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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. Applications. Paris 234 (1952), 393-395. A. Dold and R. Math. 3 (1959), 285-305. - B. J.

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. We adopt the following conventions: All group operations are written multiplicatively. The same symbol is used for a map A + B and its homotopy class in [A,B). We let C(B) denote the group of homotopy classes of homotopy equivalences of the space B, where the group operation is defined by composition of maps.

Y) = . = l. - 1). S now complete. 5. 4. Then ~(y) is finite if and only if 6 (y) = 1 for all i = l, ••• ,k. n. l. If C(y) is finite then plainly Proof. 4. If on the other hand n. l. the n. are distinct. Thus p(n ) ~ 1 for all n ~ l. n finite group. It then follows from the sequence proof that t (y) is finite if ~ (y(N» is. But 6 ~ has rank zero, and so all Bn. (y) = 1, then all l. 2 ~ (y(N» ~ is an S-group (see Chapter 2). S finite if its rank is zero. However, p( Cf«y(N») = p( C(y» k = = Thus c (y(N»