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{A, F3 'YI7}Ern+1(Ss)/(A 0 7Rn+L(Sr) + 7R,+I(Ss) 0

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{A, F3 'YI7}Ern+1(Ss)/(A 0 7Rn+L(Sr) + 7R,+I(Ss) 0 1102 MATHEMATICS: H. TODA PROC. N. A. S. Note that for fixed p = IlyII, Fp'(t) > 0 for all t. The mapping t Fp(t), for fixed p, is a C'-diffeomorphism of the t-axis onto itself. Under q D(t', y1', *... r f) > 0 (2.4) D(t, yi, . , r) The mapping v is clearly a homeomorphism of Ety onto Ety. It is of class C', and by virtue of (2.4), is also a C'-diffeomorphism. Restricted to the t-axis the mapping takes the form t -* f(t) and so carries (b, 0) into (a, 0), When p > e, Fp(t) = t, as it also does when 0 < p < e and t < a' or t > b'. This completes the proof of Lemma 2.1 and Theorem 1.2 follows. 1 Huebsch, William, and Marston Morse "An explicit solution of the Schoenflies extension problem," J. Math. Soc. Japan, 12 (1960). 2 Morse, Marston, "A reduction of the Schoenflies extension problem," Bull. Am. Math. Soc., 66, 113-115 (1960). 3 Morse, Marston, "Differentiable mappings in the Schoenflies theorem," Compositio Mathe- matica, 14,83-151 (1959). 4Huebsch, William, and Marston Morse "The dependence of the Schoenflies extension on an accessory parameter," J. d'Analyse Mathimatique, to be published. 6 Brown, Morton, "A proof of the generalized Schoenflies theorem," Bull. Am. Math. Soc., 66, 74-76 (1960). 6 Mazur, Barry, "On embeddings of spheres," Bull. Am. Math. Soc., 65, 59-65 (1959). ON UNSTABLE HOMOTOPY OF SPHERES AND CLASSICAL GROUPS BY HIROSI TODA* DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF CHICAGO AND UNIVERSITY OF KYOTO Communicated by Saunders Mac Lane, May 18, 1960 Throughout this note, p denotes an odd prime. This note proves non-triviality of the p-primary components of 7r2k(p-1)-1+i(St) for i _ 3 and r26(SU(j)) for n _ j > l, n = k(p - 1) + 1, p > . 1-. For elements acE7rr(Ss), f3Elrq(Sr) and YE7n(S6) satisfying a 0o = f3o0y = 0, a toric construction { a, f3 'YI7}Ern+1(Ss)/(a 0 7rn+l(Sr) + 7r,+i(Ss) 0 Ey) was defined in.1 The following (Al)-(A4) are some of the properties1 of toric constructions. (A1) {IaOA, a,y} C {a, 1A y, a},-Ela,f, y} c {Ea, E3,Ey}, whereEde- notes Freudenthal's suspension homomorphism. (A2) a 0 {I, y,yj = {a, f3, y} 0 (-E6). (A3) 2{ tt8+2, E2a, ttr+2} 0 if ta = 0, where Lte7r (S') denotes the class of the identity of S5. (A4) Let Y = Ss u er+1 be a cell complex and let a be the attaching class of the cell er+l. Let f:Sr+l Y be a mapping of degree t in dimension r + 1. Assume that tLr 0 Y = 0 for yE7r.(Sr), then f* (E-y)ei* {a, IL, '4I for the injection homomor- phs _ i r~ (Ss --)-- 7,+ (y) Downloaded by guest on September 24, 2021 VOL. 46, 1960 MATHEMATICS: H. TODA 1103 Consider the mod p Hopf invariant2 H,: 7r2p(S') Z,. This is an isomorphism of the p-primary components. LEMMA 1. Let p be an odd prime. There exists a sequence { at(S); k = 1, 2,... of elements ak(3) E7r2k(p-l)+2(S3) satisfying Hp(al(3)) = 1 (mod. p), pak(3) = 0 and ak4-1(3) et al(3), p2p, E2P-3 ak(3)} fork _ 1. Proof: Obviously al(3) exists. Assume that ak(3) exists. Consider an element E2P-5X of E2P5{ pt5, E2ak(3), PL2k(v-l)+4} C -1{pt2p, E2P'ak(3), PL2(k+1)(o-1)+1}- By (A3), 2E2P-5x = pE2P-5y for some y. Since pal(3) = 0, al(3) 0 (p + l)E2P-5x = al(3) o (p(p + 1)/2)E2P-y = 0. By (A2), ptIai(3), pt2p, E2P-'ak(3)} = ai(3) 0 (-{ Pt2p, E2P-3ak(3), PL2(k+1)(p1)+1}) contains ai(3) o E2P-5x= 0. Thus there exists an ak+1(3) such that Pak+1(3) = 0. Then Lemma 1 is proved by induction on k. q.e.d. We fix a sequence ak(3) and write ak(i) = E'-3ak(3)eT2k(P1)1l+i(S(), i . 