Maps from Bπ into X

Pub . Mat . UAB Vol . 30 ns 2-3 Des . 1986 Let n be a finite group and lets Bn be its classifying space . With every subgroup  yc n  there is associated a covering i(y,n) : By - " Bn . If g E n then multiplicationby g on En  induces a map cg : By - B(g -1 yg) . Let E be an infiniteloop space . Then there is the following exact sequence i*,j* (*) 0 - [Bw ;E] -~ II[Bn ;E]  II  [B(n ngn g-1) ;E] nP  n p ~gER P P P where u  is a p-Sylow subgroup of n , products  II ... and P  n nP  gtlc II . . . are over all p-Sylow subgroups for all  p  primes p , i * =  II  i(ir  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 From the sequence (*) it follows that a map from Bn to an infinite loop space is homotopic to zero if and only if its restrictions to classifying spaces of all Sylow subgroups are homotopic to zero . We want to see whether the same statement is true for an arbitrarysimply connected space . For example if n = IIx p then wehave the P following proposition MAPS FROM Bn INTO X Zdzislaw Wojtkowiak Proposition 1 . If X is simply-connected then Proof . The map VBn p - Bn is a homological equivalence . Therefore p using an obstruction theory we obtain a required isomorphism for any simply connected space X . In further considerations we restrict our attentionto a very smallclass of groups . Let np be a maximal p-Sylowsubgroup of n Let N(n p )  be a normalizor' of  np in u and let W p = N(np)/np Definition 1 . We say that n satisfies W -condition if the map p H*(n ;Z(P))  H*(ir ;Z(P)) p  is an isomorphism . Examples 1 . If n  is a normaldivisor in n then W -condition is satisfied . p  p 2 . If n  is abelian then W -condition is satisfied . p  p 3 . W p -condition is satisfied for the binaryicosahedralgroup I* 4 . If n = GL(n ;F q )  then W p -condition is satisfied for some Notation . -  means "is homotopic to" . p-Sylow subgroup of n .) 90 and allprimesp . primesp [Ba :X] ~z n[Bn p ;X] p Wehave the following sequence of cofibrations  (up is a maximal i  j  6  S (i) (**)  Bnp -  Bn  --~  Cone (i)  = C  - S (Bn p ) 1 S (Bn)  -~ " " Let  () (P)  denotes the p-completion functor and let  ( ) (P) denotes the p-localization functor . After applying ( )(P) to (**) we obtain the following sequence of cofibrations (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) -" Furtherwe shall deal onlywith a case of a fixed príme p and therefore we always drop the índex p in iP,jP,6P, . . . . Theorem 1 .  (F . Cohen [1 ))  If  n  satisfies  Wp condition then s (i)  : S(BrtP )  - S(Bn) (P)  has a left inverse  k  , bvk : C(P)v S(B7r) (P) - S(Bir P )  is a homotopy equivalence and j  :  (B n) P - C (P)  is homotop ic  to zero . Proof . Let  k = jWP1 . Every element  g EW P  induces a map h g : Bic P - Bn P  (conjugation by g ) . Let N = % S(h9) : S(BirP ) - S(B,rp) g sW and let k-N = k " id-N : S(Bn P ) - S(Bn P ) . One  P easilychecks that the natural map r=r 1 + r2 : S(Bn ) P  Tel(N)v Tel(k-N)  is a homotopy equivalence . Every element  g e WP induces also a map hg : Bn " Bit homotopicto the identity . Let  N =  S S(h9) _ geW P = k : (SBn) (P) -" (SBn) (P) . The maps Q: Tel (N)  - Tel (N)  and  r - , :  (SBn) (P)  - Tel (N)  are homotopy equivalences . Let  i 1 : Tel(N) - Tel(N)v Tel(k-N)  be the natural inclusion . One can check'that  k = .(r 2+ r1)-1o i1oe-1'r1  is a left inverse to S(i)  . Therefore  b v k  is a homotopy equivalence . It rests to show that  j -O . b  has a right inverse  t . This implies that  j -j , bot . Hence wehave that  j -0 . t Corollary 1 . If  n  satisfies  W P -condition and  X  is simply-connected and p-local then the map f : Bn - X is homotopically trivial if and only if its restriction to B7r p is homotopically trivial . Proof . If  f .