Moore spaces in proper homotopy

TSUKUBA J. $MA^{r}I^{\backslash }H$ . Vol. 19 No. 2 (1995), 305–327 MOORE SPACES IN PROPER HOMOTOPY By R. AYALA, E. DOM\’INGUEZ, A. M\’ARQUEZ and A. QUINTERO Abstract. Moore spaces are defined in proper homotopy theory. Some results on the existence and uniqueness of those spaces are proven. An example of two non properly equivalent Moore spaces is given. Introduction. The purpose of this paper is to provide the correct statements and details of the results announced in [4]. Namely, we prove the existence of proper Moore spaces of types of type $(S;n)$ for certain objects $S$ in the abelian category of towers of groups (tow-JZ $b,$ $\llcorner fb$ ) and $n\geqq 2$ (Theorem 2.9). Nevertheless objects can be of projective dimension 2 in (tow-db, $\iota Ab$ ), and this fact determines an obstruction to the uniqueness of proper Moore spaces (Theorem 3.2). In fact, an example of two non properly equivalent Moore spaces is given in Appendix A. As a consequence of Theorem 3.2 two sufficient conditions for the uniqueness of proper Moore spaces are stated (Corollary 3.7 and Proposition 3.9). The existence of Moore spaces in proper homotopy was already announced in [4], but in that paper the obstruction from Theorem 3.2 was not considered and the uniqueness of such spaces was wrongly asserted. For towers of projective dimension 1, proper Moore spaces behave in a very similar way to ordinary Moore spaces. In particular, for projective dimension 1, proper Moore spaces define proper homotopy groups, and a coefficient exact sequence which generalize the various proper homotopy groups known in the literature and their corresponding Milnor exact sequences (Examples 2.15). 1. Preliminaries and Notations CATEGORIES OF TOWERS. Given a category $c$ , the category of towers of $C$ , tow-C, is the category of inverse sequences $s=\{A_{1}\leftarrow A_{2}\leftarrow\cdots\}$ in $C$ where a (tow-C)-morphism $f:S\rightarrow S^{\prime}$ is represented by a sequence of C-morphisms 1980 Mathematics Subject Classification. $54C10,55P99,55Q70$ . Key words: Moore space, homology decomposition, proper homotopy, tower of groups. Received August 25, 1993. 306 R. AYALA, E. $D0MtNGUEZ$ , A. MARQUEZ and A. QUINTERO $f_{k}$ : $A_{n_{k}}\rightarrow A_{k}^{\prime},$ $ n_{1}<n_{2}<\cdots$ , such that given $r>s$ there exists $j>n_{r},$ $n_{s}$ making commutative the diagram $A_{n}\uparrow^{f}\underline{f_{f}}A_{r}^{\prime,}$ $A_{j}$ $A\downarrow_{n},$ $\underline{f_{1}}A$ ; where the maps without name are bonding maps. We are interested in the full subcategory of $\ovalbox{\tt\small REJECT}_{ot}$ (tow-C) whose objects are arrows $f:X\rightarrow A$ where :IE7 is a (tow-C)-object and $A$ is a C-object regarded as a constant tower whose bonding maps are the identity. This category is denoted (tow-C, $C$ ). A (tow-C, $C$ )-morphism from $f:x\rightarrow A$ to $g:\eta\rightarrow B$ can be regarded as a $C$ -morphism between $A$ and $B$ and a (tow-C)-morphism from ec to $Qf$ such that both morphisms are compatible via the bonding maps. It is convenient to represent (tow-C, $C$ ) as follows. Objects are towers $X=$ $\{X_{0}\leftarrow X_{1}\leftarrow\cdots\}$ ; a morphism consists of a map $f:X\rightarrow Y$ in tow-C, together with a compatible map $f_{0}$ : $X_{0}\rightarrow Y_{0}$ in $C$ . We shall specially use the above constructions for $c=\mathcal{F}_{0}p,$ $\mathcal{G}_{t},$ $\llcorner flb$ ; the categories of topological spaces, groups and abelian groups respectively. Since $tow- \mathcal{A}6$ and (tow-X6, a6) are abelian categories (see [1]) we can define kernels and images and state exact sequences in a natural way. In particular, we can use projective objects and define the functor Ext. See [10] for details. PROPER CATEGORY. A proper map (p-map) is a continuous map $f:X\rightarrow Y$ such that $f^{-1}(K)$ is compact for each compact subset $K\subseteqq Y$ . Proper homotopy (p-homotopy), proper homotopy equivalence, etc $\cdots$ can be defined in the natural way. We shall deal with the category $\mathcal{P}$ of $T_{2}$ -locally compact $\sigma$ -compact spaces and p-maps. One can check that $\mathcal{P}$ is a cofibration category in the sense of H. Baues (see [5] and [3]) whose cofibrations are the $p$ -maps with the Proper Homotopy Extension Property. We call these $p$ -maps proper cofibrations (p-cofibrations). It can be shown that any p-cofibration