CW SIMPLICIAL RESOLUTIONS OF SPACES WITH AN APPLICATION TO LOOP SPACES DAVID BLANC

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1 CW SIMPLICIAL RESOLUTIONS OF SPACES WITH AN APPLICATION TO LOOP SPACES DAVID BLANC Absac. We show how a ceai ye of CW simlicial esoluios of saces by wedges of shees may be cosuced, ad how such esoluios yield a obsucio heoy fo a give sace o be a loo sace. 1. Ioducio A simlicial esoluio of a sace X by wedges of shees is a simlicial sace W such ha (a) each sace W is homooy equivale o a wedge of shees, ad (b) fo each k 1, he augmeed simlicial gou k W! k X is acyclic (see x3.5 below). Such esoluios, s cosuced by Chis Sove i [S, x2], ae dual o he \usable Adams esoluios" of [BK, I, x2], ad have a umbe of alicaios: see x3.1 below ad [S, DKSS, DKS1, B1, B5, B6, B7]. Howeve, he Sove cosucio yields vey lage esoluios, which do o led hemselves eadily o comuaio, ad o ohe cosucio was hiheo available. I aicula, i was o clea whehe oe could d miimal esoluios of his ye. The uose of his oe is o show ha ay sace X has simlicial esoluios by wedges of shees, which may be cosuced fom uely algebaic daa, cosisig of a (abiay) simlicial esoluio of X as a -algeba { ha is, as a gaded gou wih a acio o he imay homooy oeaios o i (see x3.1 below): Theoem A. Evey fee simlicial -algeba esoluio of a ealizable -algeba X is ealizable oologically as a simlicial esoluio by wedges of shees. ad i fac such esoluios ca be give a coveie \CW sucue" (x3.15). Thee is a aalogous esul fo mas (Theoem 3.24). Sice o such esoluio of a o-ealizable -algeba ca be ealized (see x3.16 below), his comleely deemies which fee simlicial -algeba esoluios ae ealizable. The Theoem imlies ha i he secal sequeces of [S, B1, DKSS] we ca wok wih miimal esoluios, ad allows us o ideify he highe homooy oeaios of [B5, B1, B7] as lyig i aoiae cohomolgy gous (comae [B6, 4.17] ad [B8, x6]). A geealizaio of Theoem A o ohe model caegoies aeas i [B9]. As a alicaio of such CW esoluios, we descibe a obsucio heoy fo decidig whehe a give sace X is a loo sace, i ems of highe homooy oeaios. Oe such heoy was give i [B7], bu he ese aoach does o equie a give H-sace sucue o X, ad may be adaed also o he exisece of A -sucues (ad hus subsumes [B6]). I is summaized i Dae: Ail 23, Mahemaics Subjec Classicaio. Pimay 55Q5; Secoday 55P35, 18G55, 55Q35. Key wods ad hases. simlicial esoluio, CW objec, -algeba, highe homooy oeaio, loo sace. 1

2 2 DAVID BLANC Theoem B. A sace X wih ivial Whiehead oducs is homooy equivale o a loo sace if ad oly if he highe homooy oeaios of x5.1 below vaish coheely Noaio ad coveios. G will deoe he caegoy of gous, T ha of oological saces, ad T ha of oied oological saces wih base-oi esevig mas. The full subcaegoy of -coeced saces will be deoed by T c T. The caegoy of simlical ses will be deoed by S ad ha of oied simlicial ses by S ; we shall use boldface lees: X, S ; : : : o deoe objecs i ay of hese fou caegoies. If f : X! Y is a ma i oe of hese caegoies, we deoe by f # : X! Y he iduced ma i he homooy gous Ogaizaio. I secio 2 we eview some backgoud o simlicial objecs ad bisimlicial gous, ad i secio 3 we ecall some facs o -algebas, ad ove ou mai esuls o CW esoluios of saces by wedges of shees: Theoem A (Theoem 3.2) ad Theoem I secio 4 we dee a ceai cosimlicial simlicial sace u-o-homooy, which ca be ecied if ad oly if X is a loo sace. I secio 5 we cosuc a ceai collecio of face-codegeeacy olyheda, which ae used o dee he highe homooy oeaios efeed o i Theoem B (Theoem 5.12). We also show how he heoem may be used i evese o calculae a ceai eiay oeaio i S Ackowledgemes. I would like o hak he efeee fo his o he commes (see i aicula x3.18 below). 2. simlicial objecs We s ovide some deiios ad facs o simlicial objecs: 2.1. Deiio. Le deoe he caegoy of odeed sequeces h; 1; : : : ; i ( 2 N), wih ode-esevig mas. A simlicial objec ove a caegoy C is a fuco X : o! C, usually wie X, which may be descibed exlicily as a sequece of objecs fx k g 1 i C, equied wih face mas k dk i : X k! X k1 ad degeeacies s k j : X k! X k+1 (usually wie simly d i, s j, fo i; j k), saisfyig he usual simlicial ideiies ([Ma, x1.1]). If I (i 1 ; i 2 ; : : : ; i ) is some muli-idex, we wie d I fo d i1 d i2 d i, wih d ; : id; ad similaly fo s I. A augmeed simlicial objec is oe equied wih a augmeaio " : X! Y (fo Y 2 C), wih "d "d 1. The caegoy of simlicial objecs ove C is deoed by sc. We wie s hi C fo he caegoy -simlicial objecs ove C (ha is, objecs of he fom fx k g k, wih he eleva face mas ad degeeacies), ad deoe he ucaio fuco sc! s hi C by. Fo echical coveiece i he ex wo secios we shall be wokig maily i he caegoy of simlicial gous, deoed by G (ahe ha sg); objecs i G will be deoed by caial lees X, Y, ad so o. A simlicial objec X (X ; X 1 ; : : : ) i sg is hus a bisimlicial gou, which has a exeal simlicial dimesio (he i X 2 G), as well as he ieal simlicial dimesio k (iside G), which we shall deoe by (X ) i, if ecessay. k

