SOME HOMOTOPY GROUPS OF STIEFEL MANIFOLDS 1

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1 SME HMTPY GRUPS F STIEFEL MAIFLDS 1 BY C. S. H AD M. E. MAHWALD Counicated by W. S. Massey, February 1, 1965 Paechter [7] ade soe putations of Tk p (Vk,) where Vk, is the Stiefel anifold of fraes in k space. In this note we give a table (Table 1) extending his results in the case where is large. Since Vk, >Vki,i >S k is a fibering it is clear that Tk p (Vk,) depends only on k and p for p^ra. This is called the stable range and we feel that these stable groups are the ost iportant ones. n the other hand for sall values of ra, one of us [] has ade extensive putations and the results are available. Jaes' periodicity [5, Theore 3.1] is reflected in the table but the basic periodicity of period is also present. In [l] it is proved that if n > 1, then TT ; (5(W)) = TT/S) 7r J i(fn,n) for j<n 1. Hence it is easy to deduce the first fourteen nonstable groups of S(n) fro this table. Tables of hootopy groups are uch ore useful if generators are given. Instead of generators we settle for giving the order of the iage of i*: 7rjb p(5*) *Trkp(Vk, ) (Table ). ne can nstruct the generators fro this inforation and this ap has iportant nnections with Whitehead products []. The groups have been puted by using odified Postnikov towers [6]. An outline of the putation for one case, 6 od 3, is given. The case k = 6 od 3. This procedure is essentially the sae as the Adas spectral sequence ethod. Let = 3»6 and we suppose ra is large. Consider the fibering F3n«,7-*Fn,i-^F n,-6. We are only interested in groups in the hootopy stable range so that we can nstruct a new fibering ~ Vzn,-& ""* Vzn6,1 ~~* Vzn,l' We will build the odified Postnikov tower to this fibering. By [3] the hoology of T^n.i is given by ^'(P^n.i; Z ) =, < i < 3fl - 1. = Z, 3n - 1 ^ i ^ 3» - 1. Let hi generate H i (Vzn,i) Z ) when it is nonzero. Then Sq*hi 1 This research was supported by a grant fro the U. S. Ary Research ffice (Durha). 661

2 66 C. S. H AD M. E. MAHWALD [July I J ) J ( I J ^ J S S t-» rh Î r 1 J v T J I ^ r-1 t^ J I W r-h c* J Csl <* ^Tf î 'M v c* < f I * rh "* C* I e* I l 'W I <* C^ J cst cî T vo r-1 J l I! J tb J I I i 't J t i-h "e* I J <"<* 5(3) Z I "fcl S i t I *î vo l rh ^ I i «I J I I T I I I I < J fr I < I I ( J' I M < W < J ) "k v i» in i-h J ( I iu J. <M ^ C* I irt ^< J 9 vj I 1 "*

3 1965] SME HMTPY GRUPS F STIEFEL MAIFLDS 663 J J J I I «" l J oo J l J -* vo «a ss a ^c W < "" *->» ^ '-*» ^f S 3. r- 5» r-l l l vo vo -* vo vo I ^jp; ta "fe S V eo \o H r J l "feo "to " "^ " T J l J a s s ^ cf ci ci et «-» «a» l r-l t-i I ^ ] J J ^ * 'S? *""* ^" ^ ^-f ^C 'ir' H I-H *-» V w *_* r-l r-h r-l t/3 1-1 f J *< l J -, _. s s s s I J I I * Ç \ 3 rh «K S d H -«* r-l J J I I I I s I s a S Ci Ct o rh * r-h in i-h I I et. I «I I ««J» 1/5 «-H i-* I w w r-«"f* r-* l r-h J l ^ J vo l l CS I I I I S c- ^ oc So <*

4 66 C. S. H AD M. E. MAHWALD [July TABLE rder of i(fc et$ n )->7iï( V*n,n)) Top row is the nae of the ste»() i V 17 V p* a V 17* 77 77P rpa M i7ju r oo 1 oo 3 115(1) oo 5 53(6) 6 oo (3) 7 15 = (J)A» y. Hence the hoology of the base space is given by hnn = 5c7* 1 A n-i. We let /W-i = A. 1st level. ver the Steenrod algebra the basis for ker p* is given by {Sq 7 h, Sq s h, Sq l *h}. f these three only the first two can be spherical in the sense of [6]. Indeed using ii Vk, *BS(k) each class in Trj(Vk,) represents a fe-plane bundle over S' which bees trivial when sued with a trivial -plane bundle. It is also easy to see that the bundle is a fraed tangent bundle of S' if and only if the hoology ap is nontrivial. Since the 153«sphere has only an eight field, Sq l *h is not spherical. It is useful to kill it anyway but one has to be careful and identify the eleent at a later stage which is produced because of this. nd level. Consider the following fibering Kx(Z, 3n 5) K (Z, 3n 6) K(Z, 3n 1) with ^-invariants Sq 7 h, Sq*h and Sq h. i q ~* E 1 Vzn t l

