Moore spaces in proper homotopy

TSUKUBA J. $MA^{ }I^{ backslash }H$ . Vol. 19 No. 2 (1995), 305–327 MOORE SPACES IN PROPER HOMOTOPY By R. AYALA, E. DOM ’INGUEZ, A. M ’ARQUEZ and A. QUINTERO Abs ac . Moo e spaces a e de ined in p ope homo opy heo y. Some esul s on he exis ence and uniqueness o hose spaces a e p o en. An example o wo non p ope ly equi alen Moo e spaces is gi en. In oduc ion. The pu pose o his pape is o p o ide he co ec s a emen s and de ails o he esul s announced in [4]. Namely, we p o e he exis ence o p ope Moo e spaces o ypes o ype $(S;n)$ o ce ain objec s $S$ in he abelian ca ego y o owe s o g oups ( ow-JZ $b,$ $ llco ne b$ ) and $n geqq 2$ (Theo em 2.9). Ne e heless objec s can be o p ojec i e dimension 2 in ( ow-db, $ io a Ab$ ), and his ac de e mines an obs uc ion o he uniqueness o p ope Moo e spaces (Theo em 3.2). In ac , an example o wo non p ope ly equi alen Moo e spaces is gi en in Appendix A. As a consequence o Theo em 3.2 wo su icien condi ions o he uni- queness o p ope Moo e spaces a e s a ed (Co olla y 3.7 and P oposi ion 3.9). The exis ence o Moo e spaces in p ope homo opy was al eady announced in [4], bu in ha pape he obs uc ion om Theo em 3.2 was no conside ed and he uniqueness o such spaces was w ongly asse ed. Fo owe s o p ojec i e dimension 1, p ope Moo e spaces beha e in a e y simila way o o dina y Moo e spaces. In pa icula , o p ojec i e dimension 1, p ope Moo e spaces de ine p ope homo opy g oups, and a coe icien exac sequence which gene alize he a ious p ope homo opy g oups known in he li e a u e and hei co esponding Milno exac sequences (Examples 2.15). 1. P elimina ies and No a ions CATEGORIES OF TOWERS. Gi en a ca ego y $c$ , he ca ego y o owe s o $C$ , ow-C, is he ca ego y o in e se sequences $s= {A_{1} le a ow A_{2} le a ow cdo s }$ in $C$ whe e a ( ow-C)-mo phism $ :S igh a ow S^{ p ime}$ is ep esen ed by a sequence o C-mo phisms 1980 Ma hema ics Subjec Classi ica ion. $54C10,55P99,55Q70$ . Key wo ds: Moo e space, homology decomposi ion, p ope homo opy, owe o g oups. Recei ed Augus 25, 1993. 306 R. AYALA, E. $D0M NGUEZ$ , A. MARQUEZ and A. QUINTERO $ _{k}$ : $A_{n_{k}} igh a ow A_{k}^{ p ime},$ $ n_{1}<n_{2}< cdo s$ , such ha gi en $ >s$ he e exis s $j>n_{ },$ $n_{s}$ making commu a i e he diag am $A_{n} upa ow^{ } unde line{ _{ }}A_{ }^{ p ime,}$ $A_{j}$ $A downa ow_{n},$ $ unde line{ _{1}}A$ ; whe e he maps wi hou name a e bonding maps. We a e in e es ed in he ull subca ego y o $ o albox{ small REJECT}_{o }$ ( ow-C) whose objec s a e a ows $ :X igh a ow A$ whe e :IE7 is a ( ow-C)-objec and $A$ is a C-objec ega ded as a cons an owe whose bonding maps a e he iden i y. This ca ego y is deno ed ( ow-C, $C$ ). A ( ow-C, $C$ )-mo phism om $ :x igh a ow A$ o $g: e a igh a ow B$ can be ega ded as a $C$ -mo phism be ween $A$ and $B$ and a ( ow-C)-mo phism om ec o $Q $ such ha bo h mo phisms a e compa ible ia he bonding maps. I is con enien o ep esen ( ow-C, $C$ ) as ollows. Objec s a e owe s $X=$ $ {X_{0} le a ow X_{1} le a ow cdo s }$ ; a mo phism consis s o a map $ :X igh a ow Y$ in ow-C, oge he wi h a compa ible map $ _{0}$ : $X_{0} igh a ow Y_{0}$ in $C$ . We shall specially use he abo e cons uc ions o $c= ma hcal{F}_{0}p,$ $ ma hcal{G}_{ },$ $ llco ne lb$ ; he ca e- go ies o opological spaces, g oups and abelian g oups espec i ely. Since $ ow- ma hcal{A}6$ and ( ow-X6, a6) a e abelian ca ego ies (see [1]) we can de- ine ke nels and images and s a e exac sequences in a na u al way. In pa i- cula , we can use p ojec i e objec s and de ine he unc o Ex . See [10] o de ails. PROPER CATEGORY. A p ope map (p-map) is a con inuous map $ :X igh a ow Y$ such ha $ ^{-1}(K)$ is compac o each compac subse $K subse eqq Y$ . P ope homo opy (p-homo opy), p ope homo opy equi alence, e c $ cdo s$ can be de ined in he na u al way. We shall deal wi h he ca ego y $ ma hcal{P}$ o $T_{2}$ -locally compac $ sigma$ -compac spaces and p-maps. One can check ha $ ma hcal{P}$ is a co ib a ion ca ego y in he sense o H. Baues (see [5] and [3]) whose co ib a ions a e he $p$ -maps wi h he P ope Homo opy Ex ension P ope y. We call hese $p$ -maps p ope co ib a ions (p-co ib a- ions). I can be shown ha any p-co ib a ion is a closed embedding. We deno e p-co ib a ions by a ows angle $ igh a ow$ . Moo e spaces in p ope homo opy 307 Gl en a space $X$ in $ ma hcal{P}$ , a sys em o $ in y$ -neighbou hoods o $X$ is he objec o $ ow- ma hcal{G}_{op},$ $ epsilon(X)= {U_{1} le a ow U_{2} le a ow cdo s }$ whe e $ o e line{X-U}_{j}=K_{j}$ is compac , $K_{j} subse eqq K_{j+1}$ and $X=$ $ cup {in K_{j} ; j geqq 1 }$ . We ecall ha a CW-complex $X$ is said o be s ongly locally ini e i $X$ can be co e ed in a locally ini e way by ini e subcomplexes. In ha case, i is known ha $X$ admi s a coun able locally ini e co e by ini e subcomplexes and so he $ in y$ -neighbou hoods o $X$ can be chosen o be subcomplexes. Fini e dimensional