Maps from Bπ into X

Pub . Ma . UAB Vol . 30 ns 2-3 Des . 1986 Le n be a ini e g oup and le s Bn be i s classi ying space . Wi h e e y subg oup  yc n  he e is associa ed a co e ing i(y,n) : By - " Bn . I g E n hen mul iplica ionby g on En  indu- ces a map cg : By - B(g -1 yg) . Le E be an in ini eloop space . Then he e is he ollowing exac sequence i*,j* (*) 0 - [Bw ;E] -~ II[Bn ;E]  II  [B(n ngn g-1) ;E] nP  n p ~gER P P P whe e u  is a p-Sylow subg oup o n , p oduc s  II ... and P  n nP  g lc II . . . a e o e all p-Sylow subg oups o all  p  p imes p , i * =  II  i(i  ngn P9 -1 ; 3c )  and  j * =  II  i(gn -1  gnn " n ) oc  . (see (2]) . nP  g  P  P  n P g  P  P '  P  9 F om he sequence (*) i ollows ha a map om Bn o an in ini e loop space is homo opic o ze o i and only i i s es ic- ions o classi ying spaces o all Sylow subg oups a e homo opic o ze o . We wan o see whe he he same s a emen is ue o an a bi- a ysimply connec ed space . Fo example i n = IIx p hen weha e he P ollowing p oposi ion MAPS FROM Bn INTO X Zdzislaw Woj kowiak P oposi ion 1 . I X is simply-connec ed hen P oo . The map VBn p - Bn is a homological equi alence . The e o e p using an obs uc ion heo y we ob ain a equi ed isomo phism o any simply connec ed space X . In u he conside a ions we es ic ou a en ion o a e y smallclass o g oups . Le np be a maximal p-Sylowsubg oup o n Le N(n p )  be a no malizo ' o  np in u and le W p = N(np)/np De ini ion 1 . We say ha n sa is ies W -condi ion i he map p H*(n ;Z(P))  H*(i ;Z(P)) p  is an isomo phism . Examples 1 . I n  is a no maldi iso in n hen W -condi ion is sa is ied . p  p 2 . I n  is abelian hen W -condi ion is sa is ied . p  p 3 . W p -condi ion is sa is ied o he bina yicosahed alg oup I* 4 . I n = GL(n ;F q )  hen W p -condi ion is sa is ied o some No a ion . -  means "is homo opic o" . p-Sylow subg oup o n .) 90 and allp imesp . p imesp [Ba :X] ~z n[Bn p ;X] p Weha e he ollowing sequence o co ib a ions  (up is a maximal i  j  6  S (i) (**)  Bnp -  Bn  --~  Cone (i)  = C  - S (Bn p ) 1 S (Bn)  -~ " " Le  () (P)  deno es he p-comple ion unc o and le  ( ) (P) deno es he p-localiza ion unc o . A e applying ( )(P) o (**) we ob ain he ollowing sequence o co ib a ions (BnP) (P)  = BnP  P, (B n) (P)  p _ C (P)  - C (P)  P - S(BnP) (P) = S(Bwp) S (i) P . S(Bn) (P) = S(Bn) (P) -" Fu he we shall deal onlywi h a case o a ixed p íme p and he e- o e we always d op he índex p in iP,jP,6P, . . . . Theo em 1 .  (F . Cohen [1 ))  I  n  sa is ies  Wp condi ion hen s (i)  : S(B P )  - S(Bn) (P)  has a le in e se  k  , b k : C(P) S(B7 ) (P) - S(Bi P )  is a homo opy equi alence and j  :  (B n) P - C (P)  is homo op ic  o ze o . P oo . Le  k = jWP1 . E e y elemen  g EW P  induces a map h g : Bic P - Bn P  (conjuga ion by g ) . Le N = % S(h9) : S(Bi P ) - S(B, p) g sW and le k-N = k " id-N : S(Bn P ) - S(Bn P ) . One  P easilychecks ha he na u al map = 1 + 2 : S(Bn ) P  Tel(N) Tel(k-N)  is a homo opy equi alence . E e y elemen  g e WP induces also a map hg : Bn " Bi homo opic o he iden i y . Le  N =  S S(h9) _ geW P = k : (SBn) (P) -" (SBn) (P) . The maps Q: Tel (N)  - Tel (N)  and  - , :  (SBn) (P)  - Tel (N)  a e homo opy equi a- lences . Le  i 1 : Tel(N) - Tel(N) Tel(k-N)  be he na u al inclusion . One can check' ha  k = .( 2+ 1)-1o i1oe-1' 1  is a le in e se o S(i)  . The e o e  b k  is a homo opy equi alence . I es s o show ha  j -O . b  has a igh in e se  . This implies ha  j -j , bo . Hence weha e ha  j -0 . Co olla y 1 . I  n  sa is ies  W P -condi ion and  X  is simply-connec ed and p-local hen he map : Bn - X is homo opically i ial i and only i i s es ic ion o B7 p is homo opically i ial . P oo . I  .¡ -0  hen . he e is  ' : C - X  such ha  'oj _ . This implies ha  -O . (7 unde s and i s es ic ions o Tel(N) and Tel(k-N) . L emma 1 . Le us suppose ha X = gY . Then he e is an isomoüphism P oo . Weha e a di ec sys em o spaces The e is he ollowing exac sequence o Milno Le us no ice ha N .N = k " N ( esp . (k-N)a(k-N) = k(k-N))  implies ha ou in e se sys ems sa is y he Mi ag-Le le condi ions . This implies ha lim 1 e ms anish . 0 I  e [SBn p ;S?Y]  and  Y  is p-local hen o any  n e Z (P)  we can de ine n . in he ollowing wo ways . i)  Maps .(S 1 ;Y) = S2Y  has he same hcmo opy ype as Maps .(S( P) ;Y) Fo any ne Z (p)  he e is a map_ n : S 1 - S1 o deg ee n and we de ine n . as a composi ion no . Le us suppose ha we ha e a map : SBn P - X . We wan o [Tel(N) ( esp . Tel(k-N)) ;X] 11  lim [SBn p ;X] N( esp . k-N) SBn N( esp . k-N) P SB n  - . P O - "  lim 1  [SBn  ,  X]  - [Tel(N) ( esp .k-N)) ;X]  -  lim  [SBn P ;X]- O N( esp . k-N) P  N( esp . k-N) ii)  Slh Bn  S1P) ^ B : p  .,The map  n  :  S ~P)  - " S ~P)  induces P n : S~ p) ^ B .p  S( P) n B p . We de ine n " as a composi ion . . Le  : SBn p - " X = 4Y . Le us se  i = k " ( aN)  and  = 1 " ( - (k-N))  . Then  *  }  s lim[SB7 ;X]  and 2  k  1  kn 1 ne{1,2 . . .} N  p * = { 1 }  e lim[SBg ;X] . The e o e by Lemma 1 * and 2  kn 2 ne{1,2 . . .} k-N  p  1 2  de ine maps  * : Tel (N)  - X  and  2 : Tel (k-N)  - X  .  * * es ic ed o  SBn P (i .e .  ( i 2) o  whe e = i + 2 : SBn p - " Tel(N) Tel(k N)  is a sum o inclusions on o he i s segmen s o he mapping elescopes) is homo opic o k  -N  +-'  o (k-N)  =  P opos i ion 1 . The na u al isomo phism * : lim[SB7 ;X]®lim[SBw ;X] - " [SBwp ;X] ÑP k-N P is gi en by (( n) ;(gn)) - 1+ 9 1 . The in e se map is gi en by -  ( *~ ; 2) P oo . Themap : SBn - Tel(N) Tel(k-N)  induces a map P [Tel(N) ;X] ® [Tel(k-N) ;X] - " [SBn