Equivariant maps up to homotopy and borel spaces

Pub . Ma . UAB Vol . 28 Nó 1 Maig 1984 Equi a ian maps be weenG-spaces induce ibe p ese ing maps be ween he associa ed Bo elspaces . We will show ha no all ibe p ese ing maps be ween Bo el spaces a e induced ha way, no e en all ibe homo opyclasses o such maps . Howe e he e is a one- o-one co espondencebe weenhomo opyclasses o G. -maps (i .e . maps equi a ian up o homo opy in-á way, see sec ion 1 o de ini ions) be ween G-spaces and ibe homo opyclasses o maps be ween Bo él spaces . This one- o-one co espondence is ob ained by a unc o equi alence be ween he espec i e ca ego ies(Theo em 1 and 2 in sec ion 4) . As a esul equi a ian homo opy heo y (in amodi ied sense) is equi alen o he heo y o homo opy ib a ions . To p o e hese heo ems we ha e o includeH-spaces in o ou discussion : In ac , he unc o equi alence men ioned abo e is an ex ension o he equi alence be ween he ca ego ies o H-spaces and classi ying spaces p esen ed in [2) . The e o e we need he no ion o a Bo el space o H-spaces . EQUIVARIANT MAPS UP TO HOMOTOPY AND BOREL SPACES Ma in Fuchs The Bo elspacewe use, is associa ed wi h he modi ied Dold-Lasho cons uc ion in [3) . In sec ion se enwe p esen a numbe o examples o G-spaces wi h di e ing ix poin se s, such ha hese di e ences canno be de ec ed by s udying he cohomology o hei Bo elspaces, no by s udying he Bo el space i sel . The g oups in mos examples a e M p o S l , bu he G-spaces a eno all o ini e dimension . Thus we illus a e he limi s o heo ems like he local- iza ion heo em by Hsiang ([5], p . 47) . All he examples a ise om he ac ha i  h =  (h n )  n = o,l, . . .  is  a G .-map be ween he G-spacesX 1 and X 2 and h is an o dina y homo opy equi alen e, hen he ibe map induced be ween he Bo elspaces is a ibe homo opy equi alen e . 1 . De ini ions 1 .1 . The H-spacesH we a e using a e supposed o be s ic ly associa i e and o ha e a s ic uni elemen e . Fu he mo e we assume H has a homo opy in e se (such ha  H-=--> H x H lx ~ H xH u > H  is homo opic o  id H ) . 1 .2 . We say ha a opological space X is a G-space, i an H-space H ac s on X om he le con inuously and in a s ic ly associa i e manne . We assume ha ex = x o all x E X . 1 .3 . As usual, an H m -map h om H 1 o H 2 (o leng h ) is asequence o con inuous maps h n :  (H 1x  I ) n x H1  _# H 2 (n =  0 ,1 , 2 , . . . )  such  ha h n (g0 , i 1 .. .1 n .g n ) o  n  >  O,  g 0 . . . .,g n  E  H l ,  and  i , . . . , n  E  I = [O, ] c 7R .  I = O, hemap h0 is a homomo phism in he usualsense . 11-4 . I H 1 ac s on X 1 and H2 ac s on X 2 om he le , and i h is an H -map om H 1 o H 2 o a leng h  ,  henwe de ine aG C7 -map   om  X 1  o X 2 o leng h associa ed wi h h o be a seguenceo maps n : (H 1 x I ) n x X 1-0 X 2 (n = 0,1,2  . .) such ha o n > O n (g 0 , l  . .,g n- l, n ,x) h n-l (g 0 , l  . .,gi-lgi, . . ., n,gn) i = h i-l (g 0 , i ,. . .1g i-1 )hn-i(gi, . . . . gn )  = i =O n-1(g0, l, . . .,gi-1gi, . . .,gn-l, n,x) i = hi-l(g0, . . .,gi-1 ) n-i(gi, . . .,gn-l, n,x)  i = 0 Composi ion o H -maps and G -maps is de ined as in [3] . 