locally ini e CW-complexes and locally ini e simplicial complexes a e s ongly locally ini e (see [11]). I $ :X igh a ow Y$ is a p-map and $ {U_{j} }$ and $ {W_{j} }$ a e sys ems o $ in y$ -neighbou hoods o $X$ and $Y$ espec i ely, o each $j$ he e exis s $k(j)$ such ha $ (U_{k(j)}) subse W_{J}$ and he e o e we ge a mo phism $ epsilon( ): epsilon(X) igh a ow epsilon(Y)$ (see [10] o mo e de ails). Gi en a space $X$ in $ ma hcal{P}$ a F euden hal end o $X$ is an elemen o he se $ ma hcal{F}(X)= lim_{ le a ow} pi_{0}(U_{j})$ whe e $ pi_{0}(-)$ deno es he se o connec ed componen s. Now le $ ma hcal{P}^{J}$ be he ca ego y $ ma hcal{P}$ unde $ J=[0, in y$ ) such ha o e e y $ ma hcal{P}^{J_{-}}$ objec $J igh a ow iX,$ $ io a$ is a p-co ib a ion. The ca ego y $ ma hcal{P}^{J}$ is a ca ego y o co ib a- ions whe e he no ion o p ope wedge $(V_{p})is$ de ined in a na u al way. The se o p-homo opy classes in $ ma hcal{P}^{J}$ will be deno ed by $[$ –, $-]_{p}^{J}$ . (1.0.1) The ca ego y o e $J,$ $ ma hcal{P}_{J}$ , is again a co ib a ion ca ego y, and i allows he de ini ion o p ope quo ien s. Mo e explici ly, i $ :X igh a ow J$ is a $ ma hcal{P}_{J^{-}}$ objec , and $A succ X$ he p ope quo ien $X/pA( )$ is de ined by he push-ou $ ln ma hcal{P}$ $ $ $_{1}I_{1}|q$ $ ee|$ $i$ $J succ---------*X/pA( )$ No ice ha $ o e line{i}$ is a p-co ib a ion whose image is $q(A)$ . In pa icula , $q(A)$ is homeomo phic o $J$ . Fu he mo e, his shows ha he p ope quo ien has he same o dina y homo opy ype as he usual opological quo ien . (1.0.2) I can be shown ha any space in $ ma hcal{P}$ admi s an on o p-may $h:X igh a ow J$ ([10; 6.5.3]). Ac ually, $h$ can be chosen o be a e ac ion o a gi en p-co ib a- ion $i:J subse eqq X$ . Indeed, since all $p$ -maps on $J$ a e p-homo opic one can ind a p-homo opy $H:J imes I igh a ow J$ connec lng $h|J$ o $id_{J}$ . Then by using he p ope H. E. P. one ge s a $p$ -homo opy $H^{ p ime}$ : $X imes I igh a ow J$ ex ending $H$ , and $H_{i}^{ p ime}$ is a p- e ac- ion o $i$ . (1.0.3) The p ope homo opy ype o $X/pA( )$ does no depend on he map 308 R. AYALA, E. DOM NGUEZ, A. MARQUEZ and A. QUINTERO $ ;X igh a ow J$ . Mo eo e , gi en ano he p-map $ ^{ p ime}$ : $X igh a ow J$ , he e exis s a p-homo opy equi alence $ xi:X/pA( ) igh a ow X/pA( ^{ p ime})$ such ha $q ci c xi=q^{ p ime}$ . This is due o he homo- opy in a iance o push-ou s in co ib a ion ca ego ies ([5: II. 1. $2b$ )]), since all p ope maps on $J$ a e p ope ly homo opic. When i is clea which map $ $ is in ol ed in he quo ien , we shall d op he map $ $ om he no a ion. (1.0.4) Finally, le $ ma hcal{P}^{*}$ deno e he ca ego y unde and o e $J$ . I can be shown ha $ ma hcal{P}^{*}$ is a co ib a ion ca ego y whe e we can de ine p ope wedges and p ope quo ien s as well as p ope cones $(c_{p})$ , and p ope suspensions $( Sigma_{p})$ . Fo a space $J succ X igh a ow J$ in $ ma hcal{P}^{*}$ , hese cons uc ions do no depend on he p- e ac ion $ $ (up o p-homo opy equi alence in $ ma hcal{P}^{J}$ ). Mo eo e , he se $[ Sigma_{p}X, Y]_{p}^{J}$ is endowed o a na u al g oup s uc u e o any space $Y$ in $ ma hcal{P}^{J}$ . See [3] o de ails. A s ongly locally ini e CW-complex $X$ will be conside ed as a space in $ ma hcal{P}^{*}$ by choosing a cellula embedding $i:J subse eqq X^{1}$ , and a $p$ - e ac ion o $i$ . See (1.0.2) abo e. I (X, $ alpha$ ) is a space in $ ma hcal{P}^{J}$ he $( ow- ma hcal{G}_{ }, 9_{ })$ -objec $ Pi_{n}(X, alpha)= { pi_{n}(X, *_{0})- pi_{n}(U_{1}, *_{1})- pi_{n}(U_{2}, *_{2}) cdo s }$ $(n geqq 1)$ is called he n- h homo opy owe o he pai (X, $ alpha$ ), whe e $ alpha( _{j})=*J$ wi h $ alpha([ _{j}, in y)) subse eqq U_{ }$ and he bonding maps a e induced by he inclusions and he base- poin change isomo phisms. (1.0.5) $A$ space $X$ is said o be p ope ly k-connec ed i $ ma hcal{F}(X)= {* }$ and $ Pi_{ }(X, alpha) cong 0$ in $( ow- ma hcal{G}_{ }, ma hcal{G}_{i})(0 leqq leqq k)$ . Simila ly o p ope pai s (X, $A$ ) wi h $ ma hcal{F}(X)= ma hcal{F}(A)= {* }$ . I is wo h poin ing ou ha al hough $ alpha$ and $ alpha^{ p ime}$ ep esen he same F euden hal end, he owe s $ Pi_{n}(X, alpha)$ and $ Pi_{n}(X, alpha^{ p ime})$ need no o be isomo phic (see [23; p. 13]). Ne e heless, i $X$ is p ope ly l-connec ed he e is no dependence on he ay $ alpha$ . In ac , any $p$ -homo opy $H: alpha cong alpha^{ p ime}$ induces a p o-isomo phism $H_{ $}$ ; $ Pi_{n}(X, alpha^{ p ime}) igh a ow Pi_{n}(X, alpha)$ . In addi ion, i $H: cong g$ is a p- homo opy he e is a commu a i e diag am $ Pi_{x}(X unde line{ alpha) o e line{g^{*} _{*} sim}},’ p od_{ Pi_{n}}n(Y, ci c alpha)(Y,g ci c alpha) downa ow G_{l}$ whe e $G=H ci c( alpha imes id)$ . The owe s $ Pi_{n}(X, A, alpha)(n geqq 2)$ a e also de ined o pai s (X, $A$ ) in $ ma hcal{P}^{J}$ . In Moo e spaces in p ope homo opy 309 addi ion, i $A=J$ we ha e he iden i ica ion $ Pi_{n}(X, J; alpha)= Pi_{n}(X, alpha)(n geqq 1)$ . We ecall