3 CW RESOLUTIONS OF SPACES Simlicial ses ad gous. The sadad simlex i S is deoed by [], geeaed by 2 []. [] deoes he sub-objec of [] geeaed by d i ( i ). The simlicial -shee is S : [] [], ad he -disk is D : CS 1. Le F : S! G deoe he (dimesiowise) fee gou fuco of [Mi2, x2], ad G : S! G be Ka's simlicial loo fuco (cf. [Ma, Def. 26.3]), wih W : G! S he Eilebeg-Mac Lae classifyig sace fuco (cf. [Ma, x21]). Recall ha if S : T! S is he sigula se fuco ad k k : S! T he geomeic ealizaio fuco (see [Ma, x1,14]), he he adjoi ais of fucos (2.3) T S S G kk W G iduce isomohisms of he coesodig homooy caegoies (see [Q1, I, x5]), so ha fo he uoses of homooy heoy we ca wok i G ahe ha T Deiio. I aicula, S : FS 1 2 G fo 1 (ad S : GS fo ) will be called he -dimesioal G-shee, i as much as [S ; GX] G X [S ; X] fo ay Ka comlex X 2 S. Similaly, D : FD 1 will be called he -dimesioal G-disk Deiio. I ay comlee caegoy C, he machig objec fuco M : S o sc! C, wie M A X fo a (ie) simlicial se A 2 S ad X 2 sc, is deed by equiig: (a) M [] X : X, ad (b) if A colim i A i, he M A X lim i M Ai X (see [DKS2, x2.1]). I aicula, if A k is he subcomlex of [] geeaed by he las ( k + 1) faces (d k ; : : : ; d ), we wie M kx M A k X : exlicily, (2.6) M k X f(x k ; : : : ; x ) 2 (X 1 ) +1 j d i x j d j1 x i fo all k i < j g: ad he ma k : X! M k X iduced by he iclusio A k,! [] is deed k (x) (d k x; : : : ; d x). The oigial machig objec of [BK, X,x4.5] was M X M [] X, which we shall fuhe abbeviae o M X ; each face ma d k : X +1! X facos hough :. See also [Hi, XVII, 87.17] Remak. Noe ha fo X 2 G ad A 2 S we have M A X Hom G (FA; X) 2 G (cf. x2.2), so fo X 2 sg also (M A X) k HomG (FA; (X ) i k ) i each simlicial dimesio k Deiio. X 2 sg is called ba if each of he mas : X! M X ( ) is a baio i G (ha is, a sujecio oo he ideiy comoe { see [Q1, II, 3.8]). This is jus he codiio fo bacy i he Reedy model caegoy, (see [R]), as well as i ha of [DKS1], bu we shall o make exlici use of eihe. By aalogy wih Mooe's omalized chais (cf. [Ma, 17.3]) we have: 2.9. Deiio. Give X 2 sg, we dee he -cycles objec of X, wie Z X, o be he be of : X! M X, so Z X fx 2 X j d i x fo i ; : : : ; g (cf. [Q1, I,x2]). Of couse, his deiio eally makes sese oly whe X is ba (x2.8). Similaly, he -chais objec of X, wie C X, is deed o be he be of 1 : X! M 1 X. fo

4 C C Z Z A Z A A Z A R A ~Z A A UA R 4 DAVID BLANC If X 2 sg is ba, he ma d d j C X : C X! Z 1 X is he ullback of : X! M X alog he iclusio : Z 1 X! M X (whee (z) (z; ; : : : ; )), so d is a baio (i G), ig io a baio sequece (2.1) Z X j d! C X! Z 1 X : Poosiio. Fo ay ba X 2 sc, he iclusio : C X,! X iduces a isomohism : C X C ( X ) fo each. Poof. (a) Fis oe ha if j : A,! B is a ivial cobaio i S, he j : M B X! M A X has a aual secio : M A X! M B X (wih j id) fo ay X 2 sg: This is because by emak 2.7, (M A X ) k HomG (FA; (X ) i) fo k A 2 S; sice FA is ba i G, we ca choose a lef ivese : FB! FA fo F j : FA,! FB, so j : (M B X ) i k! (M A X ) i k has a igh ivese, which is aual i (X ) i; so hese mas k ogehe o yield he equied ma. This eed o be ue i geeal if j is o a weak equivalece, as he examle of M 1 2 X! M1 X shows. (b) Give 2 C m X eeseed by h : S m! X wih d k h (1 k ), coside he diagam: S m X X X X X h X X C X X X C R X X C X P C P P k+1 k P P P P C P M P Pq k - C X Z d k PB k M k+1 X X 1 k - 1 M1X k j (d k ; : : : ; d k ) i which j is a baio by (a) if k 1, so he lowe lef-had squae is i fac a homooy ullback squae (see [Ma, x1]). By descedig iducio o 1 k 1, (saig wih d ), we may assume k+1 h : S m! M k+1 X is ullhomooic i C, as is d k h, so he iduced ullback ma kh : Sm! M kx, is also ullhomooic by he uivesal oey. We coclude ha 1 h, ad sice 1 : X! MX 1 is a baio by (a), we ca choose h : S m! X so ha h 1. Thus h lifs o C X Fib(), 1 ad is sujecive. (c) Fially, he log exac sequece i homooy fo he baio sequece C X!X 1! M 1 X imlies ha # : C X! X is moic, so : C X! C ( X ) is, oo.

5 CW RESOLUTIONS OF SPACES Deiio. The dual cosucio a o ha of x2.5 yields he colimi L X : X 1 ; i1 whee fo ay x 2 X 2 ad i j 1 we se s j x i he i-h coy of X 1 equivale ude o s i x i he (j +1)-s coy of X 1. L X has someimes bee called he \-h lachig objec" of X. The ma : L X! X is deed x (i) s i x, whee x (i) is i he i-h coy of X algebas ad esoluios I his secio we ecall some deiios ad ove ou mai esuls o -algebas ad esoluios: 3.1. Deiio. A -algeba is a gaded gou G fg k g 1 k1 (abelia i degees > 1), ogehe wih a acio o G of he imay homooy oeaios (i.e., comosiios ad Whiehead oducs, icludig he \ 1 -acio" of G 1 o he highe G 's, as i [W, X, x7]), saisfyig he usual uivesal ideiies. See [B3, x2.1] fo a moe exlici desciio. These ae algebaic models of he homooy gous X of a sace (o Ka comlex) X, i he same way ha a algeba ove he Seeod algeba models is cohomology ig. The caegoy of -algebas is deoed by -Alg. We say ha a sace (o Ka comlex, o simlicial gou) X ealizes a (absac) -algeba G if hee is a isomohism of -algebas G X. (Thee may be o-homooy equivale saces ealizig he same -algeba { cf. [B5, x7.18]). Similaly, a absac mohism of -algebas : X! Y (bewee ealizable -algebas) is ealizable if hee is a ma f : X! Y such ha f Deiio. The fee -algeba geeaed by a gaded se T ft k g 1 is k1 W, whee W W 1 W k1 2T k S k (ad we ideify 2 T ( ) k wih he geeao of k W eeseig he iclusio S k,! W). ( ) If we le F -Alg deoe he full subcaegoy of fee -algebas, ad he homooy caegoy of wedges of shees (iside hot o hos { o equivalely, he homooy caegoy of cooducs of G-shees i hog), he he fuco :! F is a equivalece of caegoies. Thus ay -algeba mohism ' : G! H is ealizable (uiquely, u o homooy), if G ad H ae fee -algebas (acually, oly G eed be fee) Deiio. Le T : -Alg `! -Alg be he \fee -algeba" comoad (cf. [Mc, VI, x1]), deed T G `1 k1 g2g k S k. The coui " " (g) G : T G! G is deed by k 7! g (whee (g) k (g) is he caoical geeao of S k (g)), ad he comulilicaio # # G : T G,! T 2 G is iduced by he aual asfomaio # : id F! T j F deed by x k 7! k (x k ) Deiio. A abelia -algeba is oe fo which all Whiehead oducs vaish. These ae ideed he abelia objecs of -Alg { see [B3, x2]. I aicula, if X is a H-sace, he X is a abelia -algeba (cf. [W, X, (7.8)]) Deiio. A simlicial -algeba A is called fee if fo each hee is a gaded se T A such ha A is he fee -algeba geeaed by T (x3.2), ad each degeeacy ma s j : A! A +1 akes T o T +1.

