he relaon, s j0 + r n jr β j ψ pj ω 0nJ pp n C Thus, s j0 β j ψ pj ω 0n Gp, n, r 36 Defnon We defne he nvaran u r k by replacng he ndex funcons j m n he defnon of u k by he reduced ndex funcons mr j We oban for Gp, n, r he analogue of Theorem 221 for J p P n C LeM n+1 p r be as defned n 1, Defnon Theorem Gp, n, r r1 k0 Z p k +ur k The frs summand generaed by ω has order M n+1p r From Theorem 37 we wre down he decomposon of JL n p r no cyclc groups; e 38 Theorem J L n p r r1 k0 Z p k +ur k f p s odd and n 0 mod 4, r1 k0 Z p k +ur Z k 2 f p s odd and n 0 mod 4, r2 k0 Z 2 k +ur k Z 2 r1 +ur r1 +1 f p 2 The frs summand generaed by ω has order M n+1 p r As a corollary o Theorem 38, we recover 1, Proposon 738, e 39 Corollary Le n p k p 1 + r 0 r p 2 1 r k Then Z p p k +r1 Z p p k1 1 Z p p k2 1 Z p p kr+1 1 f p s odd and n 0 mod 4, J L n p r Z p p k +r1 Z p p k1 1 Z p p k2 1 Z p p kr+1 1 Z 2 f p s odd and n 0 mod 4, Z 2 2 k +r1 Z 2 2 k1 1 Z 2 2 k2 1 Z 2 2 kr+1 f p 2 Proof m r < 1 m r < <0 m r and he se Hr defned n analogy wh H consss, merely of φ and he assocaed se Φ o s empy and hence u r 0 r 1 k The order of he frs summand follows from he defnon of he number M n+1 p r eferences 1 I Dbag, Deermnaon of he J -groups of complex projecve and lens spaces, K-Theory Furher readng 1 JF Adams, On he groups JX, II, J Topology JF Adams, On he groups JX, III, J Topology JF Adams, Infne Loop Spaces, Prnceon Unversy Press, Prnceon, NJ, JF Adams, G Walker, On complex Sefel manfolds, Proc Cambrdge Phlos Soc MF Ayah, JA Todd, On complex Sefel manfolds, Proc Cambrdge Phlos Soc JC Becker, DH Goleb, Transfer map for fbre bundles, J Topology Bo, Lecures on KX, Benjamn, New York, MC Crabb, K Knapp, On he codegree of negave mulples of he Hopf bundle, Proc oy Soc Ednburgh A MC Crabb, K Knapp, James numbers, Mah Ann

15 2498 I Dbag / Topology and s Applcaons I Dbag, Degree funcons q and q on he group J SO x, Hablaonsschrf, Mddle Eas Techncal Unversy Ankara, I Dbag, Degree-heory for sphercal fbraons, Tohoku Mah J I Dbag, On he Adams conjecure, Proc Amer Mah Soc I Dbag, Inegraly of raonal D-seres, J Algebra I Dbag, J -approxmaon of complex projecve spaces by lens spaces, Pacfc J Mah K Fuj, T Kobayash, K Shmumura, M Sugawara, KO-groups of lens spaces modulo powers of wo, Hroshma Mah J Fuj, J -groups of lens spaces modulo powers of wo, Hroshma Mah J K Fuj, M Sugawara, The order of he canoncal elemen n he J -group of he lens space, Hroshma Mah J T Kambe, The srucure of K -rngs of he lens spaces and her applcaons, J Mah Soc Japan T Kobayash, S Murakam, M Sugawara, Noe on J -groups of lens spaces, Hroshma Mah J KY Lam, Fbre homoopc rval bundles over complex projecve spaces, Proc Amer Mah Soc