ha E. B own in [7] gi es a unc o $P: ow- ma hcal{G}_{s} igh a ow ma hcal{G}_{ }$ which ca ies he owe $ Pi_{n}(X, alpha)$ o he B own-G ossman g oup $ pi_{n}^{ in y}(X, alpha)$ . In a simila way we can de ine a unc o $P_{0}$ : $( ow- ma hcal{G}_{ }, ma hcal{G}_{i}) igh a ow ma hcal{G}_{ }$ which maps $ Pi_{n}(X, alpha)$ o he global B own-G ossman g oup $ pi_{n}(X, alpha)$ (see [17]). The n- h homology owe o $X$ can be de ined as he owe $(n geqq 0)$ $H_{n}(X)= {H_{n}(X)-H_{n}(U_{I})-H_{n}(U_{2})- cdo s }$ whe e he bonding maps a e induced by he inclusions. The chain complex o owe s o $X,$ $C_{*}(X)$ is $ { pa ial:C_{n}(X) igh a ow C_{n-1}(X) }$ whe e $C_{n}(X)= {C_{n}(X) le a ow C_{n}(U_{1})-C_{n}(U_{2})- cdo s }$ . Now a p ope cohomology heo y, $H^{n}$ , wi h coe icien s in a ( ow-db, $ cup b$ ) $-$ objec $S$ is de ined as he homology o he complex $... le a ow C^{n}(X) le a ow C^{n-1}(X) le a ow cdo s$ whe e $C^{n}(X)=( ow_{c}- Ab, Ab)(C_{n}(X), S)$ (see [14] o de ails). Also ela i e e sions o hese unc o s o pai s (X, $A$ ) in $ ma hcal{P}$ a e de ined. No ice ha he abo e unc o s (including $ Pi_{n}$ ) a e well de ined up o ( ow- $ ma hcal{G}_{i},$ $ ma hcal{G}_{ }$ )-isomo phisms. Fundamen al esul s on homo opy g oups like he Blake s-Massey Theo em, he F euden hal Theo em o he Hu ewicz Theo em can be ansla ed o p ope homo opy by using he ollowing p oposi ion 1.1. $p_{ROPOSlT10N}.-([2;1.1])$ Le (X, $A$ ) be a connec ed s ongly locally ini e CW-pai wi h only one F euden hal end and assume ha (X, $A$ ) is p ope ly k-connec ed. Then he e exis s a s ongly locally ini e CW-pai (X’, $A^{ p ime}$ ) such ha i) $X$ ( espec i ely $A$ ) is a s ong de o ma ion p- e ac subcomplex o $X^{ p ime}$ ( espec i ely $A^{ p ime}$ ). ii) $(X^{ p ime})^{k} subse eqq A^{ p ime}$ . In pa icula i $A=J,$ $X$ has he same homo opy ype as a CW-complex $X^{ p ime}$ wi h $(X^{ p ime})^{k}=J$ . We ecall ha o any pai in $ ma hcal{P}$ and any p ope map $ ho:X igh a ow J$ , he e is a na u al homeomo phism $J cong q(A)$ , whe e $q:X igh a ow X/pA( ho)$ is he quo ien (see (1.0.1)). The e o e, o any ay $ alpha:J igh a ow X$ he map $q$ induces mo phisms o owe s 310 R. AYALA, E. DOM NGUEZ, A. MARQUEZ and A. QUINTERO $q_{*};$ $ Pi_{ }(X, A, alpha)- Pi_{ }(X/pA( ho), q^{0} alpha)$ 1.2. THEOREM.–Le (X, $A$ ) be a s ongly locally ini e CW-pai such ha (X, $A$ ) is p ope ly n-connec ed and $A$ is p ope ly m-connec ed $(n, m geqq 1)$ . Then he mo phism $q*de ined$ abo e is an isomo phism i $2 leqq leqq m+n$ , and an epimo - phism i $ =m+n+1$ . PROOF. Fi s ly, we shall p o e he heo em when $X^{n} subse eqq A$ and $A^{m}=J$ . Le $i:J igh a ow A$ be he inclusion. Acco ding o (1.0.2), we can ind a p ope e ac ion $ :X igh a ow J$ o $i$ . Le $ {U_{j}^{ p ime} }$ be a sys em o $ in y$ -neighbou hoods o $X$ con- sis ing o subcomplexes. Wi hou loss o gene ali y we can assume ha $ (U_{j}^{ p ime})$ $ subse eqq[ _{j}, in y)$ . Le $ U_{j}=U_{j}^{ p ime} cup[ _{ }, in y$ ). I is clea ha $ {U_{j} }$ is a new sys em o $ in y-$ neighbou hoods wi h $ U_{j}^{1}=[ _{j}, in y$ ), $(U_{j}, U_{j} cap A)$ is n-connec ed and $A_{j}=U_{j} cap A$ is m-connec ed o any $j geqq 0$ . Then, i $X/pA$ is he p ope quo ien cons uc ed wi h he e ac ion $ $ , i is easily checked ha $ {U_{j}/pAj }$ is a sys em o $ in y-$ neighbou hoods o $X/pA$ , whe e $U_{ }/pAj$ is cons uc ed by using he es ic ion $ |U_{j}$ : $ U_{j} igh a ow[ _{j}, in y$ ). Since p ope quo len s has he same o dina y homo opy ype as o dina y quo ien s, we can le elwise apply he o dina y Blake s-Massey Theo em [24; 6.22] o ge isomo phisms $q_{j* ddagge } pi_{ }(U_{j}, A_{j}) igh a ow pi_{ }(U_{j}/pA )$ i $ leqq n+m$ and epimo phisms i $ =m+n+1$ . Now he esul ollows when $ alpha=i$ and $ ho= $ . Mo eo e , we can use he na u ali y o he base ay change isomo - phisms and he homo opical in a iance o p ope quo ien s (see (1.0.3), and (1.0.5)) o p o e he esul o (X, $A$ ) as abo e and a bi a y $ alpha$ and $ ho$ . The gene al case can be educed o he p e ious case by using Theo em 1.1. Using Theo em 1.2 and a p oo simila o he o dina y case ([24; 6.23]) we ob ain 1.3. THEOREM.–Le $X$ be a p ope ly n-connec ed s ongly locally ini e CW- complex. Then he e is a na u al suspension $( ow- ma hcal{G}_{ }, ma hcal{G}_{ })$ -mo phism $ Sigma_{*}:$ $ Pi_{k}(X)- Pi_{k+1}( Sigma_{p}X)$ which is an isomo phism i $k leqq 2n$ and an epimo phism i $k=2n+1$ . 1.4. REMARK.–The esul s s a ing ha $ Sigma$ and $q$ induce isomo phisms be- ween he co esponding homology owe s can be p o ed in a s aigh o wa d way wi hou using P oposi ion 1.1. Moo e spaces in p ope homo opy 311 P oposi ion 1.1 and he abo e ema k gi e an easy p oo o he ollowing heo em. 1.5. THEOREM. $-$ ([21; II.4.2.7]) Le $X$ be a p ope ly n-connec ed s ongly locally ini e CW-complex. Then he na u al Hu ewicz mo phism $h; Pi_{k}(X) igh a ow H_{k}(X)$ is an isomo phism i $k=n+1$ and an epimo phism i $k=n+2$ . Fu he mo e Theo em 1.2 and Rema k 1.4 p o ide a ela i e e sion o Theo em 1.5 o a p ope pai $(X, A)$ o s ongly locally ini e CW-complexes wi h $A$ p ope ly l-connec ed and $X$ p ope ly n-connec ed. Finally Theo em 1.5, he B own-G ossman unc o $P_{0}$ in (1.0.5), and [7; $p$ , $43J$ lead o 1.6. THEOREM.