6 6 DAVID BLANC A fee simlicial esoluio of a -algeba G is deed o be a augmeed simlicial -algeba A! G, such ha (i) A is a fee simlicial -algeba, (ii) i each degee k 1, he homooy gous of he simlicial gou (A ) k vaish i dimesios 1, ad he augmeaio iduces a isomohism (A ) k Gk. Such esoluios always exis, fo ay -algeba G cosucio i [B1, x4.3]. { see [Q1, II, x4], o he exlici 3.6. Deiio. Fo ay X 2 G, a simlical objec W 2 sg equied wih a augmeaio " : W! X is called a esoluio of X by shees if each W is homooy equivale o a wedge of G-shees, ad W! X is a fee simlicial esoluio of -algebas Examle. Oe examle of such a esoluio by shees is ovided by Sove's cosucio; we shall eed a vaia i G (as i [B7, x5]), ahe ha he oigial vesio of [S, x2], i T. (The agume fom his oi o would acually wok equally well i T ; bu we have aleady chose o wok i G, i ode o faciliae he oof of Poosiio 2.11). Dee a comoad V : G! G fo G 2 G by (3.8) whee D k+1 V G 1a k a 2Hom G (S k ;G) S k [ 1 a k a 2Hom G (D k+1 ;G) D k+1 ;, he G-disc idexed by : Dk+1! G, is aached o S k, he G-shee idexed by j D k+1, by ideifyig D k+1 : F D k wih S k (see x2.4 above). The cooduc hee is jus he (dimesiowise) fee oduc of gous; he coui " : V G! G of he comoad V is \evaluaio of idices", ad he comulilicaio # : V G,! V 2 G is as i x3.3. Now give X 2 G, dee Q 2 sg by seig Q V +1 X, wih face ad degeeacy mas iduced by he coui ad comulilicaio esecively (cf. [Go, A., x3]). The coui also iduces a augmeaio " : Q! X; ad his is i fac a esoluio of X by shees (see [S, Po. 2.6]) Remak. Noe ha we eed o use he G-shee ad disk S k ad D k of x2.4 i his cosucio; we ca elace i by ay ohe homooy equivale coba ai of simlicial gous, so i aicula by (F ^D k ; F ^S k1 ) fo ay ai of simlicial ses ( ^D k ; ^S k1 ) ' (D k ; S k1 ) The Quille secal sequece. A esoluio by shees W! X is i fac a esoluio (i.e., coba elaceme) fo he cosa simlicial objec cx 2 sg (i.e., c(x) X, d i s j id X ) i a aoiae model caegoy sucue o sg { see [DKS1] ad [B9]. Howeve, we shall o eed his fac; fo ou uoses i suces o ecall ha fo ay bisimlicial gou W 2 sg, hee is a s quada secal sequece wih (3.11) E 2 s; s( W ) ) s+ diag W

7 CW RESOLUTIONS OF SPACES 7 covegig o he diagoal diag W 2 G, deed (diag W ) k (W k ) i k (see [Q2]). Thus if W! X is a esoluio by shees, he secal sequece collases, ad he aual ma W! diag W iduces a isomohism X (diag W ). Combied wih he fac ha W is a esoluio (i s-alg) of X, his simle esul has may alicaios { see fo examle [B1], [DKSS], ad [S] Deiio. A CW comlex ove a oied caegoy C is a simlicial objec R 2 sc, ogehe wih a sequece of objecs R ( ; 1; : : : ) such ha R R q L R (x2.5), ad d i j R fo 1 i. The objecs ( R ) 1 ae called a CW basis fo R, ad d : d j R is called he -h aachig ma fo R. Oe may he descibe R exlicily i ems of is CW basis by a a (3.13) R R I2I ; whee I ; is he se of sequeces I of o-egaive ieges i 1 < i 2 < : : : < i (< ), wih s I s i s i he coesodig -fold degeeacy (if, s I id). See [B2, 5.2.1] ad [Ma,. 95(i)]. Such CW bases ae coveie o wok wih i may siuaios; bu hey ae mos useful whe each basis objec is fee, i a aoiae sese. I aicula, if C -Alg, we have he followig Deiio. A CW esoluio of a -algeba G is a CW comlex A 2 s-alg, wih CW basis ( A ) 1 ad aachig mas d : A! Z 1 A, such ha each A is a fee -algeba, ad each aachig ma d j C A is oo Z 1 A (fo, whee we le d deoe he augmeaio " : A! G ad Z 1 A : G ). Comae [B2, x5]. Evey -algeba has a CW esoluio (x3.14), as was show i [B1, 4.4]: fo examle, oe could ake he gaded se of geeaos T fo A o be equal o he gaded se Z 1 A Deiio. Q 2 sg is called a CW esoluio by shees of X 2 G if Q! X is a esoluio by shees (Def. 3.6), ad Q is a CW comlex wih CW basis ( Q ) 1 ), such ha each Q 2 F (i.e., Q is homooy equivale o a wedge of shees). The coce is deed aalogously fo X 2 S o X 2 T Remak. Closely elaed o he oblem of ealizig absac -algebas (x3.1) is ha of ealizig a fee simlicial -algeba A 2 s-alg: his is because, as oed i x3.5, evey G 2 -Alg has a fee simlicial esoluio A! G ; if i ca be ealized by a simlicial sace W 2 st c { o equivalely, via (2.3), by a bisimlicial sace o gou { he he secal sequece (3.11) imlies ha diag W G. Howeve, o evey -algeba is ealizable (see [B5, x8] o [B4, Po ]). I would eveheless be vey useful o kow he covese: amely, ha ay fee esoluio of a ealizable -algeba is iself ealizable. This was misakely quoed as a heoem i [B5, x6], whee i was eeded o make he obsucio heoy fo ealizig -algebas descibed hee of ay acical use { ad aeaed as a cojecue i [B6, x4], i he coex of a obsucio heoy fo a sace o be a H-sace. I ode o show ha his cojecue is i fac ue, we eed seveal elimiay esuls:

