1102 : 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 "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 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 . (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 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- 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 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 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)). Proof: We can see the i*a,(3) = - (p-l )b in the proof of Lemma 2. Assume that the lemma is true for some k. Let m = (k + 1) (p - 1) + 1 and con- sider (B). Then by Lemma 2, (_ 1)+l(m!/p)b~m = (_1)k t*P-lE2P-2(n!/p)bEn = D*P-'E2P-2i* aCk(3) = -i*i*'ak+1(3) + y for it:S3 C EMp_1 and ycE-1(0) n io*- (0). Then the lemma is proved by induction on k. q.e.d. THEOREM 1. Let p be an odd prime. Let m = k(p - 1) + I for some integers k . 1 and p > 1 . 1. Then the p-primary component of 7r2m(SU(j)) is not zero for m . j > 1. In fact, i**l-lak(21 + 1) is of order p for the injection homomorphism i*: r2m(SMz) 1 7r2m(SU(j)). Proof: Let n = k(p - 1) + 1 and m = k(p - 1) + 1 in (B). It follows easily from Lemma 3 that i*r* 1- ak(21 + 1) = (-1)k+I(m!/p)bio*gm and this is of order p. q.e.d. By use of the injections Sp(n) c SU(2n) and SU(n) c SO(2n) and by use of an isomorphism 7r4n+2(SU(2n + 1)) -] r4n+2(SO(4n + 2)) of p-primary components6 we have COROLLARY. Let n = k(p - 1)/2 + Ifor k _ 1 and (p - 1)/2 > 1 . 0. Then Downloaded by guest on September 24, 2021 VOL. 46, 1960 MICROBIOLOGY: H. RUBIN 1105

the p-primary component of 7r4.+2(Sp(j)) is not zero for n > j> 1. The p-primary component of 7r4±+2(SOO"')) is not zero for 4n + 2 > j' > 41 + 4. THEOREM 2. Let p be an odd prime and let k . 1 and i . 3. Then the p-primary component of r2k(Pl)-l+(Si) is not zero. Infact, ak(i) is of order p. Proof: Obviously pak(i) = 0. Let n = k(p - 1) + 1 and let m be an integer such that 2m - 2n > i - 3. Then it follows from (B) and Lemma 3 that i ak(2m - 2n + 3) = (-1)k+1(m!/p) 68m $ 0. Thus'ak(i) $ 0. q.e.d. We say that an element ae7r,(S8) is decomposable if a is the sum of some composi- tions ,l o 'y of 13er,(S-) and yiEir,(ST) for q > r > s. If a is decomposable, then so is Ea. THEOREM 3. ak(s), S _ 3, is not decomposable. Proof: Consider (B) of n = k(p - 1) + 1 and 2m - 2n > s - 3. Assume that atk(s) is decomposable: ak(s) = Z13 o -i. Then 0 $ (- 1)+l(m!/p)io*8m = i0*i*r*"-afk(2M- 2n + 3) = 2(io*i*¢*m1-nE` fi) 0 Et'yi, t = 2m - 2n - (s -3) > 0. Since E'#34 belongs to a finite group Etirr(SS) 2n - 3 + s > r > s and since 7r,+, (SU(m)) - Z or 0 for 2m > r + t, then io*i*r*mnEt 8i = 0 and thus (m!/p) do*&. = 0. But this is a contradiction. Thus ak(S) is not decomposable. q.e.d. * This research wag supported in part by the Air Force through the Office of Scientific Research of the Air Research and Development Command. I Toda, H., Memoire Univ. of Kyoto, 32,297-332 (1959). 2 Ibid., 31, 143-160 (1958). 3 Yokota, I., J. Inst. Poly. Osaka City Univ., 8,93-120 (1957). 4 Toda, H., Memoire Univ. ofKyoto, 32, 103-119 (1959). Hilton, P. J., J. London Math. Soc., 30, 154-171 (1958), and Serre, J.-P., Ann. of Math., 54, 425-505 (1951). 6 Karvaire, M., these PROCEEDINGS, 44, 280-283 (1958).

A VIRUS IN CHICK EMBRYOS WHICH INDUCES RESISTANCE IN VITRO TO INFECTION WITH ROUS SARCOMA VIRUS* BY HARRY RUBIN

DEPARTMENT OF VIROLOGY AND VIRUS LABORATORY, UNIVERSITY OF CALIFORNIA AT BERKELEY Communicated by Robley C. Williams, June 13, 1960 When chick embryo cells are infected in vitro with Rous sarcoma virus (RSV) under the appropriate conditions they grow into foci of Rous sarcoma cells. The enumeration of these foci serves as an accurate assay for the infectious titer of a RSV preparation. In occasional sets of chick embryo cultures, however, the effi- ciency of infection with a standard RSV stock is reduced by a factor 50 or more. Once established, the resistance of such cultures to RSV infection is maintained indefinitely under all physiological conditions tested. This state of cellular resistance, in which multiplication of RSV is prevented, is distinct from those physiological effects which merely suppress characteristic morphologic alterations in cells supporting multiplication of RSV.' The present paper describes the detection of resistant cultures and the characterization of the Downloaded by guest on September 24, 2021