–Gi en a p-map $ :X igh a ow Y$ whe e $X$ and $Y$ a e p ope ly 1- connec ed ini e dimensional, locally ini e CW-complexes such ha $ _{*};$ $H_{ }(X) igh a ow H_{ }( }^{7})$ is an isomo phism o each $ $ , hen he map $ $ is a p-homo opy equi alence. 2. P ope Moo e Spaces. We s a wi h some no ions in ( $ ow_{c}- Ab,$ (Ab). 2.1. DEFINITION.–A ee owe in ( ow-.A6, A6) is a owe $F( ma hcal{L})= dagge F(L_{0})-F(L_{1})- cdo s }$ whe e he ollowing ou condi ions hold; i) $ ma hcal{L}$ is a il a ion $ ma hcal{L} equi L_{0} supse eqq L_{1} supse eqq cdo s$ wi h $L_{0}$ a coun able se . ii) $ bigcap_{j=1}^{ in y}L_{j}= emp yse $ . iii) The di e ences $L_{k} backslash L_{k+1}$ a e ini e. i ) $F(L_{i})$ is he ee g oup gene a ed by $L_{i}$ and he bonding mo phism a e in- duced by he inclusions. Gi en he owe s $F( ma hcal{L})$ and $F( ma hcal{L}^{ p ime})$ , i can easily be checked ha any bijec- ion $L_{0} cong L_{0}^{ p ime}$ induces an isomo phism $F( ma hcal{L}) cong F( ma hcal{L}^{ p ime})$ in ( ow- A6, $ ma hcal{A}b$ ). So he iso- mo phism class o $F( ma hcal{L})$ is de e mined by he ca dinali y o $L_{0}$ . 2.2. REMARKS. $-a$ ) F ee owe s a e p ojec i e objec s $ ln$ ( ow-Ab, $ ma hcal{A}b$ ) (see [14]). b) Gi en a s ongly locally ini e CW-complex $X$ , and a sys em o $ in y$ -neigh- bou hoods $ {U_{j} }$ consis ing o subcomplexes, he owe o cellula n-chains o $X$ , $C_{n}(X)= {C_{n}(X)-C_{n}(U_{1})-C_{n}(U_{2})- cdo s }$ is ob iously a ee owe . Also he owe o cellula n-cycles 312 R. AYALA, E. DOMiNGUEZ, A. MARQUEZ and A. QUINTERO $Z_{n}(X)= {Z_{n}(X)-Z_{n}(U_{1}) le a ow Z_{n}(U_{2}) le a ow cdo s }$ is a ee owe . Mo e gene ally, he ke nel o any mo phism be ween wo ee owe s is always a ee owe (see [14; 5.1]). 2.3. DEFINITION.–A owe $S$ is said o be geome ically admissible i he e exis s an exac sequence in ( ow-Ab, $ ma hcal{A}b$ ) $j_{s}$ $j_{2}$ $j_{1}$ $0-F( ma hcal{L}_{3})-F( ma hcal{L}_{2})-F( ma hcal{L}_{1})-S-0$ When $F( ma hcal{L}_{3})$ is i ial we say ha $S$ has geome ical p ojec i e dimension (g.p. $d.$ ) 1. 0 he wise, we w i e $g$ . $p.d$ . $S=2$ . 2.4. REMARK.–Fo any s ongly locally ini e CW-complex $X$ he n- h homology owe o $X$ is geome ically admissible since he exac sequence o owe s $0-Z_{n+1}(X)-C_{n+1}(X)-Z_{n}(X)-H_{n}(X)-0$ is a ee esolu ion. Howe e , he sho exac sequence $0 igh a ow{ m Im} pa ial_{n+1} igh a ow Z_{n}(X) igh a ow H_{n}(X) igh a ow 0$ is no always a ee esolu ion. Indeed, le $X$ be he CW-complex ob ained om he cylinde $ S^{n} imes[0, in y$ ) by a aching an $(n+1)$ -cell a $S^{n} imes {j }$ by a map $ :S^{n} igh a ow S^{n}$ o deg ee $2^{j}(j geqq 1)$ . Then one can check ha ${ m Im} pa ial_{n+1}$ is no ee since $ pa ial_{n+1}$ has no igh in e se. 2.5. REMARK.–In [9] Dymo in oduced he no ion o cop esen a ion o a owe $S$ . Mo e explici ly, ollowing [9] we say ha $s= {G_{0} le a ow G_{1} le a ow cdo s }$ admi s a cop esen a ion i he e exis a le elwise epimo phism $ phi:F( ma hcal{L}) igh a ow S$ wi h $F( ma hcal{L})$ a ee owe and subse s $R_{ } subse eqq F(L_{ })$ wi h $R_{j}-R_{J+1}$ ini e and $ cap in y R_{j}= emp yse $ , such $j Le a ow 1$ ha $Ke phi_{j}$ is he subg oup $ langle R_{j} angle$ gene a ed by $R_{j}$ . In gene al, he owe $ langle R angle$ $ equi langle R_{0} angle le a ow langle R_{1} angle le a ow cdo s$ needs no o be p ojec i e. I is easy o check ha a owe $S$ is geome ically admissible i and only i admi s a cop esen a ion (up o isomo phism). In ac , i $ langle 9 angle igh a ow F( ma hcal{L}) igh a ow S igh a ow 0$ is a Dymo cop esen a ion, and $F(R)$ deno es he ee owe consis ing o he ee g oups $F(R_{i})$ , he e is a le elwise epimo phism $ F(R) igh a ow k langle R angle$ and an exac sequence $F(R)^{ k} igh a ow langle R angle igh a ow F( ma hcal{L})$ . By Rema k 2.2(b), $Ke ( ci c k)$ is a ee owe , and so $S$ is geome ically admissible. Con e sely, o any ee esolu ion $j_{3}$ $j_{2}$ $j_{1}$ $0 igh a ow F( ma hcal{L}_{3})-F( ma hcal{L}_{2})-F( ma hcal{L}_{1})-S-0$ Moo e spaces in p ope homo opy 313 le $ { a phi_{i} : F(L_{n(i)}^{2}) igh a ow F(L_{ }^{1}) }$ be a le elwise ep esen a i e o $j_{2}$ . Then $s^{J}=$ {Coke $ a phi_{i}$ } is isomo phic o $S$ by exac ness and admi s he ollowing cop e- sen a ion. Le $ phi:F( ma hcal{L}_{1}) igh a ow S^{ p ime}$ be he na u al le elwise quo ien mo phism. We ake $R_{i}= a phi_{i}(L_{n(i)}^{2}) subse eqq F(L_{i}^{1})$ . Now, i is clea ha $R_{i}-R_{i+1} subse eqq a phi_{i}(L_{n(i)}^{2}-L_{n i+1}^{2})$ is ini e. I $ ilde{S}^{n}$ ( $ ilde{B}^{n}$ espec i ely), is he space ob ained by a aching ini ely many copies (possibly no copy) o $S^{n}$ ( $B^{n}$ espec i ely) $(n geqq 2)$ a each $ m in N subse eqq[0, in y$ ), one checks ha $ Pi_{n}( ilde{S}^{n})( Pi_{n}( ilde{B}^{n}, ilde{S}^{n-1})$ espec i ely) can be iden i ied in a na u al way wi h some $F( ma hcal{L})$ , wi h $L_{0} subse eqq N$ . Mo eo e he ollowing esul holds: 2.6. LEMMA.