8 8 DAVID BLANC Poosiio. Evey CW esoluio A! X of a ealizable -algeba embeds i Q fo some esoluio by shees Q! X. Poof. To simlify he oaio, we wok hee wih oological saces, ahe ha simlicial gous, chagig back o G if ecessay via he adjoi ais of x2.2. Give a fee simlicial -algeba esoluio A! J wih CW basis ( A ) 1, whee J X fo some X 2 T, ad A is he fee -algeba geeaed by he gaded se T, le deoe he cadialiy of `1 `1 k k X : X W 1 W _ < D. W < D ad ^D : (This is o esue ha hee will be a leas diee eeseaives fo each homooy class i X o ^S.) By emak x3.9 above, if we use he cosucio of x3.7 i T (o i G, muais muadis) wih hese \shees" ad \disks", ad aly i o he sace X, ahe ha o X, we obai a esoluio by shees Q! X. Dee ew \shees" ad \disks" of he fom ^S : S W 1 _ ^S _ D. We dee : A,! Q by iducio o he simlicial dimesio; i suces o oduce fo each a embeddig : A,! C Q commuig wih d. If we deoe " A : A! X X by d : C A! Z 1 A : A 1 ad se 1 id X, he we may assume by iducio we have a moomohism 1 : A 1,! Q 1 (akig geeaos o geeaos, ad commuig wih face ad degeeacy mas). Fo each -algeba geeao i ( A ) k, if d ( ) 6 he 1 (d ( )) 2 Z 1 k Q is eeseed by some g : ^S k! Q 1, ad we ca choose disic (hough ehas homooic) mas g fo diee geeaos by ou choice of ^S k. The by (3.8) hee is a wedge summad ^S k g i Q V Q 1 (wih o disks aached), ad he coesodig fee -algeba cooduc summad ^S k g i Q, geeaed by g, has d ( g ) [g] 2 k Q 1 ad d i ( g ) di1 g 2 k Q 1 fo 1 i by x3.7, sice [g] 1 (d ( )) 2 Z 1 k Q ad hus d i [g] [d i g], ad shees idexed by ullhomooic mas have disks aached o hem. We see ha g 2 C k Q, so we may dee ( ) g. If d ( ), he all we eed ae eough disic -algeba geeaos i Z Q : we cao simly ake g fo ullhomooic g : S k! Q 1, because of he aached disks; bu we ca oceed as follows: Sice ^D k C ^S k _ D k ad X X W 1 W _ i < Di, we have disic ozeo mas F : ^D k! X wih F j C ^S. Dee k H + F, H ; he S k : H ^D k H + k1 [^S H 2 k Q ^D k is, u o homooy, a shee wedge summad i Q H, ad hus is a -algeba geeao maig o ude he augmeaio. Similaly, ^D k i Q G 1 by G +, G k whee k is a dee S k G : ^D k G + k1 [^S homoeomohism oo he summad D k i ^D k. The G H ad G 6 bu H G ; hus H is a -algeba geeao i Z 1 k Q. By hus aleaig he + ad we oduce disic -algeba geeaos i Z Q fo each Remak. The efeee has suggesed a aleaive oof of his Poosiio, which may be easie o follow: ahe ha \faeig" he shees ad disks, we ca modify he Sove cosucio of (3.8) by usig coies of each shee o disk fo each 2 Hom G (S k ; G) o 2 Hom G (D k+1 ; G), esecively. The oof of [S, Po. 2.6] sill goes hough, ad so does he agume fo embeddig A i Q above Poosiio. Ay fee simlicial -algeba A has a (fee) CW basis ( A ) 1.

9 - - CW RESOLUTIONS OF SPACES 9 Poof. Sa wih A A. Fo 1, assume A `1 k 2T k S k. By Deiio 3.5, T T S [ SI2I ; ^T (as i x3.13), so we ca se ^A `1 ` k 2 ^T k S k ; bu d i j ^A eed o vaish fo i 1. Howeve, give 2 ^T k, we may dee i 2 (A ) i k iducively, saig wih, by i+1 i s i1 d i 1 i (face ad degeeacy mas ake i he exeal diecio); we d ha : is i C A. If we dee ' : ^T! A by '(), by he uivesal oey of fee -algebas his exeds o a ma ' : ^A! A, which ogehe wih he iclusio : L A,! A yields a ma : A! A which is a isomohism by he Huewicz Theoem (cf. [B7, Lemma 2.5]). Thus we may se A : '( ^A ), ha is, he fee -algeba geeaed by fg 2 ^T. Comae [K, x3] Theoem. Evey fee simlicial -algeba esoluio A! X of a ealizable -algeba X is iself ealizable by a CW esoluio R! X i sg. Poof. By Poosiios 3.17 ad 3.19 we may assume A has a (fee) CW basis ( A ) 1, ad ha hee is a esoluio by shees Q! X (i sg) ad a embeddig of simlicial -algebas : A! Q. We may also assume ha Q is ba (x2.8), wih " Q : Q! X a baio. We shall acually ealize by a ma of bisimlicial gous f : R! Q. Noe ha oce R has bee deed hough simlicial dimesio, fo ay k we have a commuaive diagam ` (d ) k C R - # k Z 1 R (j 1 ) - # k C 1 R (d ) - # k Z 2 R (j 2 ) - # k C 2 R - 1 C k R d Z 1 k R ic. C 1 k R d 1-2 Z 2 k R ic. C 2 k R (obaied by ig ogehe hee of he log exac sequeces of he baios (2.1)). The veical mas ae iduced by he iclusios C R,! R, ad so o { see Poosiio The oly diculy i cosucig R is ha Poosiio 2.11 does o hold fo Z { i.e., he mas i he above diagam i geeal eed o be isomohisms { so we may have a eleme i Z A eeseed by 2 C R C R wih (d ) # () 6 (bu of couse (j 1 ) # (d ) # () ). I his case we could o have 2 C +1 R C +1 A wih (j ) # (d +1 ) # (), so R would o be acyclic. I is i ode o avoid his diculy ha we eed he embeddig, sice by deiio his cao hae fo Q : we kow ha d : C Q! Z 1 Q is sujecive fo each >, so 1 : Z 1 Q! Z 1 Q is, oo, which imlies ha fo each > : (3.21) Imf(d +1 ) # : C +1 Q! Z Q g \ Kef(j ) # : Z Q! C Q g which we shall call Poey (3.21) fo Z Q. (This imlies i aicula ha Z Q Kef(d ) # : C Q! Z 1 Q g.) Noe ha give ay ba K 2 sg havig Poey (3.21) fo Z m K fo each < m, if we coside he log exac sequece of he baio d m : C m K!

10 - 1 DAVID BLANC Z m1 K : (3.22) (d : : : k+1 C m K m ) #! k+1 Z m1 K m1 (j m1 ) #! k Z m K! k C m1 K : : : ; we may deduce ha (3.23) m j Im( m1 ) is oe-o-oe, ad sujecs oo Im( m ) fo < m. We ow cosuc R by iducio o he simlicial dimesio: (i) Fis, choose a baio " R : R! X ealizig " A : A! X. By x3.2, hee is a ma f : R! Q ealizig, so " Q f "R ; sice " Q is a baio, we ca chage f o f : R! Q wih " Q f " R. (ii) Le Z R deoe he be of " R. Sice " R # "A is a sujecio, we have Z R Ke(" R # ) Z A, ad d A mas C 1 A oo Z A, so d A : A1! A facos hough Z R, ad we ca hus ealize i by a ma d R : R 1! Z R. Se R 1 : R1 q L 1 R (so R 1 A 1 ), wih 1 : R 1! M 1 R R R equal o ( d R ; ), ad chage 1 o a baio 1 : R 1! M 1 R. Agai we ca ealize 1 : A 1! Q 1 by f 1 : R 1! Q 1 wih Q 1 f 1 f R 1, sice Q 1 is a baio; so we have deed 1 f : 1 R! 1 Q ealizig 1. (iii) Now assume we have f : R! Q ealizig, wih Poey (3.21) holdig fo Z m R fo < m <. Fo each -algeba geeao 2 A+1 (i degee k, say), (3.21) imlies ha d +1 () 2 Ke(d ) Ke((d R ) # ) (C A ) k k C R, so by he exacess of (3.22) we ca choose 2 k Z R such ha (j ) # d +1 (). This allows us o dee d R : R+1! Z R so ha (j ) # ( d R ) # ealizes (ic.)d A : A+1! C A, as well as f+1 : R +1! C Q ealizig +1 j A+1. Because A +1 R+1 is a fee -algeba, his imlies he homooy-commuaiviy of he oue ecagle i f +1 R - +1 C +1Q d R Z f Z R d Q Z Q j R C f C R - j Q C Q (as well as he lowe squae, by he iducio hyohesis). Thus j Q Z f d R j Q dq f +1, so (j Q) # (Z f) # ( d R ) # (j Q) # (d Q ) # ( f +1 ) #. By (3.21) his imlies (Z f) # ( d R ) # (d Q ) # ( f +1 ) #, so (sice R+1 is a fee -algeba) also Z f d R dq f +1 { which meas ha we ca choose f +1 so ha Z f d R d Q f +1 (sice d Q is a baio). Thus if we se +1 R : R +1! M +1 R o be ( d R ; ; : : : ; ), we have M +1 f +1 R +1 Q f +1.