1 .5 . I is a G -map om X 1 o X 2 associa ed o he H -map h om H 1 o H 2 , hen is called a m G- -homo opy equi alence i he e exis s an H - -map k om H 2 o H 1 and a G. -map g om X 2 o X 1 associa ed wi h k such ha  g o  and  og a e G -homo opiC o and 1  . . 2 o he H m -homo opies be ween  k oh  espec i ely  h o k and  id H  espec i ely  id H ) . 1  2 We a e going o use he heo em om [4] : Theo em . I - H 1 ac son X 1 and H2 ac s on X 2 and i  h :H 1 -o H 2 is an H m -map such ha  h 0 is an o dina y homo opy equi alence, and i :X 1 -4 X 2 is a G . -map associa ed wi h h such ha 0 is an o dina y homo opy equi alence, hen  h  is an H m -homo opy equi alence and is a G -homo opy equi alence associa ed o h . 1 .6 . H-spaces and H -maps o m he ca ego y W and G-spaces and G_-maps o m he ca ego yJ , .  The associa ed homo opy ca ego ies a é deno ed by 1,! and .1 . 2 . Co : . .s uc ion o he Bo el Space In his sec ion we ely hea ily on [3J, whe e many addi ional .de ails can be ound . 2 .1 . Le (p, ) be an H-p incipal ib a ion E xH  E P l~ ~P E  B espec i ely (associa ed as desc ibed in [3] and le X be a G-space wi h espec o  H  wi hac ion  s : H xX -+ X .  Assume ha  pX : EX -+ B is a ib a ion wi h ibe X associa ed o p : E - i B  in he ollowing sense : 1) The wo ib a ions a e ibe homo opy i ial wi h espec o he same nume able co e ingZ o B and e e y U E 21 is con ac ible in  B .  2) The e is , a map  X : E XX -+ EX  such ha o each U F . n! he diag am is commu a i e ((a,(3,aX,PX) maps) .  In addi ion we wan o be commu a i e . UxHxX-- . 1xs~U,~X 1  l (U) xX  ---~ P X p  (U) x E x H x X E xX X ->  EX a e he ob ious coo dína e 2 .2 . Fo he gene al s ep o he Bo el space cons uc ion we look a he H-p incipal ib a ion (p, ) as desc ibed in [3], p . 329-331 . The base space B o henew ib a ion is he mapping coneo p :E -i B wi h he coo dína e opology . We conside he co e ing o B consis ing o B1 = (y J, 1 '3]  and  B 2= (y1 l Le pl : E l -o B 1  espec i ely  plX : E l X -o B 1 be he ib a ions inducedby  (yl ) = p(y),  he map collapsing B 1 o he ange space B o he mapping cone B . p1X is associa ed o pl i we de ine 1X : E l x X -o  E 1 X  by u he mo e le E 2 = B 2 xH  and  E 2 X=B 2 XX . De ine  2X (Y .L ,h,x) = (y .i ,hx) .  Ob iously hese ib a ions a e associa ed . We ecall om [3), p . 330, ha he map F :p -1 (B 1 (1 B 2 )  -o p-1 (B 1 ()  B 2 )  de ined by is a s ic ly equi a ian ibe homo opy equi alence . We de ine he associa ed map F X : p2X (B 1 (1 B 2 ) p -1 (B1 n B2) by Fx is a map o e B 1 (1 B 2 and a homo opy equi alence on each ibe ( his ollows om diag am (1) and he ac ha H has a homo opy in e se) and hence is a ibe homo opy equi alence acco ding o Theo em 6 .3 in [1] . 