–Gi en a space $X$ in $ ma hcal{P}^{J}$ , he e is a na u al bijec ion $ ho:[ ilde{S}^{n}, X]_{p}^{J}$ $ cong( ow- A6, Ab)(F( ma hcal{L}) igh a ow Pi_{n}(X))(n geqq 2)$ , gi en by $ ho([ ])= * cdo $ Mo eo e , i $Z$ is he mapping cone $X bigcup_{ } ilde{B}^{n+1}$ in he co ib a ion ca ego y $ ma hcal{P}^{J}$ , hen $ *can$ be also ega ded as he bounda y ope a o $d_{n+1}$ ; $ Pi_{n+1}(Z, X) igh a ow Pi_{n}(X)$ . PROOF.–Take $ a phi:F( ma hcal{L}) igh a ow Pi_{n}(X)$ . A e iden i ylng $ Pi_{n}( ilde{S}^{n})$ wi h $F( ma hcal{L})$ , le $ pi_{n}( ilde{S}^{n})- pi_{n}( ilde{S}_{k(1)}^{n})- pi_{n}( ilde{S}_{k(2)}^{n}- cdo s$ $ downa ow$ $ downa ow$ $ downa ow$ $ pi_{n}(X)- pi_{n}(U_{1})- pi_{n}(U_{2})- cdo s$ be a le elwise ep esen a i e o $ a phi$ , whe e $ ilde{S}_{k( )}^{n}$ is ob ained by dele ing om $ ilde{S}^{n}$ he copies, $S_{j}^{n}$ , o $S^{n}$ placed a he poin s $1 leqq in <k( ),$ $ k(1)<k(2)< cdo s$ . We de ine $ |S_{j}^{n}$ as a ep esen a i e o $ a phi_{h(j)}[l_{ }]$ whe e $l_{j}$ : $S_{j}^{n} subse eqq ilde{S}_{k( )}^{n},$ $ k( ) leqq$ $j<k( +1)$ . One easily checks ha $ _{*}= a phi$ . This p o es ha $ ho$ is on o. The injec i i y ollows in a simila way. Finally, we ha e he diag am $ Pi_{n+1}(Z, X) Pi_{n+1}( ilde{B}^{n+1} unde line{p_{ cong^{*}}} ilde{S}^{n})$ $ Pi_{n} downa ow_{(X)}$ $- Pi_{n}( ilde{S}^{n})=F( ma hcal{L}) downa ow cong$ whe e $p: ilde{B}^{n+1} igh a ow Z$ is he canonical p-map and $p_{*}$ is an isomo phism by Theo em 1.2 since $ ilde{B}^{n+1}/p ilde{S}^{n} cong ilde{S}^{n+1} cong Z/pX$ . Fo he sake o simplici y, we shall use he single no a ion $ ilde{S}^{n}( ilde{B}^{n})$ o all he “s ings” o sphe es (balls) desc ibed abo e. The pa icula objec s $ ilde{S}^{n}( ilde{B}^{n})$ we a e using in he u u e will be clea om he con ex . Simila ly o $F( ma hcal{L})$ . 320 R. AYALA, E. DOMiNGUEZ, A. MARQUEZ and A. QUINTERO is he p-map de ined by he cha ac e is ic maps o he $(n+1)$ -cells o $X$ . We easily check ha $ _{n+1*}= a phi_{n+1}$ . When one ies o go u he an obs uc ion appea s as ollows. In he diag am $ Pi_{n+2}(X^{n+2}, X^{n+1})$ $ Pi_{n+1}(X^{n+1})$ $ pa ial_{n+2}- Pi_{n+1}(X^{n+1}, X^{n})$ $ a phi_{n+2} downa ow$ (1) $ downa ow _{n+1*}$ $ downa ow _{n+1*}= a phi_{n+1}$ $ Pi_{n+2}(Y^{n+2}, Y^{n+1})$ $ Pi_{n+1}(Y^{n+1})- Pi_{n+1}(Y^{n+1} pa ial_{n+2}^{ p ime}Y^{n})$ he squa e (1) needs no be commu a i e. One de ines an elemen in $( ow- Lambda b, Lambda b)$ $(F( ma hcal{L}_{s}); Pi_{n+1}(Y^{n+1}))$ by he di e ence $ be a( _{n+1})= _{n+12} pa ial_{n+2}- pa ial_{n+2}^{ p ime} a phi_{n+2}$ Since he o he squa e is commu a i e we ha e, by de ini ion o $Ke j_{*}$ in ( ow- $ llco ne Ab,$ $ ma hcal{A}b$ ) ha $ be a$ can be ega ded as a mo phism $ be a( _{n+1}):F( ma hcal{L}_{ he a}) igh a ow Ke j_{*}= Gamma_{n+1}Y$ . Ob iously $ be a$ is a cocycle and de ines a class $c( a phi) in H^{n+2}(X, Gamma_{?l+1}Y)$ . The nex lemma shows ha $c( a phi)$ is a well de ined obs uc ion. 3.3. LEMMA. $-1$ ) $c( a phi)$ does no depend on he mo phisms $ a phi_{i}(i=n, n+1, n+2)$ . 2) $c( a phi)$ is an obs uc ion o ealizing $ a phi$ . PROOF. 1) Le $ { a phi_{i}^{ p ime} }$ be ano he mo phism such ha he diag am $(^{*})$ com- mu es and le $ _{n+1}^{ p ime}$ : $X^{n+1} igh a ow Y^{n+1}$ be a $p$ -map ealizing $ a phi_{n+1}^{ p ime}$ . I is a well-known ac om Homological Algeb a in abelian ca ego ies ha $ { a phi_{i} }$ and $ { a phi_{i}^{ p ime} }$ a e homo opic chain mo phisms. Thus, he e exis mo phisms $ { alpha_{i} : C_{i}(X) igh a ow C_{i+1}(Y) }$ $(i=n, n+1, n+2)$ such ha he ollowing equali ies hold a) $ a phi_{n}^{ p ime}- a phi_{n}=d_{n+1}^{ p ime} ci c alpha_{n}$ ; b) $ a phi_{n+1}^{ p ime}- a phi_{n+1}= alpha_{n} ci c d_{n+1}+d_{n+2}^{ p ime} ci c alpha_{n+1}$ ; and c) $ a phi_{n+2}^{ p ime}- a phi_{n+2}=d_{n+3}^{ p ime} ci c alpha_{n+2}+ alpha_{n+1} ci c d_{n+2}$ . By c) and he de ini ion o $ be a( _{n+1})$ we ha e (I) $ be a( _{n+1}^{ p ime})- be a( _{n+1})=( _{n+1*}^{ p ime}- _{n+1^{*}}- pa ial_{n+2}^{ p ime} ci c alpha_{n+1} ci c j_{*}) ci c pa ial_{n+2}$ whe e $j_{*}:$ $ Pi_{n+1}(Y^{n+1}) igh a ow C_{n+1}(Y)$ . Now, a) p o ides a $p$ -homo opy $H:X^{n} imes I$ $ igh a ow Y^{n+1}$ be ween $ _{n}$ and $ _{n}^{ p ime}= _{n+1}^{ p ime}|_{X^{n}}$ . As in o dina y homo opy heo y, $H$ yields a “di e ence” mo phism $ Del a=d( _{n+1}^{ p ime}, H, _{n+1});C_{n+1}(X)- Pi_{n+1}(Y^{n+1})$ (see [15]). Mo eo e , $H$ can be chosen in such a way ha $ in * ci c Del a= _{n+1*}^{ p ime}- _{n+15}$ $- alpha_{n} ci c d_{n+1}$ . Take $ be a= Del a- pa ial_{n+2}^{ p ime} ci c alpha_{n+1}$ . By b), $j_{*}( be a)=0$ . And he p ojec i eness o $C_{n+1}(X)$ allows us o ega d $ be a$ as an elemen in ( ow-.A6, $ cup b$ ) $(C_{n+1}(X); Gamma_{n+1}(Y))$ since $Ke j_{*}= Gamma_{n+1}(Y)$ . Moo e spaces in p ope homo opy 321 Finally, one can eadily check om he de ini ions ha $ _{n+1}^{ p ime} cong p _{n+1}+$ $( be a+ pa ial_{n+2}^{ p ime} ci c alpha_{n+1})$ and hen, by Lemma 3.1 he igh side in he equali y (I) is $ be a^{Q}j_{*}o pa ial_{n+2}= be a ci c d_{n+2}= del a be a$ . This p o es $[ be a( _{n+1})]=[ be a( ex {{ i ’{n}}}_{+1})] in H^{n+2}(X; Gamma_{n+1}(Y))$ . 