11 CW RESOLUTIONS OF SPACES 11 If R +1 : R +1 R +1 (i he oaio of x2.5 & 2.12) we se R +1 : R +1 q L +1 R, ad dee : +1 R! M +1 +1R, ad f : +1 R! Q esecyively by : ( +1 R R ) ad f : ( f +1 L +1 f). We see ha (f+1) # +1 ad M +1 f +1 Q +1 f+1, ad his will sill hold if we chage +1 io a baio, ad exed f +1 o f +1 : R +1! Q +1. This dees +1 f : +1 R! +1 Q ealizig +1. (iv) I emais o veify ha +1 R so deed saises (3.21). Howeve, (3.23) imlies ha we have a ma of sho exac sequeces: Im( 1 ) ic. - R k Z R f Im( 1 ) ic. - Q k Z Q (Z f) # Im((j R ) #) Z A - Z Im((j Q ) #) Z k Q - i which he lef veical ma is a isomohism ad he igh ma is oe-o-oe, so (Z f) # is oe-o-oe, oo. Theefoe, Ke((j R ) # ) Ke((j R ) # ) \ Z R, which imlies ha Poey (3.21) holds fo Z R, oo. This comlees he iducive cosucio of R. We also have a aalogous esul fo mas: Theoem. If K " K "! X ad L L! Y ae wo fee simlicial -algeba esoluios, g : X! Y is a ma i G, ad ' : K! L is a mohism of simlicial -algebas such ha " L ' g " K, he ' is ealizable by a ma f : A! B i sg. Poof. Choose fee CW bases fo K ad L, ad ealize he esulig CW esoluios by A ad B esecively, whee (as i he oof of Theoem 3.2) we may assume d : C B! Z 1 B is a baio fo each. f : A! B will be deed by iducio o : ' : K! L may be ealized by a ma f : A! B (x3.2), ad sice " B is a baio ad " B f g " A, we ca choose a ealizaio f fo ' such ha " B f g " A. I geeal, ' ' j K : K! C L may be ealized by a ma f : A! C B (Poosiio 2.11), ad sice d : C B! Z 1 B is a baio, we may choose f so d f Z 1 f d : A! Z 1 B. By iducio his yields a ma f L f f : A L A q A! L B q B B such ha B f M f A : A! M B, so f is ideed a simlicial mohism (ealizig ). 4. The simlicial ba cosucio As a alicaio of Theoem 3.2, we descibe a obsucio heoy fo deemiig whehe a give sace X is, u o homooy, a loo sace (ad hus a oological gou { see [Mi1, x3]). I he ex wo secios we o loge eed o wok wih simlicial gous, so we eve o he moe familia caegoy of oological saces; we ca sill uilize he esuls of he evious secio via he adjoi ais of (2.3) Deiio. A -cosimlicial objec E ove a caegoy C is a sequece of objecs E ; E 1 ; : : :, ogehe wih coface mas d i : E! E +1 fo 1 1

12 12 DAVID BLANC saisfyig d j d i d i d j1 fo i < j (cf. [RS]). Give a odiay cosimlicial objec E (cf. [BK, X, 2.1]), we le E deoe he udelyig -cosimlicial objec (obaied by fogeig he codegeeacies) The cosimlicial James cosucio. Give a sace X 2 T, we dee a -cosimlicial sace U U(X) by seig U X +1 (he Caesia oduc), ad d i (x ; : : : ; x ) (x ; : : : ; x i1 ; ; x i ; : : : ; x ). Noe ha colimu(x) JX (he James educed oduc cosucio), ad 4.3. Fac. If hx; mi is a (sicly) associaive H-sace, we ca exed U o a full cosimlicial sace U by seig s j (x ; : : : ; x ) (x ; : : : ; m(x j ; x j1 ); : : : ; x ) Deiio. Le A be a CW -esoluio of he -algeba X U, as i x3.14. We cosuc a -cosimlicial augmeed simlicial -algeba (E )! U, such ha each E is a CW -esoluio of U (X +1 ), wih CW -basis fe g 1. We sa by seig E C A fo all, ad he dee E by a double iducio (o ad he o ) as (4.5) E a a [ C ] I ; I2I ; whee I ; is as i (3.13) ad C m C fo all m;. The coface mas d i : E 1! E ae deemied by he cosimlicial ideiies ad he equieme ha d i j [ C ] (i1 be a isomohism oo [ C ;:::;i) ] (i1 ;:::;i;i) if i > i. The oly summad i (4.5) which is o deed is hus [ C ] ;, which we deoe simly by C. We equie ha i be a -h coss-em i he sese ha d j C does o faco hough he image of ay coface ma d i : E 1 1! E1. Ohe ha ha, C may be ay fee -algeba which esues ha (4.5) dees a CW -basis fo a CW -esoluio E! U. We shall call he double sequece (( C )1 1 )1 a 1 coss-em basis fo (E ). Noe ha A is a eac of E 2 i wo diee ways (ude he wo coface mas d, d 1 ), coesodig o he fac ha X is a eac of X X i wo diee ways; he esece of he coss-ems C 2 idicaes ha A A is a esoluio of X 2, bu o a fee oe, while A q A is a fee simlicial -algeba, bu o a esoluio. Similaly, X X embeds i X 3 i hee diee ways, ad so o Examle. Fo ay A! X we may se C 2 1 `S x,!a () Sy q,!a (1) wih [ x ; y ] (i he oaio of x3.3). The highe coss-ems C 1 d j S +q1 (x;y) ` S +q1, (x;y) fo 3, sice ay k-h ode coss-em eleme z i ` j A(j) (k 3) is a sum of elemes of he fom z # [: : : [[ 1 ; (x 1 ) 2 ]; (x 2 ) 3 ]; : : : ; (x 3 ) k (x k )], ad he z d ( # [: : : [ (x 1 ;x 2 ) ; s 3 (x 3 ) ]; : : : ; s k (x k ) ]): 4.7. Deiio. Le h (W )! U be he -cosimlicial augmeed simlicial sace u-o-homooy which coesods o (E )! U via x3.2. Thus he vaious (co)simlicial mohisms exis, ad saisfy he (co)simlicial ideiies, oly i

