By Theodor Bröcker, Klaus Jänich
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1 A nilpotent group G is P-local if the function x ~ x p, x ~ G, is bijective for all primes homomorphism ~: G * K p of nilpotent groups di~oint from P-localizes if P. K A is P-local and ~*: Hom(K,L) -~ Hom(G,L) is hljective for all P-local nilpotent groups L. 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.
Mod p Postnikov system of X is determined While the spectral sequence is not absolutely necessary for such an argument, it organizes the computations and reveals the basic facts in a sharp straightforward manner. REFERENCES [0] J. F. Adams, On hhe structure and applications of the Steenrod algebra, [I] W. D. Barcus, Comment. Math. Helv. 32 (1958), 180-214. On a theorem of Massey and Peterson, Quart. J. Math. 19 (1968), 33-41. [2] A. K. Bousfield and D. M. Kan, The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 (1972) , 79-106.
To kill that we have finally embedded Yi is the projection by a fibration WnX/Fi+iWnX qi: Yi ~ Yi-l" G i = riWnX/ri+Iwnx, so that K(Gi,n), so that qi n, where the induced homomorphism >> ~nX/ri~nX, Replace the inclusion Then the fibre of niX = wiYi_l It then follows of course that ~i X qi is Yi ~ Z K(Gi,n), where operates trivially on G i. operates trivially on the homology of is induced by gi: Yi-I ~ K(Gi'n+l)--we take the (negative) transgression of the fundamental class in the fibre. it is clear that the map argument shows that so Z, such that the effect on the in a space homotopy groups occurs only in dimension Wn+l' Wn+2' "''' si qi is n-connected, so that an easy obstruction factors, up to homotopy, uniquely, as Yi qi~ Thus we continue until we have factored qi s i = Si_lq i, Si-l~ Yi-I with all the flbre maps Moreover induced.