2 .3 . As in [3), p . 330 we now o m he mappingcylinde o F and o FX and cons uc he H-p incipal ib a ion p :É -o B and simila ly he associa ed ib a ion 84 l(Y 1 .Yl ,x) = (Y 1 , x (Yl ,x)) Y(Y l . ,n) = (Y 1 ,YIIi F X (y .~ ,x) = (y 1 , X (Y,x) ) pX : EX -+ BX .  Wi h he help o  Xl  and  X2  we in he ob ious manne . No X :EXX cons uc p oblem a ises since he diag am (y i ,h,x)  2X - >(y -, ,hx) (y 1 ,yh,x)  - 1 (y 1 , (yh,x) ) lX  X commu es as a consequence o diag am (2) . So i is easy o see ha E and EX a e associa ed . 2 .4 . To cons uc he Bo el space o X we s a ou wi h  pO : EO -" B O ,  whe e  EO = H  and  BO =  [*) = poin , and wi h poX :E O X -+ B O , whe e EOX =X . F om pn and pnX we cons uc pn+1 and pn+l,X by le ing E .+l = E n' B n+l - Bn and E n+1 X=En X . Ob iously p . 333 in [3] we use elescopes o inally ge he uni e sal H-p incipal ib a ion p H :EH 4 BH and he associa ed ib a ion pX :EX -+ BH . We call EX he Bo elspace o X and pX he Bo el ib a ion . o X . No ice ha i ial ib a ion wi h ibe h ough hemap he di ec limi o he maps because we used he elescope cons uc ion . con inui y o H in [3], p . 333) . -i 2 3< < 3 pn+1 -Pn and pn+1,X- pnX a e associa ed . As on pX is a nume able, locally ibe homo opy X associa ed wi h pH X is essen ially n,X , and i is con inuous (Compa e he X -EHXX -i 3 . InducedM aps Be ween Bo el-Spaces 3 .1 . Be o e we can discuss G-Spaces, we ha e o know mo eabou H-Spaces . So le h :H 1 -+ H 2 be an H e -map Be ween he H-Spaces  H 1 and  H 2 .  We de ine a G m -map E0 h :E 0 H 1 -+ E0 H 2 as  E0h = h .  (No e ha all he Spaces EH ha e a igh ac ion, so he no ion o G -map has o n  m be modi ied acco dingly) .  Alsowe le  B 0h :B 0 H 1 -+ B0H 2 be he i ial map . Assume ha Eh has been ex ended o a G -map O  °' Enh :E nH 1 -+ EnH 2 associa ed wi h  h  and  B0h  has been ex ended o Bh such ha n (We will call a G -map wi h his p ope y ibe p ese ing) . On E nl H l we de ine and p n2 0 E n hk (Y . l.gl .. . . . k .gk) = Bnh ° pnl(Y) . . Fi s we ex end B nh om Bn H 1 o BnH 1 by de ining E n1 h0 (Y 1 .Y 0 ) = (E nh0 (Y) 1 . Enh0 (Y O )) En1h k (Y 1 ,y 0 , i , 91 .. . . . k . gk) o k = 1,2, . . . Bn h(y ~ . ) = (E n h0 (Y) 1 ) = (E nh0 (Y) 1 , E nhk (Y 0 . 1.91 .. . ., k .g k ) Recall  ( om  [3],  p .  330)  ha  En2H1 =  (Bn2 H 1 XH1)  U (B ni  n B n2 XI X H1)  and de ine Eñ 2 hk(y l ,T,g o . l   k,gk) ((E n h0 (y) l , hk (g 0 ' l   k .g k )) (E n h0 (y)i . , 2T,h k (g 0 . l   k .gk) when O S T S 2 and 3 < < 3 (E n h0 (y) x , Enhk+l(y,2T - l,g0 . l . .. . . k .gk)) when Z S T S1  and 3 < T < 3 ' (when T =1 we use ha Enhk+l(y,l,g0, 1, . . .) E n hk (yg 0 , i . . . .) . Hence Eñ2hk and E nl h oge he induce a G_-map  E nh  om  En H 1  o  E nH2  which sa is ies all he condi ions men ioned be o e and hence we ge E n+1 h ' E n+1 H 1 -4 E n+1 H2  oge he wi h  B n+l h .  