On stable James numbers of quaternionic projective spaces. Osaka Journal of Mathematics. 12(1) P.209-P.213

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1 Title Author(s) On stable James numbers of quaternionic projective spaces Ōshima, Hideaki Citation Osaka Journal of Mathematics. 12(1) P.209-P.213 Issue Date 1975 Text Version publisher URL DOI /4783 rights

2 Oshima, H. Osaka J. Math. 12 (1975), ON STABLE JAMES NUMBERS OF QUATERNIONIC PROJECTIVE SPACES HIDEAKI OSHIMA (Received March 12, 1974) In [4], we have defined the stable James numbers k s {X, A) some finite CW-pairs (X, A) and computed d c (ή)=k s (CP", CP 1 ). In this note we estimate djan)=k s (HP", HP 1 ), where HP" denotes the quaternionic projective space of topological dimension 4n. We obtain Theorem. For ^ (0) dχ(n) is a factor of (2n)\ (2re 2)! 4!, in particuler none of the prime factors of d^n) is greater than 2n, 2 +l<ί "^K))= 3/+3 «= J \vjddn)) 3j+6 (ii) 2j^v 3 (dn(n))^2j+2 3^ ^H(«))^2/+4 (iii) /or αprime p^s v p {d H {n)) = 2j p* ^v p (d H (n)u2j+2 /or />(/«) denotes the exponent of p in the prime factorization of m. Recall that ^w^the index of the image of ί*: {HP", S 1 } -* {S\ S 4 }, where {X, Y} denotes the set of stable homotopy classes of stable maps X-> Y and i: S ι =HP i^hp n the natural inclusion. Then obviously <*#(!)= 1.

3 210 STABLE JAMES NUMBERS OF QUATERNIONIC PROJECTIVE SPACES 1. Lower bound of d^ji) In this section we use ίγ-theories. We introduce the following notations: Λ =the canonical quaternionic line bundle over HP"; g H =ξ 1 ί^ksp(s 4 ); g R =ghag H^κδ-\S 4 );ξ n =g H A(ξ n -l)^κδ-\hp n ); τ?=the canonical complex line bundle over S 2 = CP 1 ; g c = V~ 1 G K(S 2 ); S: KO*( )->K*( ), the complexification c: KSp( )^K( ), the sealer restriction; ch: K*( )->ίf*( Q), the Chern character; y 2k =gr k <=KO 8k ; y 2k+1 <=KO 8fc+ \ the generator such that % + 1 )=2^- 2 ; z n = c{ξ n -\)<ΞK{HP n ); t^h\hp»; Z), the first symplectic Pontrjagin class of ξ n. Then we have ch(z n ) = exp (\/T)+ exp ( \/Ύ) 2, and K0-\HP n ) is the free group with basis ξ H, yffi,,j Λ - I n, and K(HP n ) is the truncated polynomial ring over Z with generator z n and the relation # +1 =0. Choose /<Ξ \HP n, S 4 } such that the composition S 4^HP"^S 4 d H (ή). Put is of degree y i And put 2a 2j =b 2j and a 2J+1 =b 2J+1. Then, by the above equations, we have <)* =f*-ch-6(g R ) = ch-s f*{g R ) = Σ bj fexp (VT)+exp (->/T)-2y in H*(HP"; Q) and hence b x =d H {n). Put *=* 2, then we have Differentiating this equation twice, we have (-*)-2)' mod Λ: 2 2b, = 2b,+ Σ 0" 8 *y+2(2/+l)0"+l)*y+i)(«p (*)+exp (-x)-2y mod * 2 «. Hence and theree j 2 bj-\-2(2j-\-l)(j-{-ί)bj +1 = 0 j^n 1, That is, we have yi b, = (-1)^.3.5 (2y+i)0'+i)fty +1 (2;)! d H {n) = 2^^5-(4j+ί)(2j+l)a 2J+1

