By Oscar Zariski
Zariski presents a superb advent to this subject in algebra, including his personal insights.
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4. Let Y be a I-connected N-dimensional CW-complex whose rational cohomology is an exterior algebra on generators of odd dimension 't (y) has finite rank if and only if all the n. , i = 1, ••• ,k. Then ~ ~ distinct. In this case, k p( ~ (Y» = L (8 i=l ni 8n. -th Betti number or Y. where ~ ~ Proof. 1T n ~ 1 iCY) (Y ) -1 ) Clearly all the n. are distinct if and only if the rank of ~ = 1T (y). Thus if n. = n. for i i: J, then n n ~ J contains a free subgroup on at least two generators by the preceding for all n ~ N, where 1T Proposition.
The classical and exceptional Lie groups for which this is the case are enumerated in Theorem 5 of [Arkowitz-Curjel, 1963I~. 5 does for topological groups what results of James did for spheres [195'U, [1959~. James considers commutative and homotopy.. n n n commutat~ve products S ~ S ~ S whose type ~s def~ned by means of the suspension structure of Sn. He establishes the existence of integers to which our N and Nh are the analogues. It is amusing to note that c c James' and our work meet ~n S3 and Sl, the only spheres which are topological groups.
Hath. I (1955), 169-187. D. Amer. Soc. 77 (195 4 ), 520-551. D. Puppe, Homotopiemengen und ihre induzierten Abbi1dungen I, Math. Zeitschrift 69 (1958), 299-344. M. J. (1935), 50-69. -P. Serre, Homo10gie singuliere des espaces fibres, Ann. of Math. ~ (1951), 425-505. R. Thom, L'homo10gie des espaces fonctionne1s, Co110que de Topo10gie A1gebrique, pp. 29-39, Louvain, 1956. C. Soc. (1949), 213-245. L2 H. , 1958. - 36 - Beschaffenheit der Manuskripte Die Manuskripte werden photomechanisch vervielfliltigt; sie mussen daher in sauberer Schreibmaschinenschrift geschrieben sein.
An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski