Download e-book for kindle: Localization of nilpotent groups and spaces (Amsterdam NH by Peter John Hilton, etc.

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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Example text

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.

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