13 CW RESOLUTIONS OF SPACES 13 he homooy caegoy (we may choose eeseaives i T, bu he he ideiies ae saised oly u o homooy). Each W is homooy equivale o a wedge of shees, ad has a wedge summad W,! W coesodig o he CW -basis fee -algeba summad E,! E. We le C deoe he wedge summad of W coesodig o C,! E Deiio. A simlicial sace V 2 st is called a ecicaio of a simlicial sace u-o-homooy h W if V ' W fo each, ad he face ad degeeacy mas of V ae homooic o he coesodig mas of h W. See [DKS, x2.2], e.g., fo a moe ecise deiio; fo ou uoses all we equie is ha V be isomohic (as a simlicial -algeba) o ( h W ). Similaly fo ecicaio of (-)cosimlicial objecs, ad so o. By cosideig he oof of Theoem 3.2, we see ha we ca make he followig 4.9. Assumio. (E ) mas moomohically io V (U ), ad h (W )! U ca be ecied so as o yield a sic -cosimlicial augmeed simlicial sace (W )! U ealizig (E )! U Deiio. Now assume ha X is a abelia -algeba (Def. 3.4) { his is he ecessay -algeba codiio i ode fo X o be a H-sace { ad le : X X! X be he mohism of -algebas deed levelwise by he gou oeaio (see [B6, x2]). This is of couse associaive, i he sese ha (; id) (id; ) : (X 3 )! X, so i allows oe o exed he -cosimlicial -algeba F : (U ) o a full cosimlicial -algeba F, deed as i x4.3. Sice E! F U is a fee esoluio of -algebas, he codegeeacy mas s j : F! F 1 iduce mas of simlicial -algebas s j : E! E 1, uique u o simlicial homooy, by he uivesal oey of esoluios (cf. [Q1, I, & II, x2, Po. 5]). Noe, howeve, ha he idividual mas s j : E! E1 ae o uique, i geeal; i fac, diee choices may coesod o diee H-mulilicaios o X. These mas s j make (E )! F io a full cosimlicial augmeed simlicial -algeba E! F, ad hus h W! U io a cosimlicial augmeed simlicial sace u-o-homooy (fo which we may assume by 4.9 ha all simlicial ideiies, ad all he cosimlicial ideiies ivolvig oly he coface mas, hold ecisely) Poosiio. The cosimlicial simlicial sace u-o-homooy h W of x4.1 may be ecied if ad oly if X is homooy equivale o a loo sace. Poof. If X is a loo sace, i has a sicly associaive H-mulilicaio m : X X! X which iduces o () (cf. [G, Po. 9.9]), so U exeds o a cosimlicial sace U by Fac 4.3. Alyig he fucoial cosucio of [S, x2] o U yields a (sic) cosimlicial augmeed simlicial sace (V )! U, ad sice we assumed W embeds i V fo each, h W may also be ecied. Covesely, if W is a (sic) cosimlicial simlicial sace ealizig E, he we may aly he ealizaio fuco fo simlicial saces i each cosimlicial dimesio o obai kwk ' U X +1 (by x3.1). The ealizaio of he codegeeacy ma ks k : kwk 1! kwk iduces : (X 2 )! X, so i coesods o a H-sace mulilicaio m : X 2! X (see [B6, Po. 2.7]).

14 14 DAVID BLANC The fac ha kwk is a (sic) cosimlicial sace meas ha all comosie codegeeacy mas ks s j 1 s j 1 k : kwk! kwk ae equal, ad hus all ossible comosie mulilicaios X +1! X (i.e., all ossible backeigs i (2.6)) ae homooic, wih homooies bewee he homooies, ad so o { i ohe wods, he H-sace hx; mi is a A 1 sace (see [S3, Def. 11.2]) { so ha X is homooy equivale o loo sace by [S3, Theoem 11.4]. Noe ha we oly equied ha he codegeeacies of h W be ecied; afe he fac his esues ha he full cosimlicial simlicial sace is eciable. I summay, he quesio of whehe X is a loo sace educes o he quesio of whehe a ceai diagam i he homooy caegoy, coesodig o a diagam of fee -algebas, may be ecied { o equivalely, may be made 1-homooy commuaive. 5. Polyheda ad highe homooy oeaios As i [B5, x4], hee is a sequece of highe homooy oeaios which seve as obsucios o such a ecicaio, ad hese may be descibed combiaoially i ems of ceai olyheda, as follows: 5.1. Deiio. The N-emuohedo P N is deed o be he covex hull i R N of he ois ((1); (2); : : : ; (N)), whee ages ove all emuaios 2 N (cf. [Z, x9]). I is (N 1)-dimesioal. Fo ay wo ieges < N, he coesodig (N; )-face-codegeeacy olyhedo P N is a quoie of he N-emuohedo P N obaied by ideifyig wo veices ad o a sigle veex of P N wheeve (i; i+1), whee (i; i + 1) is a adjace asosiio ad (i); (i + 1) >. Sice each face A of P N is uiquely deemied by is veices (see below), he faces i he quoie P N ae obaied by collasig hose of P N accodigly. Noe ha P N N1 is he N-emuohedo P N, ad i fac he quoie ma q : P N! P N is homooic o a homeomohism (hough o a combiaoial isomohism, of couse) fo 1. O he ohe had, P N is a sigle oi. Fo o-ivial examles of face-codegeeacy olyheda, see Figues 1 & 2 below Fac. Fom he desciio of he faces of he emuohedo give i [GG], we see ha P N has a edge coecig a veex o ay veex of he fom (i;i+1) (uless (i); (i + 1) >, i which case he edge is degeeae). Moe geeally, le be ay veex of P N. The faces of P N coaiig ae deemied as follows: Le P h1; 2; : : : ; `1 j `1 + 1; : : : ; `2 j : : : j `i1 + 1; : : : ; `i j : : : j `1 + 1; : : : ; N i be a aiio of 1; : : : ; N io cosecuive blocs, subjec o he codiio ha fo each 1 j < a leas oe of (`i), (`i+1 ) is. Deoe by i he umbe of j's i he i-h bloc (i.e., `i1 + 1 j `i) such ha (j). The P N will have a subolyhedo Q(P) (coaiig ) which is isomohic o he oduc P`1 1 P`2`1 2 P`i`i1 i P N`1 : This follows fom he desciio of he faces of he N-emuohedo i [B5, x4.3].