¡ -0  then .there is  f' : C - X  such that  f'oj _ f . This implies that  f -O . (7 understand its restrictions to Tel(N) and Tel(k-N) . L emma 1 . Let us suppose that X = gY . Then there is an isomoüphism Proof . Wehave a directsystem of spaces There is the following exact sequence of Milnor Let us notice that N .N = k " N (resp . (k-N)a(k-N) = k(k-N))  implies that our inverse systems satisfy the Mitag-Leffler conditions . This implies that lim 1 termsvanish . 0 If  f e [SBn p ;S?Y]  and  Y  is p-local then for any  n e Z (P)  we can define n .f in the following two ways . i)  Maps .(S 1 ;Y) = S2Y  has the same hcmotopy type as Maps .(S( P) ;Y) For any ne Z (p)  there is a map_ n : S 1 - S1 of degree n and we define n .f as a composition nof . Let us suppose that we have a map f : SBn P - X . We wantto [Tel(N) (resp . Tel(k-N)) ;X] 11  lim [SBn p ;X] N(resp . k-N) SBn N(resp . k-N) P SB n  - . P O - "  lim 1  [SBn  ,  X]  - [Tel(N) (resp .k-N)) ;X]  -  lim  [SBn P ;X]- O N(resp . k-N) P  N(resp . k-N) ii)  Slh Bn  S1P) ^ B :r p  .,The map  n  :  S ~P)  - " S ~P)  induces P n : S~ p) ^ B .p  S( P) n Bf p . We define n " f as a composition f . . Let  f : SBn p - " X = 4Y . Let us set  f i = k " (faN)  and f  = 1 " (f- (k-N))  . Then  f*  f }  s lim[SB7r ;X]  and 2  k  1  kn 1 ne{1,2 . . .} N  p f* = { 1 f }  e lim[SBg ;X] . Therefore by Lemma 1 f* and 2  kn 2 ne{1,2 . . .} k-N  p  1 f2  define maps  f* : Tel (N)  - X  and  f2 : Tel (k-N)  - X  .  f* v f* restricted to  SBn P (i .e .  (fi v f2) or  where r = ri + r 2 : SBn p - " Tel(N)v Tel(k\N)  is a sum of inclusions onto the first segments of the mapping telescopes) is homotopic to k  f -N  +-'  f o (k-N)  =  f Propos ition 1 . The natural isomorphism r* : lim[SB7r ;X]®lim[SBwr ;X] - " [SBwp ;X] ÑP k-N P is given by ((fn) ;(gn)) - f1+ 9 1 . The inverse map is given by f -  (f*~ ;f2) Proof . Themap r : SBn - Tel(N)v Tel(k-N)  induces a map P [Tel(N) ;X] ® [Tel(k-N) ;X] - " [SBn p ;X]  which is given by the sum of restrictions to the first segments of the telescopes . This shows the first part of the proposition . By the previous discussions f - (f*,f2)  defines a map in th e opposite direction which is the inverse of r* . Corollary 2 . If f-N is homotopic to k " f then iii)  for  any  gE W P  we have  that  f oS (h 9 ) - f Proof . i)  follows from the definition of E* . We have ehat fob ^, (f1 V f2) oreó ^ . fiorob -f* .P -1.Qor l on  fioe -lor L oS(i) oil ^ .O ii) implies that there is  fl : SBR - X  such that  f' .S(i) - f . This implies that  foS (h g ) - f - El Corollary_ 3 . If  X = 0 2 Y  and  X  is simply connected then i : Bn - Bn induces an isomorphism p W L(B)r) p ;X] _ [Bnp ;X] p . . Proof . Wehave that .z,oS(i)-f o r 1  . r l  and  Q  are homotopy equivalences . Therefore it is enough to show that W r* : [Tel(N) ;X] = lim[SBi ;QY] -+ [SBn p ;QY] p NW is an isomorphism . Let us suppose that  f e[SBn P ;4Y] P  . Then f*  _ { ñ 1  k  f .N  lim[SBn P= ;oY] and  r*(f1)  f . This } n,{1,2, ... } e  - implies that r* is an epimorphism . r* is also a monomorphism and therefor=_ it is an isomorphism . Theorem 2 . If X is a nilpotent, p-local space and if n satisfies W p -condition then the natural map W [Bn ;X] -» [Bn p ;X] P is a surjection . If X is a loop space then W [Bn ;X]  [Bn p ;X] P is a bijection . roof . We have alreadyproved theorem  when X is a double loop space . Let us suppose that X is a loop and that X has only a finite number ofnon-trivialhomotopy groups . Let us consider a part of the Postnikoff tower of X , - . S?X n- , L  " -  K( nn,n)  -1 X n  c  Xn-1  d  K(ic n n+1) Let us suppose that the theorem is true for Xn-l . We have the following commutative diagram [Bic,4X n-1 1 : [Brt ;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  1f k  ti¡ Q  11 ; m w 1  W b  W c i  w d 1 W [BnP ,ox n-1 1  p  [Bit P ;K(n n 'n) 1  P -+ [Bn P ;Xn1  P y [Brc P;Xn-1  P 1 P_ . [Bic ;K(jc n ;n+1) ] P we must show that k is a bijection . If k(x) = k(y)  then c(x)  = c(y)  . Hence there exists  zE[Bu ;K(n n ;n)]  such that  z = x -1 . y . This  implies that .  