2) I $c( a phi)=0$ , le $w in( ow- Ab, llco ne Ab)(C_{n+1}(X), Gamma_{n+1}(Y))$ be such ha $ be a( _{n+1})=$ $ del a w=w ci c d_{n+2}=w ci c j_{*} ci c pa ial_{n+2}$ . Take $ o e line{ }_{n+1}= _{n+1}+w$ . By Lemma 3.1, $ be a( o e line{ }_{n+1})=$ $( _{n+1*}-w ci c j_{*}) pa ial_{n+2}- pa ial_{n+2}^{ p ime} ci c a phi_{n+2}=0$ , and $ o e line{ }_{n+1}$ ex ends o a p-map $ _{n+2}$ : $X igh a ow Y$ wi h $ _{n+2*}= a phi:H_{n}(X) igh a ow H_{n}(Y)$ . 3.4. REMARKS. $-a$ ) The obs uc ion $c( a phi)$ was al eady conside ed by J. H. C. Whi head in o dina y homo opy (see [25; S 6]) and i can be de ined wi hin he gene al se lng o co ib a lon ca ego ies (see [5; VII. 1.13]). b) Fo any $n geqq 3$ he e a e wo non p ope ly equi alen Moo e spaces o ype $(S;n)$ . The examples a e gi en in Appendix A. As an immedia e consequence o Theo em 3.2 we ha e 3.5. COROLLARY.–I $S$ is a owe $ w io a$ h g.p.d. $S=1$ hen he e exis s (up o p-homo opy) a unique p ope Moo e space o ype $(S, n)(n geqq 2)$ . Mo e gene ally, we can s a e 3.6. COROLLARY. –I $S$ is a geome ically admz ssible owe wi h $Ex ^{2}(S; Gamma_{n}(S))$ $=0$ , hen he e exis s a unique p ope Moo e space o ype $(S, n)(n geqq 2)$ . He e $ Gamma_{n}(S)$ deno es he owe ob ained om $S$ by applying le elwise he algeb aic Whi ehead $ Gamma_{n}$ - unc o (see $ lceil_{-}26$ ; Ch. II] o [5; IX. 4]). I is known ha $ Gamma_{n}=- o imes Z_{2}$ when $n geqq 3$ . So, (3.6) yields 3.7. COROLLARY. –I $S$ is a geome ically admissible owe wi h $Ex ^{2}(S;S o imes Z_{2})$ $=0$ hen he e exis s a unique p ope Moo e space o ype $(S, n)$ o all $n geqq 3$ . PROOF $0F(3.6)$ . Le $Y$ be a p ope Moo e space o ype $(S, n)$ cons uc ed as in he p oo o Theo em 2.9. Le $ Y supse eqq U_{1} supse eqq cdo s supse eqq U_{n} cdo s$ be a sys em o $ in y-$ neighbou hoods such ha each $U_{j}$ is a subcomplex. Mo eo e , each $U_{j}$ is $(n-1)$ -connec ed by cons uc ion. Thus by [19; VIII. 2.4] o $n geqq 3$ and [26; III. 14] o $n=2$ we ha e $ Gamma_{n+1}U_{j} cong Gamma_{n}(H_{n}(U_{j}))$ and he e o e $ Gamma_{n+1}Y cong Gamma_{n}(S)$ . I $X$ is ano he p ope Moo e space o ype $(S;n)$ we can ealize id: $S igh a ow S$ by a $p$ -map $ :X igh a ow Y$ by Theo em 3.2 since $H^{n+2}(X; Gamma_{n+1}Y)=Ex ^{2}(S; Gamma_{n+1}Y)=0$ . By Theo em 1.6 $ $ is ac ually a p-homo opy equi alence. 322 R. AYALA, E. $DoMINGUEZ$ , A. M ’ARQUEZ and A. QUINTERO 3.8. REMARK.–The owe $s= {Z_{2} le a ow Z_{4}p_{1} le a ow Z_{8} le a ow p_{2} ldo s }$ whe e $p_{i}(1)=1$ , has geo- me ical p ojec i e dimension 2 since $S$ is he n- h homology owe o he CW- complex gi en in Rema k 2.4. Ne e heless $ Gamma_{n}S$ is he cons an owe $Z_{2}$ when $n geqq 3$ , and $ Gamma_{2}S$ is isomo phic o $S$ since $ Gamma_{2}Z_{2n}=Z_{4n}$ acco ding o [26; II.(B)]. Then, one can check as in Appendix A ha $Ex ^{2}(S; Gamma_{n}S)=0$ . So, he e is a unique p ope Moo e space o ype $(S;n)(n geqq 3)$ by Co olla y 3.7. The same esul holds o $n=2$ . Ano he su icien algeb aic condi ion on $S$ o he uniqueness o p ope Moo e spaces o ype $(S;n)n geqq 3)$ is he ollowing. 3.9. $p_{ROPOSITION}.$ –Le $S$ be a geome ically admissible owe such ha $To ^{1}(S;Z_{2})=0$ . Then he e is a unique Moo e space o ype $(S;n)(n geqq 3),$ uni- que up o p-homo opy. Be o e s a ing he p oo o P oposi ion 3.9 we shall ix no a ion and p o e a lemma whose p oo $1S$ simila o he p oo o [14; Lemma 2]. Le $( ow- Z_{2}, Z_{2})$ deno e he abelian ca ego y de ined in he same way as ( $ ow- ma hcal{A}b$ , A6) by using $Z_{2}$ - ec o spaces ins ead o abelian g oups. Gi en a ee owe $F( ma hcal{L})$ , le $Z_{2}(L_{i})$ deno e $F(L_{i}) o imes Z_{2}$ . Then 3.10. LEMMA.–Any owe $ {V_{0} le a ow V_{1} le a ow V_{2} cdo s }$ wi h $V_{i} subse eqq Z_{2}(L_{i})$ and wi h bond- ing mo phisms he co esponding es ic ions is p ojec i e in $( ow- Z_{2}, Z_{2})$ . PROOF. We may ind a basis $T_{i}$ o $V_{i}/V_{i+1}$ such ha any elemen o $T_{i}$ is ep esen ed by a linea combina ion o elemen s in $L_{i}-L_{i+1}$ . Le $B_{i}$ be he union $ cup {T_{j} ; j geqq I }$ . I is easy o check ha $B_{i}$ is a basis o $V_{i}$ . Since $ B_{0} supse eqq B_{1} cdo s$ and $ cap B_{i}= emp yse $ i is s aigh o wa dly shown ha $ {V_{0} le a ow V_{1} le a ow V_{2} cdo s }$ is p ojec i e. PROOF OF PROPOSITION 3.9. Le $d_{2}$ $d_{2}$ $d_{0}$ $0-C_{2}-C_{1}-C_{0}-S-0$ be a ee esolu ion o $S$ . Since $S o imes Z_{2}$ is a owe o g oups o o de 2 we