3. The stable element lim ak(i) coincides with the element ak of.' Denote by M. the complex projective space of complex n dimension, and by EkMn the k-fold iterated suspension of MIn(EkMV, = Sk+2). Elf,, is canonically embedded3 in SU(n + 1). There exist cellular mappings4 -: E'M,, -3 EM,,+, such that the degree of r*: H2i+l(E'M.) -* H2i+l(EM,,+,) is i for 1 < i < n + 1. Denote by Er: E3+kM,, -> El +kAI,+1 the k-fold iterated suspension of r and write vk = r o E2r 0 ... o E2(k-) :E2k+,M, -* EM,,+k(r = identity). The degree of *k:H2i+l(E2k+lMn) -> H2i+l(EM,+k) is i!/(i - k)! for k < i . n + k. Consider the following homomorphisms of exact sequences: 7r2n+l(EM-n) i 7r2n+l(EMny EMn-1) 72n (EMn-l) I io* ffI1 io* I io* 7r2,+l(SU(n + 1)) L*7r2nf+l(SU(n + 1), SU(n)) -> r2,,(SU(n)) 7r2n(SU(n + 1)) = O. An orientation of EMn - EMn-1 = e2n+l gives a generator tn of 7r2n+l(EMn, EMn-1) Z such that j*'{ "-'lS2n+l} = n!gn. j* I 8fn-lS2fn+l} generates4 an infinite cyclic group 7r2.+l(SU(n + 1)). Then it follows from the above diagram that 7r2n(SU(n)) = {biO*tn} Zni, and 7r2n(EM-,l) has a direct factor I(,}{ Zn! LEMMA 2. The sum (P-l IS2P+')*ak(2p + 1) + i*Ck+l(3) vanishes under homo- morphism g* induced by any mapping g of EMp_1 into any H-space G, where i is the injection: SI c EMp-1. In particular the sum vanishes under io*: 7r2n(EM,-,)-l r2n(SU(p)) and under the suspension E: r2n(EMl) -- 7r2n+l(E2Mpil), n = (k + 1) (p-1) + 1. Proof: Let T = SV ... VS2P-1. ri'-S2i+ I for j = 1, 2, .. , p - 1 defines a mapping {': T -- EMp_1 which induces mod p isomorphisms of homology and homotopy. Since the order of - (p- 1) !8,n is p and since i*: r2p(S') - 2p(T) is a mod p isomorphism (cf.5), - (p -1) !,, = i* (x a,(3)) for some integer x. At- taching a (2p + 1)-cell by xa,(3), we have a complex Y = S U e2P+l and set L = Y U T. Then t' can be extended to L EMp such that the mapping degree in (2 p + l)-cells is -(p - 1)!. From the definition of Hp, we may compute that Hp(xai(3)) = 1 and thus xa,(3) = al(3). Since t induces mod p isomorphisms t* of homology and homotopy, there exist an integer r 1 (mod p) and an element Downloaded by guest on September 24, 2021 1104 MATHEMATICS: H. TODA PROC. N. A. S. 3E7r2p+i(L) such that -r{IP-1 IS2P+1} = t*O. Let K be the product Y X S' X ... X S2P-1, then we have a split extension 0 7r2p+2(K, L) a ir2p+i(L) -r2,p+l(K) O 0 and a decomposition ,B = il*pl*i3 + i2*32 + ... + ip-10*3p- - b-ay for the injections ii: Y c L, ij:S2J+1 c L, the projection pi:L Y and for some JdEr2w+l(S2i+1) and YEr72p+2 (K,L). For 2 . j < p - 1, the p-component of 7r2p+l(S2J+l) vanishes, hence fj 0 ak(2p + 1) = 0. Then P( 1 |S2P+1)*ak(2p + 1) = rr{v-1 IS2P+1} 0 ak(2p + 1) = t*i1*(p143 0 ak(2p + 1)) - *by, where y' = Ybo'-ak(2p + 1) for bo:lri+i(E2P+2, S2P+1) rl(S2P+1). Apply (A4)pi*,f,for where s = 3, r = 2p, and t = rp. Then pi*13 0 ak(2p + 1)ei*{ai(3), rpL2p, ak(2p)} = i*lrai(3), Pt2p, ak(2p)} = i*{ ai(3), PL2p, ak(2p)} = i*tk-1(3) + i*al(3) 0 T2(k+1)(p-1)+2 (S2V+1) + i*72p+l(S3) 0 a)k(2p + 1) = i*ak+1(3) + {O}. Since t IS3 is the injection, we have ( P-1 S2P+1)*ak(2p + 1) + i*ak(3) = (*ay' for the injection i:S3 c EM i1. Consider g o : L -- G. By the multiplication in G, g o tdefines a mapping g:K -0 G such that g IL is homotopic to g o t. From the commutativity of the diagram 7-i+i(K, L) i7r(L) I * I 9*t* 0 = ir+1(G, G) 7ri(G) it follows that g*( *by') = 0. q.e.d. Consider the following commutative diagram for m . n> 1. i* 7r2n+l(EMny EMn-1) a 72n(EMnA l 72n (SI) E2(m-n) 2(m-n) E2(m-n) (B) 721(2m+1 (E2m-2+lM 2-2n+lM ) a r2m(E2m-2 +lM ) 7r2m(S2m-2n+3) g*m-n wr¢ -n Ir w -n 72m+l(EMm, EMm-i) 7r2m(EMm-1) . 7-ir2m(EMm-n+l) Then ;*,-nE2'm-n)n = (m!/n!) &m and this implies ¢*`E2(m-n)6&n = (m!/n!) 6gm. LEMMA 3. Let n = k(p - 1) + 1. Then the sum i*ak(3) + (-l)k(n!/p)b~n vanishes under the suspension E and under io*: 72n(EMn-1) -7 r72n(SU(n)).
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