By Peter John Hilton, etc.
Hilton P., Mislin G., Roitberg J. Localization of nilpotent teams and areas (Amsterdam NH 1975)(ISBN 0720427169)
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Additional resources for Localization of nilpotent groups and spaces (Amsterdam NH 1975)(ISBN 0720427169)
There is a d u a l theorem t o Theorem 2 . 7 concerning t h e upper c e n t r a l series of G which, however, r e q u i r e s more d i f f i c u l t t o prove. G t o be f i n i t e l y generated and is We c o n t e n t o u r s e l v e s h e r e w i t h a s t a t e m e n t of t h e r e s u l t , r e f e r r i n g t o [34 ] f o r d e t a i l s . 8. i z ( e ) = el z (G) i e: G -f is P-localization, Gp i i z (G) i n t o z ( G ~ ) . Moreover, carries if G Z (G) P-localizes and G € N If i 21 then t h e r e s t r i c t i o n z i ( e l : z i (GI + z i ( G ~ ) is f i n i t e l y generated.
We proceed as usual by induction on nil(w). 13) yields an exact sequence by the half-exactness of F. 3. 16. Note that, in fact, nil (Fo) Let w € AV(Q,A) Then the induced actions of C nil w. and l e t B be an arbitrary abeZian group. Q on A 8 B, Tor(A,B) and H,(K;A), K any group & t h t r i v i a l action on A, are nilpotent. 17. If w € Av (Q,A) , then the induced action of Q on H,(A;C) ,c triu-ial A-moduZe, is nilpotent. ) Proof. In case nil(w) = 1, the result is clear. 13). We have A/A2 acting trivially on H,(A2;C).
X X C H1, If and i f P is a family of primes, we say t h a t is P-zocal i f the homotopy groups of W e say t h a t f: X + Y P-localizes H1 in X a r e a l l P-local a b e l i a n groups. X if Y i s P-local and* f*: [Y,Z] z [X,Zl f o r a l l P-local 2 C H1. Of course t h i s u n i v e r s a l property of c h a r a c t e r i z e s i t up t o canonical equivalence: both P-localize H1 with in H1. X hfl = f 2 . if fi: X -+ then t h e r e e x i s t s a unique equivalence Yi, f i = 1, 2 , h : Y1 PI Y2 in W e w i l l prove t h e fallowing two fundamental theorems The f i r s t a t t e s t s t h e e x i s t e n c e of a l o c a l i z a t i o n theory i n H1 and the second a s s e r t s t h a t we may d e t e c t t h e l o c a l i z a t i o n by looking a t induced homotopy homomorphismor induced homology homomorphisms.