p ;X]  which is gi en by he sum o es ic ions o he i s segmen s o he elescopes . This shows he i s pa o he p oposi ion . By he p e ious discussions - ( *, 2)  de ines a map in h e opposi e di ec ion which is he in e se o * . Co olla y 2 . I -N is homo opic o k " hen iii)  o  any  gE W P  we ha e  ha  oS (h 9 ) - P oo . i)  ollows om he de ini ion o E* . We ha e eha ob ^, ( 1 V 2) o eó ^ . io ob - * .P -1.Qo l on  ioe -lo L oS(i) oil ^ .O ii) implies ha he e is  l : SBR - X  such ha  ' .S(i) - . This implies ha  oS (h g ) - - El Co olla y_ 3 . I  X = 0 2 Y  and  X  is simply connec ed hen i : Bn - Bn induces an isomo phism p W L(B) ) p ;X] _ [Bnp ;X] p . . P oo . Weha e ha .z,oS(i)- o 1  . l  and  Q  a e homo opy equi a- lences . The e o e i is enough o show ha W * : [Tel(N) ;X] = lim[SBi ;QY] -+ [SBn p ;QY] p NW is an isomo phism . Le us suppose ha  e[SBn P ;4Y] P  . Then *  _ { ñ 1  k  .N  lim[SBn P= ;oY] and  *( 1)  . This } n,{1,2, ... } e  - implies ha * is an epimo phism . * is also a monomo phism and he e o =_ i is an isomo phism . Theo em 2 . I X is a nilpo en , p-local space and i n sa is ies W p -condi ion hen he na u al map W [Bn ;X] -» [Bn p ;X] P is a su jec ion . I X is a loop space hen W [Bn ;X]  [Bn p ;X] P is a bijec ion . oo . We ha e al eadyp o ed heo em  when X is a double loop space . Le us suppose ha X is a loop and ha X has only a ini e numbe o non- i ialhomo opy g oups . Le us conside a pa o he Pos niko owe o X , - . S?X n- , L  " -  K( nn,n)  -1 X n  c  Xn-1  d  K(ic n n+1) Le us suppose ha he heo em is ue o Xn-l . We ha e he ollow- ing commu a i e diag am [Bic,4X n-1 1 : [B ;K(Rn ,n) 1 b [Bu ;X n 1 -S- [Bn,Xn-11 d [B . jc ;K(7c n 'n+1) 1 114 i  1ijj  1 k  i¡ Q  11 ; m w 1  W b  W c i  w d 1 W [BnP ,ox n-1 1  p  [Bi P ;K(n n 'n) 1  P -+ [Bn P ;Xn1  P y [B c P;Xn-1  P 1 P_ . [Bic ;K(jc n ;n+1) ] P we mus show ha k is a bijec ion . I k(x) = k(y)  hen c(x)  = c(y)  . Hence he e exis s  zE[Bu ;K(n n ;n)]  such ha  z = x -1 . y . This  implies ha .  1 . 3 (z)  = k(x)  k (y)  . The e o e he e  is w E [B3  ; 4X n-1 ]  such  ha  al (w)  = j (z)  .  Le  w1= k  E  woh g  .  Then gEw p W a l (w 1 ) = j (z)  and  w 1 E [Bnp ;QXn-11 P . The e is  E [Bn ;QX n-1 1 such ha  i ( )  = w 1  .  we ha e  j (a ( ))  = a l (i ( ) )  = a l (w 1 )  = j (z)  .  This implies ha  a ( )  = z  and he e o e  x=y W Le us suppose ha xE [BKp ;Xh1 P and le y E F-1 (c 1 (x) ) The e exis s z such ha c(z) = y because d(y) = O . We ha e ha  c1 (k(z))  = c 1 (x)  . The e o e he e is  w E [Bn p ;  K(n n ;n) 1  such ha  b1 (w)  =x " k 1 (z)  .  