ha e an isomo phism ( ow- A6, $Ab$ ) $(C_{i} ; S o imes Z_{2}) cong( ow- Z_{2}, Z_{2})(C_{i} o imes Z_{2} ; S o imes Z_{2})$ The e o e, $Ex ^{2}(S;S o imes Z_{2}) cong Ex ^{1}({ m Im} d_{1} ; S o imes Z_{2}) cong Coke ((d_{2} o imes 1)^{*})$ Moo e spaces in p ope homo opy 323 whe e $*$ s ands o he dual mo phism. Now om he exac sequence $0-C_{2}-C_{1}-{ m Im} d_{1}-0$ we ge he exac sequence $0-C_{2} o imes Z_{2}$ $d_{2} o imes 1 igh a ow C_{1} o imes Z_{2}{ m Im} d_{1} o imes Z_{2} unde line{d_{1} o imes 1} igh a ow 0$ since he owe $Ke (d_{2} o imes 1)$ is isomo phic o he owe $ {To ({ m Im} d_{1}^{i} ; Z_{2}) }$ which is i ial because each componen ${ m Im}[d_{1}^{i} : C_{1}^{k(i)} igh a ow C_{0}^{i}]$ o he owe ${ m Im} d_{1}$ is a ee abelian g oup. Thus, Coke $((d_{2} o imes 1)^{*}) cong Ex _{Z_{2}}^{1}({ m Im} d_{1} o imes Z_{2} ; S o imes Z_{2})$ whe e he igh side is he $Ex 1$ unc o in he ca ego y $( ow- Z_{2}, Z_{2})$ . On he o he hand we ha e he commu a i e diag am $0-C_{2} o imes Z_{2}$ $C_{1} o imes Z_{2}{ m Im} d_{1} o imes Z_{2}$ $d_{2} o imes 1 unde line{d_{1} o imes 1}-0$ $ Ve $ $ Ve $ $0-c_{2} o imes z_{2^{-}}^{d_{2} o imes 1}$ $C_{1} o imes Z_{2} igh a ow^{d_{1} o imes 1}{ m Im}(d_{1} o imes 1)-0$ whe e he uppe ow is exac as i was p o en abo e. And he lowe ow is also exac since $To ^{1}(S;Z_{2})=Ke (d_{1} o imes 1)/{ m Im}(d_{2} o imes 1)=0$ by hypo hesis. Thus ${ m Im} d_{1} o imes Z_{2} cong{ m Im}(d_{1} o imes 1)$ and hence $Ex _{Z_{2}}^{1}({ m Im} d_{1} o imes Z_{2} ; S o imes Z_{2}) cong Ex _{Z_{2}}^{1}({ m Im}(d_{1} o imes 1);S o imes Z_{2})$ . Now he o me e m anishes because ${ m Im}(d_{1} o imes 1)$ is p ojec i e by Lemma 3.10. This yields $Ex ^{2}(S;S o imes Z_{2})=0$ and he uniqueness ollows om Co olla y 3.7. FINAL REMARK. The ca ego y o ees o abelian g oups (see [12]) seems o be he igh algeb aic amewo k o a gene aliza ion o he esul s o his pape o spaces wi h many F euden hal ends. Appendix A. Two non p ope ly equi alen p ope Mo e spaces o ype $(S;n),$ $n geqq 3$ . Le $S$ be he owe $ { bigoplus_{1}^{ in y}Z_{2} oplus Z_{2} igh a ow^{k_{1} oplus 1} bigoplus_{2}^{ in y}Z_{2}Z_{2} }$ $k_{2} oplus 1$ $ ldo s$ in ( ow-di, $ ma hcal{A}b$ ), wi h $k_{j}$ s anding o he na u al inclusion mo phism. A ee esolu ion o $S$ is 324 R. AYALA, E. $DoM NGUEZ$ , A. M ’ARQUEZ and A. QUINTERO $ pa ial_{2}$ $ pa ial_{2}$ $0-F( ma hcal{L}_{3})-F( ma hcal{L}_{2})-F( ma hcal{L}_{1})-S-0$ whe e $ ma hcal{L}_{3}$ is $ L_{1}^{3} supse eqq L_{2}^{3} supse eqq cdo s$ wi h $L_{j}^{3}= { alpha_{i} ; i geqq] }$ . $ ma hcal{L}_{2}$ is $ L_{1}^{2} supse eqq L_{2}^{2} supse eqq cdo s$ wi h $L_{j}^{2}=$ $ { ho_{i}, mu_{i}, sigma_{i} ; i geqq j }$ and $ ma hcal{L}_{1}$ is $ L_{1}^{1} supse eqq L_{2}^{1} supse eqq cdo s$ wi h $L_{j}^{I}= { epsilon_{i}, gamma_{i} ; i geqq j }$ . And he mo - phisms a e gi en by $ pa ial_{2}( alpha_{i})= mu_{i+1}-2 sigma_{i}- mu_{i}$ ; $ pa ial_{1}( ho_{i})=2 epsilon_{i},$ $ he a_{1}( mu_{i})=2 gamma_{i}$ and $ pa ial_{1}( sigma_{i})=$ $ gamma_{i+1}- gamma_{i}$ . Clea ly, $S=S_{1} oplus S_{2}$ , whe e $S_{1}$ is he owe $ { bigoplus_{1}^{ in y}Z_{2} bigoplus_{2}^{ in y}Z_{2} } unde line{k_{1}} unde line{k_{2}} ldo s$ and $S_{2}$ is he cons an owe $ {Z_{2}=Z_{2}= cdo s }$ . A. 1. LEMMA. $-Ex ^{2}(S;S) neq 0,$ $Ex ^{2}(S_{1} ; S_{1})=Ex ^{2}(S_{2} ; S_{2})=0$ . PROOF. By he na u ali y o $Ex $ ’ and $ oplus$ we ha e $Ex ^{2}(S;S)= oplus$ $ {Ex ^{2}(S_{i} ; S_{j});i, j leqq 2 }$ . On he o he hand, i is easy o check ha $g$ . $p$ . $d$ . $S_{1}=1$ , and so $Ex ^{2}(S_{1} ; S_{j})$ $=0$ . Wi h he abo e no a ions, $S_{2}$ admi s he ee esolu ion $ pa ial_{2}$ $ pa ial_{1}$ $0 igh a ow F( { alpha_{i} })-F( { mu_{i}, sigma_{i} })-F( { gamma_{i} })-S_{2}-0$ and by he s anda d Hom-Ex exac sequence we ge $Ex ^{2}(S_{2} ; S_{2})=Ex ^{1}({ m Im} pa ial_{1} ; S_{2})=0$ . Indeed, o any $ a phi in( ow- Ab, d6)(F( { alpha_{i} }), S_{2})$ we may de ine $ o e line{ a phi} in$ ( $ ow- Ab,$ Ab) ( $F( { mu_{i}, sigma_{i} }, S)$ by $ o e line{ a phi}( sigma_{i})=0, o e line{ a phi}( mu_{1})=0$ and $ o e line{ a phi}( mu_{J})= Sigma { a phi( alpha_{i});i leqq j-1 }$ . Then $ o e line{ a phi} ci c pa ial_{2}$ $= a phi$ . Finally $Ex ^{2}(S_{2}, S_{1})=Ex ^{1}({ m Im} pa ial_{1} ; S_{1}) neq 0$ since $ xi:F( { alpha_{i} }) igh a ow S_{1}$ gi en by $ xi( alpha_{i})=$ $ epsilon_{i} in y 1$ de ines a non- i ial elemen . 