4 H. OSHIMA 211 (2/-1)1 d H (n) = -2 2^.3.5-(4j-l)(2j)2a 2J j [=»] Put τ p (n) = Then obviously Tp(ri)^v p (d H (n)). Elementary calculation shows that and an odd prime p τ 2 (n) = 2j-\-1 2 J^n< 2 J+1 and 2/ 2/+1 Thus we obtain the lower estimates of Theorem. 2. Upper bound of dj^n) The canonical fibration S An ~ 1 -^HP n ~ 1 factorizes as the composition of the canonical fibrations s^-'^lcp 2 "- 1 >HP n '\ The order of p 2n _ λ as a stable map is (2n)l [2], [5]. Hence the stable order of pn-\ is a factor of (2ή)\. Theree d H (ή) is a factor of (2n)l d H (n \). This implies Theorem (0). Lemma 1. Let X be a simply connected finite CW- complex with a base point. Then the natural inclusion SP m (X)^SP (X) induces isomorphisms π k (SP m (X))^π k (SP (X)) k^m, where SP M (X) and SP~(X) denote the m-fold symmetric product of X and the infinite symmetric product of X respectively [1], Proof. There is a commutative diagram [3] /^1 Hj(SP m (X)) ~ Hj(SP~(X)) 1 I ±Hj(SP k (X) y SP k '\X)) - where the bottom map is the natural inclusion. Then, it follows from Hj{SP k {X), SP k -\X))=0 l^j<k that ι m *: Hj{SP m (X))-^Hj{SP-{X)) are isomorphic j^m. Since SP m (X) is simply connected, the result follows

5 212 STABLE JAMES NUMBERS OF QUATERNIONIC PROJECTIVE SPACES from the theorem of J.H.C. Whitehead. The obstructions to extending the natural inclusion S 4 -+SP 4n ~ 1 (S 4 ) over HP" lie in H 4 \HP n, S 4 ) π Aj.lSP 4n -\S 4 )) 2^j^n. Lemma 1 shows that SP 4n -\S 4 ) and SP (S 4 )=K(Z y 4), the Eilenberg-MacLane complex, have the same 4w-type. Hence, in particuler, we have π j^1(sp 4rt 4 ~ 1 (S 4 ))=0 j^n. Theree we have a map /: HP n^sp 4n -\S 4 ) which factorizes S'-^SP 4 "- 1^4) as S'aHP^SP 4 *-'^4). This implies that d H (n) is a factor of ki"- u4 = k s (SP 4n -\S 4 ), S 4 ). k?- 1 ' 4 and k?' 1 * are factors of k?' 1Λ and k 4 "- 1 ' 5 respectively [4]. Hence djn) is a factor of k 4n ~ u \ We require the following theorem of Ucci [6]: v 2 {ht^) =S φ(2t)β 2 (m), vp(k m ' 2t+1 ) = tβ p {m) an odd prime p where βp(m) is defined by p β ρ ίfn^^m<.p β ρ cmw and φ(s) is the number of integers u such that 0<u^s and MΞO, 1, 2, or 4 mod 8. By this theorem, we have ) ^ 3β 2 (4n-ί), v p (d H (n)) ^ 2^(4^ 1) an odd prime/). Then the following lemma completes the proof of Theorem. Lemma 2. (i) «4.-i)_ (ii) β 3 (4n I) (iii) a prime p ^ lit* j +i 1 Ί 1 / 1 / I /or re = V /or 2^ Vι 4 β P (4n-ί) = 3 pi: jor 4 The proof of this lemma is easy, and we omit it. OSAKA CITY UNIVERSITY

6 H. OSHIMA 213 References [1] A. Dold and R. Thorn: Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. 67 (1958), [2] E. Dyer: Chern characters of certain complexes, Math. Z. 80 (1963), [3] M. Nakaoka: Cohomology of symmetric products, J. Inst. Polytech. Osaka City Univ. 8(1957), [4] H. Oshima: On the stable James numbers of complex projective spaces, Osaka J. Math. 11 (1974), [5] H. Toda: On unstable homotopy of spheres and classical groups, Proc. Nat. Acad. Sci. USA 46 (1960), [6] J. Ucci: Symmetric maps of spheres of least positive James number, Indiana Univ. Math. J. 21 (1972),

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