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By L. F. McAuley

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4p + 2q - 3 > N integer satisfying all of the following; A sufficient where q is some 46 (a) 0

J. Hilton and Joe Roitberg, On principal S3-bundles over sphere, Ann. of Math. 90 (1969), 91-107. 7. M. Mimura, G. Nishida and H. Toda, Localization of CW-complexes and its applications, J. Math. Soc. Japan, 23 (1971), 593-624. 8. G. Mislin, The genus of an H-space, Lecture Notes in Mathematics 249, Springer (1971), 75-83. 9. A. Sieradski, Square roots up to homotopy type, Amer. J. Math. 94 (1972), 73-81. i0. J. Stasheff, Manifolds of the homotopy type of (non-Lie) groups, Bull. A. M. S. 75 (1969), 998-1000.

2 Every nilpo~ent group admit8 a P-localization. 3 n ~ I. A map Let X E N. f: X ~ Y in Then N X is P-local if P-localizes if f*: [Y,Z] ~ [x,z] for all P-local Z in N. Y ~ X is P-local for all n is P-local and 34 We now come to the main theorems of the paper. Theorem 4A Every X in N admits a P-localization. Theorem 4B Let f: X ~ Y (i) f (ii) in N. P-localizes X; Y P-localizes for all n ~ i; Hnf: HnX -~ H Y P-localizes for all n ~_ i. ~ f: , X - ~ n (iii) Then the following statements are equivalent: n n n The pattern of proof of these theorems will closely:resemble that of Theorems 2A, 2B.

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