In he ob ious manne we ob iain he G 0 -map  Eh :EH 1 - i EH 2 associa ed wi h h . Because o ou de ini ion o Eñ 2 hk on he mapping cylinde pa o EnH, . we only ge E(h oh') is G -homo opic o Eh oEh'  and simila ly B(h oh') y W Bh o Bh' .  In ac he G-homo opymen ioned is ibe p ese ing . We ge he implies ha LpH oK is a ibe homo opy equi alence and hence SO is a homo opy equi alence . Lemma 2 . S O can be ex ended o an H -map . m P oo :  Le K ¡E 0 H = K IH = K0 .  Thenwe ha e o ind maps  S 1 , S2 ,. . .  which make  S O = Lp H o KO : H -+ nBH in o an H -map .  Assume we al eady cons uc ed m S i = Lp H o K i (i = O,l, . . .,n) .  Then  Sn+l  and hence K n+ 1  is de inedon  ~H (n + 1)  h ough he maps  S i  and K i espec i ely (i = 0, . . .,n) . and Associa ed wi h K i a e he maps De ine k i  : H (i)  X 7R +  -4  EH T4( ;1 . . io+ y ] . wi h  ki (g0, l, . . . . i,g i~0)  = *  and k i (g0 " i, . . ., ,,g,,,) = g 0 ... g i o Z i(g0, l, . . ., i'gi) . These maps de ine kn+l and n+1  espec i ely on  -3H(n + 1) .  Since  ]R + is con ac ible we can ex end  n+l  o all o  H(n + 1) . Then we can ex end  k n+l  o all o  H(n + 1)  such ha kn+l(g0' l " .. ' n+l'gn+l'0) _ and kn+1(g0' i, . . . . n+1,gn+1' n+l( .. .)) = g0 ... gn+l' Since EH is .con ac ible . K n+l  (kn+l' n+l)  and S n+l = L PH oK n+ 1 Fo u hé de ailscompa e [2J, p . 214-215 . (No e he addi ion o pa hs on p . 213 shouldbe e e sed .) 5 .2 . P oQosi ion . S is a na u al ans o ma ion be ween 1 y and QB . P oo : In he diag am S L(EH ;EH,*) LP H h LEh H' K L(EH' ;EH',*) L PH' íé(BH,*)  nBh ) C(BH' .*) he lowe po ion commu es o all he maps o LEh . To see ha he uppe po ioncommu es up o an H m - homo opy, onehas o look again a he associa edmaps in o EFI' . Since EH' is con ac ible, all ex ensions necessa y o cons uc he H m -homo opy be ween  LEh o K and Koh can be ca ied ou . Fu he de ails in [2] . (In [2) he G -map Eh was no discussed . Ins ead he no ion o a " egula " H-homomo phism had o be used . Now EH p o ides he homo opy be ween o mula 2 and 2a on p . 217 in 2 , ansla ed om igh o le ac ions .) 5 . 3 . Wi h S ou o he way we de ine o any G-space X : We al eady know ha  T 0 (y) _ (*,y)  is a homo opy equi alence .  We de ine  Tn : (H xI) n x X -+ WE  as wi h  O S n S n-1 (g o l. ., ' n-1,gn-1 )  and O S 6 S n-l - n .  Recall  X : EH x X -+ EX .  We ha e WB . Tn(g0 ; i, . . . . n .x) =  (pKn-1 (g o , . . . . n-1 'g n-1 )( n+ a), -1(go, . . . . n-1'gn-1 )( n),x)) Tn(go, 1, . . . . gn-l, n,x) T0 : X -+ WE  as  T0 = T I X 6 . P oo o Theo em 2 I( S n-1(g0 . V . . . . g n _ 1 ) .x)  n = O * .9 09 1  . . .  g n-l , x)  n =  The "G . -homo opy" be ween  LEh oK  and  K oh  implies ha T is a na u al ans o ma ionbe ween 1 2( and 66-1 . Le .T * be he ca ego y o based opological spaces X, which ha e a nume ableco e ing 2I such ha e e y U E 2I is con ac able in X, and based con inuous maps . Le T * be he associa ed homo opy ca ego y . Rema k . I is easy o see ha o e e y H in Al he classi ying space BH is in T * . . In p epa a ion o he p oo o Theo em 2 we lis h ee uni e sal ib a ions wi h ibe n(X,*) o X E a * . a) Applica ion o he modi ied Dold-Lasho cons uc ion o he i ial ib a ion (2(X,*) -+ leads o p C2X : ESZX -i B 2X b) I is well-known ha P I, : L(X ;X,*) -0 X also classi ies nume able n(X,*)- ib a ions . c) I we apply he modi ied Dold-Lasho cons uc ion o pL o b), we ge again a uni e sal ib a ion P EL : ELX -4 BLX All h ee cons uc ions induce unc o s om 7* o 6 .2 . The inclusion o C(X,*) as dis inguished ibe o pL L(X ;X,*) 4 X can be in e p e ed as a p incipal map o p incipal ib a ions and hence i induces he ibe map T *, . E(QX) p OX B(CIX)  B(LX) Le g be a homo opy in e se o T . which is a p incipal ibe homo opy equi alence ; ( , ) is an inclusion, hence P OX is p incipal ibe homo opy equi alen o he pullpack o p LX . Fo uni e sal ib a ions his implies is a homo opy equi alence . As a esul , ( , ) ep esen s a unc o equi alence be ween he unc o s om T * o 91 * induced by a) and c) . 6 .3 . The inclusion L(X ;X,*) k -j ELX kj BLX is a ibe homo opy equi alence by he same easoning as desc ibed in 6 .2 . So (k,k) ep esen s a unc o equi alence be ween he unc o s a ising om b) and c) . 6 .4 . Now conside a ib a ion p :E -4 X om he ca ego y 7 * .  The associa ed Hu ewicz- ib a ion p :E-+X admi s a map 0 : L(X ;X,*) ;c WE  E de ined h ough he addi ion o pa hs, whichmakes E a look alike o a Bo el space associa ed o WE . Assigning o p he Hu ewicz ib a ion p induces a unc o H  on ;  which is ob iously equi alen o id ~ .  .  We a e now going o show  BW - .H .  Conside he ,, *  _- diag am o Bo el spaces : K is i duced by applying he Bo el space cons uc ion o p (an ob ious modi ica ion) and G is induced,by L(X ;X,*) xWE  k  ELX xWE  g x l ~E,-?X xWE p g, he homo opy in e se o om 6 .2 . (K,k) and (G,g) ep esen unc o equi alences associa ed o he equi alences (k,k) and (g,g) discussed in 6 .2 and 6 .3 . Since he igh sideo he diag am ep esen s BW and he le side ep esen s H ,  he p oo is comple e . 7 . Two Applica ions K ~ G E  --j EL(WE) --~ E(WE) g 7 .1 .  Le  G =  IR1 and  X = IR 2 .  Cons ide he wo IR 1 -spaces  X 1 and  X 2 de ined by he wo ac ions p l : IR l x IR 2  -~  IR 2 ,  p 1 ( , e lcp )  =  e l (g+ ) ~2 : IR l x IR 2 -+ IR2 ,  P2 ( , e lqp )  = el (9+ (1- ) ) The ix poin se o  pl  is jus he o igin o  IR 2 and he ix poin se o p2 is he o igin and he uni ci cle . Ob iously we couldde ineac ionswi h mo e complica ed ix poin se s . The cons an map om one o hesespaces o he o igin o he o he is an equi a ian map which is also an o dina y homo opy equi alen e . I induces (acco ding o sec ion ou ) ahomo opy equi alen e be ween he Bo el spaces o he wo spaces . 7 .2 . a)  Le P be an acyclic ini e polyhed on wi h non i ial undamen al g oup .  