By Oscar Zariski

Zariski offers a superb creation to this subject in algebra, including his personal insights.

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On the number of mUltiplications of an H-space, Topology ~ (1963), 205-208. Amer. Soc. 70 (1964), 293-296. I. Berstein and T. Math. (1961), 99-130. A. Borel w. L 1953: Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115-207. 1960: Seminar on transformation groups, Ann. Studies No. , 1960. Browder, Homotopy commutative H-spaces, Ann. of Math. 75 (1962), 283-311. H. -P. Serre, Espaces fibres et groupes d'homotopie, II.

In addition to topological groups this class includes products of odd dimensional spheres, quaternionic, complex and some real Stiefel varieties and other homogeneous spaces. Aside from any intrinsic interest that the results in this section may have, they are included to illustrate how one can make statements on the rank of certain groups arising in homotopy theory whose group operation, being derived from composition of maps, 1S of a very different nature from those considered in the preceding chapter.

N : Hn(K;Q) ~ Hn(y(n) ;Q). 6" a has finite - 30 - order. Consequently we have shown that a is an element or rinite order en (a) = Idoc = Id. 3. F~onamorphism. This completes the Next we state a Proposition whose proor we omit because it ~s based largely on arguments not pertaining to the subject or these notes. ) Ir Proposition. p( 1T (y» > 1 ror some n L N, then n '€ (y) contains a rree subgroup on at least two generators. We next prove the main theorem or this chapter. 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.

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