IntroductionLet p be a prime number.A simply connected CW-complex X is called mod p decomposable into r spaces if there exist simply connected CW'-complexes X t (l) such that Jf*(J^; Zp) 7^=0, and if there exists a ^-equivalence f: JJ Xi->X.A mod p decomposition JJ X^-^X is l^i^r l-torsion, then as is well known H*(G; Zp)=A(Xi, •••,^t) is the exterior algebra with degx t =2n t -1.We define an integer r(G) to be the number of n t 's which are distinct in Z P -I.Main Theorem.Let G be a simply connected, simple Lie group -without p-torslon.Then if G=^Spi?i (2;/), G is irreduclbly modp decomposable into r(G) spaces and Sphi(2n) is irredncibly modp decomposable into r(G) +1 spaces.

On the cohomology ring of some homogeneous spaces.

Poincaré poylnomial iswhere (2m, -1, ...

IntroductionF o r a fin ite H -com p lex X , th e classical H o p f theorem states that the rational cohomology H *(X ; Q ) is isomorphic to A(xi, the exterior algebra over Q with deg xi o d d .W e c a ll 1 the rank o f X and (deg ..., deg xi) the typ e of X.In the present paper we will consider the homotopy type classification for 1-connected, finite H-complexes of rank 2. In the case --1 * (X ; Z) has no 2-torsion, the classification has been given by Hilton-Roitberg [6] and Zabrodsky [21] as follows:T h eo rem .T h e co m p le te lis t o f homotopy types of 1-connected, 2-torsion free, fin ite H-complexes of ra n k 2 is the f o l l o w i n g : S 3 x S 3 , SU(3), E k (k = 0 ,1 ,3 ,4 ,5 ), S 7 X S 7 , w h ere E k is the p r in cip a l S 3 -bundle o v er S 7 w ith the ch a ra cteristic cla ss k w E ir7(BS 3 ) Z i 2 , w a gen era tor.Thus our object is to classify H-spaces o f rank 2 with 2-torsion.L e t X be a 1-connected, finite H-complex o f ran k 2 such that H *(X ; Z) has 2 -torsion .According to J. R. Hubbuck [7], H*(X ; Z2) ---.-H*(G2;Z 2 ) as Hopf algebras, where 62 is the compact, exceptional L ie group o f rank 2.Let f : V7,2 ->BS 3 be the classifying map of 6 2, ÇO: T A 7,2 -0 ' V 7,2V -5 1 1 th e suitable shrinking m ap , an d a a generator o f 7rii(B5 3 ) suitably

Cohomology mod 3 of the classifying space BF4 of the exceptional group F4

Without Abstract

and T .,"=-(-1 )"'" i n [ S " " ,Apparently, the com position, the sm ash product an d E n a r e compatible with th e homotopy, and (1.1) holds fo r th e homotopy classes of th e maps.