0 he wise, $ xi= au ci c pa ial_{2}$ o some $ epsilon;F( { mu_{i}, sigma_{i} })$ $ igh a ow S_{1}$ and $ au$ yields he equali ies $ epsilon_{i} o imes 1= au( mu_{i+1})- au( mu_{i})(i geqq 1)$ . As $ au$ is a p o- mo phism one can induc i ely p o e ha $ au( mu_{i}) in bigoplus_{k geq 1}Z_{2}$ and he sequence $ { epsilon_{i} o imes 1 }$ would ep esen he i ial elemen $ ln lim^{1}S_{I}$ and i is a well-known ac ha i does no . Lemma A.1 and Co olla y 3.7 yield ha $R$ ( $S_{1}$ ; n) and $R$ ( $S_{2}$ ; n) a e uniquely de e mined up o p-homo opy $(n geqq 3)$ . Ac ually hese ypes a e ep esen ed by $ ilde{W}$ and $ W imes[0, in y$ ), whe e $W^{ app ox}$ is ob ained by a aching one copy o $W$ a each na u al coo dina e o $[0, in y$ ) and $W=S^{n} bigcup_{2}e^{n+1}$ is he n-sphe e wi h an $(n+1)-$ cell a ached by a map o deg ee 2. Thus, $X=R(S_{1} ; n)_{p}R$ ( $S_{2}$ ; n) is a ep e- sen a i e o $R(S;n)$ by Co olla y 2.10. A.2. LEMMA. The na u al map $[X; X]_{p}^{J} igh a ow( ow- AA, Lambda b)(S;S)$ is on o. Moo e spaces in p ope homo opy 325 PROOF. By P oposi ion 2.14 $[R(S_{1} ; n);X]_{p}^{J} igh a ow[ ci c$ is on o. On he o he hand, ( ow-db, $Ab$ ) $(S_{2} ; S_{1})= lim S_{1}=0 le a ow$ and $[R(S_{2} ; n);X]_{p}^{J} igh a ow$ $[R(S_{2} ; n);R(S_{2} ; n)]_{p}^{J} cong[W;W]^{J} igh a ow_{ ze a}A6(Z_{2} ; Z_{2}) cong( ow- Ab, ma hcal{A}b)(S_{2} ; S_{2})$ . whe e he i s bijec ion is gi en by he Edwa ds-Has ings embedding Theo em ([10; 6.27]). Finally he na u al bijec ion $[X; X]_{p}^{J} cong[R(S_{1} ; n);X]_{p}^{J} imes$ $[R(S_{2} ; n);X]_{p}^{J}$ comple es he p oo . Now we choose a non- i ial elemen $ alpha in H^{n+2}(X; Gamma_{n+1}X) cong Ex ^{2}(S;S) neq 0$ . By Rema k 2.7 we may assume $X= ilde{B}^{n+2} bigcup_{h_{0}}X^{n+1}$ . Now, he commu a i e dia- g am $C_{n+2}(X)= Pi_{n+2}(X, X^{n+1} Pi_{n+2}( ilde{B}^{n+1} ec{h} cong_{*}S^{n+1})$ $ downa ow d_{n+2}$ $ cong downa ow d_{n+2}$ $ Pi_{n+1}(X^{n+1}) Pi_{n+1}(S^{n+1}) unde line{h_{0*}}$ allows us o iden i y he bounda y ope a o $d_{n+2}$ wi h he mo phism $h_{0*}(h$ is he cha ac e is ic map $h: ilde{B}^{n+2} igh a ow X$ ). The isomo phism $d_{n+2} ci c h_{*}^{-1}$ also gi es he iden i ica ion (I) ( ow-Ab, $ cup b$ ) $(C_{n+2}(X); Pi_{n+1}(X^{n+1}))$ $ cong( ow- Ab, ma hcal{A}b)( Pi_{n+1}( ilde{S}^{n+1}); Pi_{n+1}(X^{n+1})) cong[S^{n+1} ; X^{n+1}]_{p}^{J}$ whe e he second isomo phism is gi en by Lemma 2.6. Thus, i $ alpha=[a],$ $a$ can be ega ded as a p-map $g;S^{n+1} igh a ow X^{n} subse eqq X^{n+I}$ . Le $ o e line{h}_{0}$ be a ep esen a i e o $[h_{0}]+[g] in[S^{n+1} ; X]_{p}^{J}$ and le $Y$ be he p ope cone o $ o e line{h}_{0}$ . Since ${ m Im} g subse eqq X^{n}$ , he complexes owe s o cellula chains o $X$ and $Y$ a e he same. Bu A.3. LEMMA.–The obs uc ion $c(id) in H^{n+2}(X; Gamma_{n+1}Y)$ gi en in Theo em 3.2 o id: $H_{n}(X)=S igh a ow S=H_{n}(Y)$ is non- i ial. PROOF. Smce $c(id)$ does no depend on he mo phisms $ a phi_{i}$ : $C_{i}(X) igh a ow C_{i}(Y)$ inducing id: $S igh a ow S$ (see Lemma $3.3(1)$ ), one can conside $ a phi_{i}=id$ o each $i=n$ , $n+1,$ $n+2$ . So, $c(id)$ is ep esen ed by $ be a(id)=d_{n+2}-d_{n+2}^{ p ime}$ , whe e $d_{n+2}$ is gi en in he abo e diag am o $X$ . Simlla ly $d_{n+2}^{ p ime}$ o $Y$ . Bea ing in mlnd he iden i ica ion (I) $ be a(id)$ is ega ded as $h_{*}-h_{*}-g_{*}=-g_{*}$ . Then $ be a(id)$ is ac ually $-a$ and $c(id)=- alpha neq 0$ . Finally we ge , 326 R. AYALA, E. DOMiNGUEZ, A. M ’ARQUEZ and A. QUINTERO A.4. $p_{ROPOSITION}.-X$ and $Y$ a e no p-homo opically equi alen . PROOF. I $h:X igh a ow Y$ is a p-homo opy equi alence, le $h^{ p ime}$ be a $p$ -homo opic in e se o $h$ . The mo phism $h_{*}^{ p ime}$ : $H_{n}(Y)=S igh a ow H_{n}(X)=S$ can be ealized by a h- map $ :X igh a ow X$ acco ding o Lemma A.2. Then $h ci c :X igh a ow Y$ is a $p$ -map wi h $(h ci c )_{*}=id:H_{n}(X)=s igh a ow s=H_{n}(1^{ nea ow})$ , and his canno happen by Lemma A.3. Re e ences [1] M. A in and B. Mazu , E ale homo opy, Lec u e No es in Ma hs ol. 100, Sp inge 1969. [2] R. Ayala, E. Dominguez and A. Quin e o, Hu ewicz Theo em o homology a in- ini y, Qua . J. Ma h. Ox o d 44 (1993), 139-153. [3] R. Ayala, E. Dominguez and A. Quin e o, A heo e ical amewo k o P ope Homo opy Theo y, Ma h. P oc. Camb. Philos. Soc. 107 (1990), 475-482. [4] R. Ayala, E. Dominguez, A. M ’a quez, A. Quin e o and S. 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Hil on, An in oduc ion o Homo opy Theo y, Camb idge Uni . P ess, 1966 [20] I.M. James, Gene al Topology and Homo opy Theo y, Sp inge 1984. Moo e spaces $ ln$ p ope homo opy 327 [21] S. Ma desic and J. Segal, Shape heo y, No h Holland 1982. [22] T. Po e , Homo opy g oups in s ong shape and p ope homo opy, Suppl. Rendi- con i Ci c. Ma . Pale mo. Se ie II, 4 (1984), 101-111. [23] L. C. Siebenmann, The obs uc ion o inding a bounda y o an open mani old o dimension g ea e han i e, Thesis. P ince on Uni e si y, 1965. [24] R. Swi ze , Algeb aic Topology-Homo opy and Homology, Sp inge 1975. [25] J.H. C. Whi ehead, Combina o ial Homo opy, II, Bull. Ame . Ma h. Soc. 55 (1949), 453-496. [26] J.H. C. Whi ehead, A ce ain exac sequence, Ann. Ma h. 52 (1950), 51-110. R. Ayala, A. M ’a quez, A. Quin e o Facul ad de Ma em ’a lcas Dp o. Algeb a, Compu aci ’on, Geome ia y Topologia Apdo 1160 41080-SEVILLA (SPAIN) E. Dom ’inguez Dp o $Ingenie ^{ p ime}1a$ El ’ec ica $e$ In o m ’a ica Facul ad de Ciencias Ciudad Uni e si a ia 50009-ZARAGOZA (SPAIN)