15 CW RESOLUTIONS OF SPACES 15 We deoe by (P N ) (k) he uio of all faces of P N of dimesio k. I aicula, fo 1 we have P N : (P N ) (N2) S N2, sice he homeomohism ~q : P N! P N eseves P N Facoizaios. Give a cosimlicial simlicial objec E as i x4.1, ay comosie face-codegeeacy ma : E +k! m+` Ek` has a (uique) caoical facoizaio of he fom, whee : E +k! m+` Ek m+` may be wie s j 1 s j 2 : : : s j fo j 1 < j 2 < : : : < j < +k ad : E k! m+` Ek` may be wie d i1 d i2 d i fo i 1 < i 2 < : : : < i m + `. Le D( ) deoe he se of all ossible (o ecessaily caoical) facoizaios of as a comosie of face ad codegeeacy mas: +m 1. We dee ecusively a bijecive coesodece bewee D( ) ad he veices of a ( + m)-emuohedo P +m, as follows (comae [B5, Lemma 4.7]): The caoical facoizaio d i1 d i2 d i s j 1 s j 2 s j coesods o he veex id. Nex, assume ha he facoizaio +m : : : 1 coesods o. The he facoizaio coesodig o, fo (i; i+1), is obaied fom 1 +m by swichig i ad i+1, usig he ideiy s j s i s i1 s j fo i > j if i ad i+1 ae boh codegeeacies, ad he ideiy d i d j d j1 d i fo i < j if hey ae boh face mas. Passig o he quoie face-codegeeacy olyhedo, we see ha he veices of ae ow ideied wih facoizaios of of he fom P +m (5.4) E +k m+` s j! E +k1 m+` : : : E +1 m+` s j 1! E m+`! E m : : : E 1 m 1 j s 1! : : : E +1 m 1 j s! E m 1! E m whee i is a comosie of face mas (i.e., we do o disiguish he diee ways of decomosig i as d k1 : : : d k ). The collecio of such facoizaios of will be deoed by D( ), whee is he obvious equivalece elaio o D( ). We shall deoe he face-codegeeacy olyhedo P +m wih is veices so labelled by P +m ( ). A examle fo d d 1 s s 1 aeas i Figue Noaio. Fo : E +k! m+` Ek` as above, we deoe by C( ) he collecio such ha is of he of all comosie face-codegeeacy mas : E ()+k()! m()+`() Ek() `() fom s (1 s ) fo some decomosiio 1 s j s j 1 1 s j 1 s j of (5.4). Tha is, we allow oly hose subsequeces b ; : : : ; a of a facoizaio +m 1 i D( ) which ae comaible wih he equivalece elaio i he sese ha b+1 ad b ae o boh face mas, ad similaly fo a1 ad a. Such a will be called allowable Highe homooy oeaios. Give a cosimlicial simlicial sace u-ohomooy h W as i x4.2, we ow dee a ceai sequece of highe homooy oeaios. Fis ecall ha he half-smash of wo saces X; Y 2 T is X Y : (X Y)(X fg); if X is a susesio, hee is a (o-caoical) homooy equivalece X Y ' X ^ Y _ X Deiio. Give a comosie face-codegeeacy ma : W +k m+`! Wk` as above, a comaible collecio fo C( ) ad h W is a se fg g 2C( ) of mas

16 CC C C C C H H H H J J J J Q J J Q Q J J J J J J J J J J 16 DAVID BLANC s dd1 s 1 s dd s 1 ds d s 1 s ds 1 d1 ds d1s 1 1 ds s d1 s ds 1 d H H ds s 1 d d s s d1 Q Q s ds d1 s 1 s s dd1 s dd1 1 s s dd s s dd s dd1s dd1 1 s s dd1s s 1 dds s d ds s 1 s ds d d s s d ds d1s s dd s ds ds Figue 1. The face-codegeeacy olyhedo P 4 2 (d d 1 s s 1 ) g : P ()+m() () W ()+k() m() codiio:! m()+`() Wk() `() fo each 2 C( ), saisfyig he followig Assume ha fo such a 2 C( ) we have some decomosiio 1 s j : : : s j 1 1 s j 1 s j i D(), as i (5.4), ad le P h1; : : : ; `1 j : : : j `i1 + 1; : : : ; `i j : : : j `1 + 1; : : : ; i be a aiio of (1; : : : ; ) as i x5.2, yieldig a sequece of comosie facecodegeeacy mas i 2 C() C( ) fo i 1; : : : ;. Le Q(P) P`1 1 ( 1 ) P`i`i1 i ( i ) P `1 ( ) be he coesodig sub-olyhedo of P ()+m() (). The we equie ha g j m() Q(P)W be he ()+k() m()+`() comosie of he coesodig mas g i i he sese ha (5.8) g (x 1 ; : : : ; x ; w) g 1 (x 1 ; g 2 (x 2 ; : : : ; g (x ; w) : : : )) fo x i 2 P`i`i1 i ( i ) ad w 2 W ()+k(). m()+`() We fuhe equie ha if 1 is of legh 1, he g mus be i he escibed homooy class of he face o codegeeacy ma 1. Thus i aicula, fo each veex of P +m ( ), idexed by a facoizaio 1 i D( ), he ma g j f gw +` eeses he class [ 1 ]. m+k 5.9. Fac. Ay comaible collecio of mas fg g 2C( ) fo C( ) iduces a ma f f : P +m W +k! m+` Wk` (sice all he faces of P +m ae oducs of face-codegeeacy olyheda of he fom P ()+m() () fo 2 C( ), ad codiio () (5.8) guaaees ha he mas g agee o iesecios).

17 CW RESOLUTIONS OF SPACES Deiio. Give h W as i x4.1, fo each k 2 ad each comosie facecodegeeacy ma : W +k m+`! Wk`, he k-h ode homooy oeaio associaed o h W ad is a subse h i of he ack gou [ +m2 Wm+`; +k Wk` ], deed as follows: Le S [P +m Wm+`; +k Wk` ] be he se of homooy classes of mas f f : P +m W +k! m+` Wk` which ae iduced as above by some comaible collecio fg g 2C( ) fo C( ). Now choose a sliig (5.11) P +m ( ) W +k m+` S +m2 W +k ' m+` (S+m2 ^ W k` ) _ W k` ad le h i [ +m2 Wm+`; +k Wk` ] be he image of he subse S ude he esulig ojecio. I is clealy a ecessay codiio i ode fo he subse h i o be o-emy ha all he lowe ode oeaios hi vaish (i.e., coai he ull class) fo all 2 C( )f g { because ohewise he vaious mas g : P ()+m() () W ()+k()! W k() m() m()+`() `() cao eve exed ove he ieio of (). A suce codiio is ha P ()+m() m() he oeaios hi vaish coheely, i he sese ha he choices of comaible collecios fo he vaious be cosise o commo subolyheda (see [B5, x5.7] fo he ecise deiio, ad [B5, x5.9] fo he obsucios o coheece). O he ohe had, if h W is he cosimlicial simlicial sace u-o-homooy of x4.4 (coesodig o he cosimlicial simlicial -algeba (E ) wih he CW -basis fe g 1 ;), he he vaishig of he homooy oeaio h j C i { wih esiced o he (; )-coss-em { imlies he vaishig of h i, fo ay : W +k! m+` Wk` (assumig lowe ode vaishig). This is because ouside of he wedge summad C, he ma is deemied by he mas 2 C( ) ad he coface ad degeeacy mas of h W, which we may assume o 1-homooy commue by iducio ad 4.9 esecively. We may hus sum u he esuls of his secio, combied wih Poosiio 4.11, i: Theoem. A sace X 2 T, fo which X is a abelia -algeba, is homooy equivale o a loo sace if ad oly if all he highe homooy oeaios h j C i deed above vaish coheely Remak. As obseved i x4.2, fo ay X 2 T he sace JX is he colimi of he -cosimlicial sace U(X), ad i fac he -h sage of he James cosucio, J X, is he (homooy) colimi of he ( 1)-coskeleo of U. Thus if we hik of he sequece of highe homooy oeaios \i he simlicial diecio" as obsucios o he validiy of he ideiy [B7, Thm. 5.7()] (u o 1-homooy commuaiviy), he he -h cosimlicial dimesio coesods o veifyig his ideiy fo f i A : A! FB of James laio + 1 (cf. [J2, x2]). I aicula, if we x k `, 1 ad oceed by iducio o m, we ae comuig he obsucios fo he exisece of a H-mulilicaio o X, as i [B6]. (Thus if X is edowed wih a H-sace sucue o begi wih, hey mus all vaish.) Obseve ha he face-codegeeacy olyhedo P 1 is a ( 1)-cube, as i Figue 2, ahe ha he ( 1)-simlex we had i [B6, x4] { so he homooy oeaios we obai hee ae moe comlicaed. This is because hey ake value i he homooy gous of shees, ahe ha hose of he sace X.