Le  w 1 = B  wah g  . Then  b 1 (w 1 )  _  (x " k -1 (z)) k . g EW I OllOws om he s anda d  P p ope ies o ib a ions há 1  w (x " k (z)  k )  lies  in he cen e o  [B7cP ;Xn1  P  . The e o e b1 (k w 1 )  =x " k(z) _1 . We ha e also ha  b 1 ( k w 1 )  - k(b ( k w 1 ) )  " This implies ha x Eim k . I es o show he heo em o an a bi a y nilpo en , p-local space X . We use once mo e he Pos niko owe o X and h e same diag am as be o e . Themap i is an isomo phism because QX n-1 is a loop space . We assume, ha Q is su jec i e . To show ha k is su jec i e wemus use he ollowing lemma . Lemma 2 . Le M be a ini ely gene a ed  Z (P) -module . Le us suppose ha he abelian g oup . M ac s on a se X in such a way ha iso- opy subg oups a e  Z(P)-submodules o  M . Wedeno e ' his ac ion by * . Le us suppose u he ha a ini e g oup G ac s on M and on X , he ac ion o G on M is . Z(P)-linea , he o de o G  is  k e z* p)  and  h 9 *xg  =  (h*x) g I  x,x 1 E X G and  w*x = x l  hen  ( £  k W 9 ) * X = xl g EG P oo . w*x = x 1 and  x,x 1 EX G  imply ha  wg *x = x1  o each  9E G wg* (w-wg) *x))  = wg*x  implies ha  (w-w 9 ) *x = x  o each  g E G The e o e  (k £ (w-w g ))*x = x . Weha e ha k £ wg+ k £ (w-w g )= w- gEG  gEG gEG This implies  (k  £  w g ) *x = x l . I~ gEG The ac ion o  [Bi p ;K(n n ;n)]  on  [B p ;Xn ]  sa is ies he assump ions o Lemma 2 . Wep o e ha k is su jec i e in he same way as o a double'loop space . Weha e ha  c 1 (k (z) )  = c 1 (z)  . The e- o e he e is w such ha wkk(z) = x . I ollows om Lemma 2 ha (  £  k (w,h  ) ) *k (z)  = x  .  (  £  k (w,h  ) )  =  j (wl)  implies  ha 9 EW  9  9EW  9 P  P k(w i* z) = x . The spaces Bn and Bi Pha e only ini e homology g oups he e o e weha e isomo phisms 96 [Bn ;X]  lim[Bu,X n 1 and  . [Bn p ;X]  Z lim[Bn p ,X n ]  .  (I  {X n } nEN  'is n  n an in e se sys em o p-comple e spaces hen he unc o lim[ ;X ] nn is ep esen able by Sulli an i .e .  lim[ ;Xn ] _ [ ;Z]  and Z = holim X n . n In ou case holim(X)  = X  and np [B n  (o  Bn p ) ;  x  (o  X n ) ]  _  [B n  (o  Bi p ) ;X p (o (X n)p) ] because Bi and Bn p ha e ini e homo opy g oups .) Weha e he ollowing commu a i e diag am [B ;X] lim[Bi ;X R : n] n W p  p 1  W [Bn p , X]  3 lim[Bnp  n ;X] p n p is an isomo phism, b is an epimo phism ( esp . isomo phism i X is a loop space) and p 1 is a monomo phism . This implies ha a is an epimo phism ( esp . isomo phism i X is a loop space) . This inishes he p oo o Theo em 2 . I we analize he p oo s ca e ully hen i appea s ha in ac weha e p o ed much mo e gene al esul . Le us suppose ha a ini e g oup G ac s homo opically on a space  X  ,i .e . he e is a homomo phism  G - n 0(e (X) )  whe e  e(X) is he space o all homo opy equi alences o X . Le us suppose ha 1 G 1 = k , X is p-local and k E Z( p) . By he esul o Cooke he e is a space X 1 wi h a ee ac ion o G and a homo opy equi alence i : X - " X 1 which is homo opy equi a ian wi h espec o he homo opy ac ion o G .