1 . 3 (z)  = k(x)  k (y)  . Therefore there  is w E [B3t  ; 4X n-1 ]  such  that  al (w)  = j (z)  .  Let  w1= k  E  woh g  .  Then gEw p W a l (w 1 ) = j (z)  and  w 1 E [Bnp ;QXn-11 P . There is  v E [Bn ;QX n-1 1 such that  i (v)  = w 1  .  we have  j (a (v))  = a l (i (v) )  = a l (w 1 )  = j (z)  .  This implies that  a (v)  = z  and therefore  x=y W Let us suppose that xE [BKp ;Xh1 P and let y E F-1 (c 1 (x) ) There exists z such that c(z) = y because d(y) = O . We have that  c1 (k(z))  = c 1 (x)  . Therefore there is  w E [Bn p ;  K(n n ;n) 1  such that  b1 (w)  =x " k 1 (z)  .  Let  w 1 = B  wah g  . Then  b 1 (w 1 )  _  (x " k -1 (z)) k . g EW It fOllOws from the standard  P properties of fibrations thát 1  w (x " k (z)  k )  lies  in the center of  [B7cP ;Xn1  P  . Therefore b1 (k w 1 )  =x " k(z) _1 . We have also that  b 1 ( k w 1 )  - k(b ( k w 1 ) )  " This implies that x Eim k . It rest to show the theorem for an arbitrary nilpotent, p-local space X . We use once more the Postnikoff tower of X and th e same diagram as before . Themap i is an isomorphism because QX n-1 is a loop space . We assume,that Q is surjective . To show that k is surjective wemust use the following lemma . Lemma 2 . Let M be a finitely generated  Z (P) -module . Let us suppose that the abelian group . M acts on a set X in such a way that isotropy subgroups are  Z(P)-submodules of  M . Wedenote ' this action by * . Let us suppose further that a finite group G acts on M and on X , the action of G on M is . Z(P)-linear, the order of G  is  k e z* p)  and  h 9 *xg  =  (h*x) g If  x,x 1 E X G and  w*x = x l then  ( £  k W 9 ) * X = xl g EG Proof . w*x = x 1 and  x,x 1 EX G  imply that  wg *x = x1 for each  9E G wg* (w-wg) *x))  = wg*x  implies that  (w-w 9 ) *x = x  for each  g E G Therefore  (k £ (w-w g ))*x = x . Wehave that k £ wg+ k £ (w-w g )= wgEG  gEG gEG This implies  (k  £  w g ) *x = x l . I~ gEG The action of  [Bit p ;K(n n ;n)]  on  [B p ;Xn ]  satisfies the assumptions of Lemma 2 . Weprove that k is surjective in the same way as for a double'loop space . Wehave that  c 1 (k (z) )  = c 1 (z)  . Therefore there is w such that wkk(z) = x . It follows from Lemma 2 that (  £  k (w,h  ) ) *k (z)  = x  .  (  £  k (w,h  ) )  =  j (wl)  implies  that 9 EW  9  9EW  9 P  P k(w i* z) = x . The spaces Bn and Bit Phave only finite homology groups therefore wehave isomorphisms 96 [Bn ;X]  lim[Bu,X n 1 and  . [Bn p ;X]  Z lim[Bn p ,X n ]  .  (If  {X n } nEN  'is n  n an inverse system of p-complete spaces then the functor lim[ ;X ] nn is representable by Sullivan i .e .  lim[ ;Xn ] _ [ ;Z]  and Z = holim X n . n In our case holim(X)  = X  and np [B n  (or  Bn p ) ;  x  (or  X n ) ]  _  [B n  (or  Bir p ) ;X p (or(X n)p) ] because Bit and Bn p have finite homotopy groups .) Wehave the following commutative diagram [Brr ;X] r lim[Bir ;X R : r n] n W p  pr1  W [Bn p , X]  3 lim[Bnp  n ;X] p n pr is an isomorphism, b is an epimorphism (resp . isomorphism if X is a loop space) and pr 1 is a monomorphism . This implies that a is an epimorphism (resp . isomorphism if X is a loop space) . This finishes the proofof Theorem 2 . If we analize the proofs carefully then it appears that in fact wehave proved much more general result . Let us suppose that a finite group G acts homotopically on a space  X  ,i .e . there is a homomorphism  G - n 0(e (X) )  where  e(X) is the space of all homotopy equivalences of X . Let us suppose that 1 G 1 = k , X is p-local and k E Z( p) . By the result of Cooke there is a space X 1 with a free action of G and a homotopy equivalence i : X - " X 1 which is homotopy equivariant with respect to the homotopy action of G .