18 18 DAVID BLANC s dd1d s ddd s ddd1 s ddd2 s dd1d2 s dd1d1 d s d1d dds d ds dd ds dd1 ds d d2 ds d1d2 ds d1 d1 ddd2 s dd1 s d dd1d s dd1s d1 d ds d1 ddds dds d2 dd1s d2 dd1d2 s dd1d1s ddd1 s Figue 2. The face-codegeeacy olyhedo P 4 1(d d 1 d 2 s ) As a coollay o Theoem 5.12 we may deduce he followig esul of Hilo (cf. [H, Theoem C]): Coollay. If hx; mi is a (1)-coeced H-sace wih i X fo i 3, he X is a loo sace, u o homooy. Poof. Choose a CW -esoluio of X which is ( 1)-coeced i each simlicial dimesio, ad le E be as i x4.4. By deiio of he coss-em -algebas C i x4.4, hey mus ivolve Whiehead oducs of elemes fom all lowe ode cossems; bu sice X is a H-sace by assumio, all obsucios of he fom h j C i 1 vaish (see x5.13). Thus, he lowes dimesioal obsucio ossible is a hid-ode oeaio h j C i ( 2), which ivolves a ile Whiehead oduc ad hus akes 2 value i i W k` fo i 3. If we aly he (31)-Posikov aoximaio fuco o h W i each dimesio, o obai h Z, all obsucios o ecicaio vaish, ad fom he secal sequece of x3.1 we see ha he obvious ma X kw 1k! kz1 k iduces a isomohism i i fo i < 3. Sice kz 1 k is a loo sace by Theoem 5.12, so is is (3 1)-Posikov aoximaio, amely X Examle. The 7-shee is a H-sace (ude he Cayley mulilicaio, fo examle), bu oe of he 12 ossible H-mulilicaios o S 7 ae homooyassociaive; he s obsucio o homooy-associaiviy is a ceai \seaaio eleme" i 21 S 7 (cf. [J1, Theoem 1.4 ad Coollay 2.5]). Sice S 7 is a fee -algeba, i has a vey simle CW -esoluio A! S 7, wih A S 7 (geeaed by 7 ), ad A fo 1. A coss-em basis (x4.4) fo he cosimlicial simlicial -algeba E of x4.1 is he give i dimesios < 24 by: C 1 1 S 13, wih d 13 [d 7 ; d 1 7 ]; C 2 2 S 19, wih d 19 [d 13 ; s d 2 d 1 7 ] [d 1 13 ; s d 2 d 7 ] + [d 2 13 ; s d 1 d 7 ]; C is a leas 24-coeced fo all ohe,. We se s j j C fo all 2; his deemies E i degees 21 ad cosimlicial dimesios 2.

19 CW RESOLUTIONS OF SPACES 19 By Remak 5.13, he wo secoday oeaios hd s j C 1 1i ad hd 1s j C 1 1 i mus vaish; o he ohe had, by Coollay 5.14 all obsucios o S 7 beig a loo sace ae i degees 21, so he oly eleva coss-em is C 2 2, wih hee ossible hid-ode oeaios h jc 2 2i, fo d d 1 s s 1, d d 2 s s 1, o d 1 d 2 s s 1. The coesodig face-codegeeacy olyheda P2 4 ( ) is as i Figue 2. I is saighfowad o veify ha he oeaios h j C 2 2i ae ivial fo d d 2 s s 1 o d 1 d 2 s s 1 (i fac, may of he mas g, fo 2 C( ), may be chose o be ull). O may also show ha hee is a comaible collecio fg g 2C(') fo ' d d 1 s s 1, i he sese of x5.7, so ha he coesodig subse h'jc 2 2i 21S 7 is o-emy; i fac, i coais he oly ossible obsucio o he 21-Posikov aoximaio fo S 7 o be a loo sace. The exisece of he eiay oeaio h'jc 2 2 i coesods o he fac ha he eleme [[ 7 ; 7 ]; 7 ] [[ 7 ; 7 ]; 7 ] + [[ 7 ; 7 ]; 7 ] 2 21 S 7 is ivial \fo hee diee easos": because of he Jacobi ideiy, because all Whiehead oducs vaish i S 7, ad because of he lieaiy of he Whiehead oduc { i.e., [; ]. O he ohe had, we kow ha hee is a 3-imay obsucio o he homooyassociaiviy of ay H-mulilicaio o S 7, amely he eleme # S 7 (see [J1, Theoem 2.6]). We deduce ha 62 h'jc 2 2 i, ad i fac (modulo 3) his eiay oeaio cosiss exacly of he elemes 14 # 7. Fo a deailed calculaio of such highe ode oeaios usig simlicial esoluios of -algebas, see [B6, x4.13] Remak. Ou aoach o he quesio of whehe X is a loo sace is clealy based o, ad closely elaed o, he classical aoaches of Sugawaa ad Sashe (cf. [S1, S2, Su]. Oe migh wode why Sashe's associaheda K i (cf. [S1, x2,6]) do o show u amog he face-codegeeacy olyheda we descibe above. Aaely his is because we do o wok diecly wih he sace X, bu ahe wih is simlicial esoluio, which may be hough of as a \decomosiio" of X io wedges of shees. Refeeces [B1] D. Blac, \A Huewicz secal sequece fo homology", Tas. AMS 318 (199) No. 1, [B2] D. Blac, \Deived fucos of gaded algebas", J. Pue Al. Alg. 64 (199) No. 3, [B3] D. Blac, \Abelia -algebas ad hei ojecive dimesio", i M.C. Tagoa, ed., Algebaic Toology: Oaxeec 1991 Coem. Mah. 146, AMS, Povidece, RI [B4] D. Blac, \Oeaios o esoluios ad he evese Adams secal sequece", Tas. AMS, 342 (1994), No. 1, [B5] D. Blac, \Highe homooy oeaios ad he ealizabiliy of homooy gous", Poc. Lod. Mah. Soc. (3) 7 (1995), [B6] D. Blac, \Homooy oeaios ad he obsucios o beig a H-sace", Maus. Mah. 88 (1995) No. 4, [B7] D. Blac, \Loo saces ad homooy oeaios", Fud. Mah. 154 (1997), [B8] D. Blac, \Homooy oeaios ad aioal homooy ye", ei [B9] D. Blac, \Algebaic ivaias fo homooy yes", ei [BS] D. Blac & C.S. Sove, \A geealized Gohedieck secal sequece", i N. Ray & G. Walke, eds., Adams Memoial Symosium o Algebaic Toology, Vol. 1, Lod. Mah. Soc. Lec. Noes Se. 175, Cambidge U. Pess, Cambidge, 1992,

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