Then he suspension  E P is a con ac ible 2Z2 -space wi h ix poin se  P,  and he join  P * S 1 is a cons ac ible  S1 o Z? p -space (p y 2) wi h ix poin se P in he ob iousmanne (no ice  P *S 1 - E 2 p) . b) Le P be any ini e polyhed on . The ob ious 2Z2 -ac ion on  E P  can be ex ended o  E 2 P  e c .  so ha  lim En P  is a con ac ible 2Z2 -space wi h ix n~ poin se P . sameby ei e a ing he join wi h  S 1 . G-space wi h nonemp y ix poin se F, e .g . le Y be one o he spaces men ioned abo e . The one poin union Id o X and - Y o medby iden i ying wo Fo  G = 2Z P  (p T 2)  and  G = S 1 we can do he 7 .3 . Le G be ei he 2Z  o S1 andle X be a P G-space wi h ix poin s . Le Y be a con ac ible ix poin s is a new G-space in he ob ious manne and he inclusion o X in o I^1 is an equi a ian map and also an o dina yhomo opy equi alence . By he heo em in [4] he inclusion ep esen s an isomo phism in .1 and induces a ibe homo opy equi alence be ween BX and BW by sec ion 4 . Hence he cohomology o hese Bo el spaces ca ies no in o ma ion abou F . 7 .4 .  Assume  G  is ei he  2Z kk o  (S1 ) k and  X1 ,X2 a e G-spaces which sa is y he assump ions o Bo el's heo em as desc ibed in P oposi ion 1 o Chap e IV in [5J, i .e ., le X l ,X 2 be pa acompac G-spaces wi h ini e cohomology dimension . Le :X 1 -# X2 be an equi a ian map which is also an o dina yhomo opy equi alence .  Again  E :EX 1i EX 2  is a ibe homo opy equi alence be ween Bo elspaces . E induces isomo - phisms be ween H G (X 2 ) and HG (X1 ) as H (BG) modules . Hence P oposi ion 1 on p .45 in [5] ells us, ha IF 1 :F . 1 =# F 2 induces an isomo phism o he cohomology ings  H * (F 2 ) (9 k RG and  H * (F 1 ) ® kRG o he ix poin se s F 1 and F 2 . T . Pe ie in [7] and elsewhe e, Ch . N . Lee and A . Wasse man in [6] Na e cons uc ed exampleso such maps which do no ha e equi a ian homo opy in e ses . Hence he ibe homo opy in e se o E  is no induced by an equi a ian map om X2 o X 1 . This answe s he opening s a emen o he in oduc ion o his pape . Re e ences Dold, A ., Pa i ions o uni y in he heo y o ib a ions, Ann . o Ma h . 78, 223-255 (1963) . 2 .  Fuchs, M.,Ve allgemeine e Homo opie-Homomo phismen und klassi izie ende Ráume, Ma h . Ann . 161, 197-230 (1965) . 3 . Fuchs, M ., A modi ied Dold-Lasho cons uc ion ha does classi y H-p incipal ib a ions, Ma h . Ann . 192, 328-340 (1971) . 4 . Fuchs, M ., Homo opy equi alences in equi a ian opology,P oc . Ame . Ma h . Soc . 58, 347-352 (1976) . 5 . Hsiang, W .Y ., Cohomology heo yo opological ans o ma ion g oups, Sp inge -Ve lag 1975 . 6 . Lee,Ch . N . and Wasse man, A .G ., On he g oups JO(G), Memoi s Ame . Ma h . Soc . 159 (1975) . 7 . Pe ie, T ., Smoo h S1 ac ions and bilinea o ms, Bull . Ame . Ma h . Soc . 79, 1056-1059 (1973) . Rebu e1' 15 de~se embAe del 19&3 Depa men o Ma hema ics MichiganS a e Uni e si y Eas Lansing, MI 48824 USA