5 I96j] SME HMTPY GRUPS F STIEFEL MAIFLDS 665 * «o ) J 5 * ^ eo^ ff 1 i i S tiikov tower 1 ified Post A od jr f c? 1 of y cr. CJ. ' < er. Cl * ir a» 59- ' T er ' 5 r esr <*, " 1 QQ. Cl eo P- 1/5 1r 1 CJ tf \ -1 S 1 "Sr CJ^ eo * «y Cl cî * i-t *3r cr M eo* ^cr CI Ir j. cô* «g* x 5 1r cî 1 "cr x* eo^ l cr Ê CJ <* eo "è èfs f f 1 cç iô A cfi <ô 1-» t. > x-r* Ir «r «S 1 «S * <3^ rs " -l i "L "* î^ x x 3 t» ^5cr x * o * t^

6 666 C. S. H AD M. E. MAHWALD [July PRPSITI. A class vgh 3 \E 1 1Z ) suchthatvç iq*andj^6n 3 satisfies: i*v?-i &&% where ai is the fundaental class of Ki and /3* is an eleent of the Steenrod algebra such that pisqtfasqtfizsq 1 *, as an eleent in the Steenrod algebra, has only classes of length or ore in its Cartan basis representation. Using this representation of H*(E ) it is now just a lengthy but straight forward putation to verify that the classes in Table 3, lun do for a basis over the Steenrod algebra for H J '(E l ) if 3w7^j = 3tt1. 3rd level. Consider the fibering with ^-invariants given by Table 3. We use & to represent also the fundaental class of Ki. The value of w,- can be inferred fro the table. Consider the diagra i * à* H*(E ) -> H*(TKi(Z,»<)) -* H*(E\ wki) \ î^ T H*(E l ) -? H*(Kx K K z ) PRPSITI. A classv(elh 3 '(E ), 7 j 3n 1, by a su ]C?-i a A satisfying: (1)*>= J^i-idiPiand () I(*ÎT&)=. is defined uniquely This is a special case of 3.3. of [6]. Using this proposition the hoology of E? in the interesting range can be puted. Another lengthy putation shows that lun 3 of Table 3 is a basis over the Steenrod algebra for Hi(E ), 7 ël/ 3n = 1. th and higher levels. The putations are ade as in the third level, using 3.3. of [6]. othing unusual happens. The class rresponding to 76SB is the extraneous class produced by killing Sq l %. This follows fro Toda []. It is ausing to note that the forula of Adas [o] Sqi* = J^aij tz <l>u with efficients, for exaple, 3,3,3 = Sq 1 and #1,3,3 = Sq 7 Sq*Sq Sq l, essentially given by ye.

7 1965] SME HMTPY GRUPS F STIEFEL MAIFLDS 667 BIBLIGRAPHY. J. F. Adas, n the non-existence of eleents of Hopf invariant one, Ann. of Math. () 7 (196), M. G. Barratt and M. E. Mahowald, The etastable hootopy of (n), Bull. Aer. Math. Soc. 7 (196), , The etastable hootopy of S n (to appear). 3. A. Borel, La hoologie od de certains espaces hoogènes, Coent. Math. Helv. 7 (1953), C. S. Hoo, Hootopy groups of Stiefel anifolds, Ph.D. Thesis, Syracuse University, Syracuse,. Y., 196 (ieographed notes, orthwestern University). 5. I. M. Jaes, Cross-sections of Stiefel anifolds, Proc. London Math. Soc. (195), M. E. Mahowald, bstruction theory in orientablefiberbundles, Trans. Aer. Math. Soc. 11 (196), G. F. Paechter, The group 7r r (F, ). I, Quart. J. Math. xford Ser. 7 (1956), H. Toda, Vectorfieldson spheres, Bull. Aer. Math. Soc. 67 (1961), -1. UIVERSITY F ILLIIS AD RTHWESTER UIVERSITY