is a closed embedding. We denote p-cofibrations by arrows \rangle $\rightarrow$ . Moore spaces in proper homotopy 307 Glven a space $X$ in $\mathcal{P}$ , a system of $\infty$ -neighbourhoods of $X$ is the object of $tow- \mathcal{G}_{op},$ $\epsilon(X)=\{U_{1}\leftarrow U_{2}\leftarrow\cdots\}$ where $\overline{X-U}_{j}=K_{j}$ is compact, $K_{j}\subseteqq K_{j+1}$ and $X=$ $\cup\{intK_{j} ; j\geqq 1\}$ . We recall that a CW-complex $X$ is said to be strongly locally finite if $X$ can be covered in a locally finite way by finite subcomplexes. In that case, it is known that $X$ admits a countable locally finite cover by finite subcomplexes and so the $\infty$ -neighbourhoods of $X$ can be chosen to be subcomplexes. Finite dimensional locally finite CW-complexes and locally finite simplicial complexes are strongly locally finite (see [11]). If $f:X\rightarrow Y$ is a p-map and $\{U_{j}\}$ and $\{W_{j}\}$ are systems of $\infty$ -neighbourhoods of $X$ and $Y$ respectively, for each $j$ there exists $k(j)$ such that $f(U_{k(j)})\subset W_{J}$ and therefore we get a morphism $\epsilon(f):\epsilon(X)\rightarrow\epsilon(Y)$ (see [10] for more details). Given a space $X$ in $\mathcal{P}$ a Freudenthal end of $X$ is an element of the set $\mathcal{F}(X)=\lim_{\leftarrow}\pi_{0}(U_{j})$ where $\pi_{0}(-)$ denotes the set of connected components. Now let $\mathcal{P}^{J}$ be the category $\mathcal{P}$ under $ J=[0, \infty$ ) such that for every $\mathcal{P}^{J_{-}}$ object $J\rightarrow iX,$ $\iota$ is a p-cofibration. The category $\mathcal{P}^{J}$ is a category of cofibrations where the notion of proper wedge $(V_{p})is$ defined in a natural way. The set of p-homotopy classes in $\mathcal{P}^{J}$ will be denoted by $[$ –, $-]_{p}^{J}$ . (1.0.1) The category over $J,$ $\mathcal{P}_{J}$ , is again a cofibration category, and it allows the definition of proper quotients. More explicitly, if $r:X\rightarrow J$ is a $\mathcal{P}_{J^{-}}$ object, and $A\succ X$ the proper quotient $X/pA(r)$ is defined by the push-out $\ln \mathcal{P}$ $r$ $_{1}I_{1}|q$ $\vee|$ $i$ $J\succ---------*X/pA(r)$ Notice that $\overline{i}$ is a p-cofibration whose image is $q(A)$ . In particular, $q(A)$ is homeomorphic to $J$ . Furthermore, this shows that the proper quotient has the same ordinary homotopy type as the usual topological quotient. (1.0.2) It can be shown that any space in $\mathcal{P}$ admits an onto p-may $h:X\rightarrow J$ ([10; 6.5.3]). Actually, $h$ can be chosen to be a retraction of a given p-cofibration $i:J\subseteqq X$ . Indeed, since all $p$ -maps on $J$ are p-homotopic one can find a p-homotopy $H:J\times I\rightarrow J$ connectlng $h|J$ to $id_{J}$ . Then by using the proper H. E. P. one gets a $p$ -homotopy $H^{\prime}$ : $X\times I\rightarrow J$ extending $H$ , and $H_{i}^{\prime}$ is a p-retraction of $i$ . (1.0.3) The proper homotopy type of $X/pA(r)$ does not depend on the map 308 R. AYALA, E. DOMfNGUEZ, A. MARQUEZ and A. QUINTERO $r;X\rightarrow J$ . Moreover, given another p-map $r^{\prime}$ : $X\rightarrow J$ , there exists a p-homotopy equivalence $\xi:X/pA(r)\rightarrow X/pA(r^{\prime})$ such that $q\circ\xi=q^{\prime}$ . This is due to the homotopy invariance of push-outs in cofibration categories ([5: II. 1. $2b$ )]), since all proper maps on $J$ are properly homotopic. When it is clear which map $r$ is involved in the quotient, we shall drop the map $r$ from the notation. (1.0.4) Finally, let $\mathcal{P}^{*}$ denote the category under and over $J$ . It can be shown that $\mathcal{P}^{*}$ is a cofibration category where we can define proper wedges and proper quotients as well as proper cones $(c_{p})$ , and proper suspensions $(\Sigma_{p})$ . For a space $J\succ X\rightarrow J$ in $\mathcal{P}^{*}$ , these constructions do not depend on the pretraction $r$ (up to p-homotopy equivalence in $\mathcal{P}^{J}$ ). Moreover, the set $[\Sigma_{p}X, Y]_{p}^{J}$ is endowed of a natural group structure for any space $Y$ in $\mathcal{P}^{J}$ . See [3] for details. A strongly locally finite CW-complex $X$ will be considered as a space in $\mathcal{P}^{*}$ by choosing a cellular embedding $i:J\subseteqq X^{1}$ , and a $p$ -retraction of $i$ . See (1.0.2) above. If (X, $\alpha$ ) is a space in $\mathcal{P}^{J}$ the $(tow- \mathcal{G}_{t}, 9_{t})$ -object $\Pi_{n}(X, \alpha)=\{\pi_{n}(X, *_{0})-\pi_{n}(U_{1}, *_{1})-\pi_{n}(U_{2}, *_{2})\cdots\}$ $(n\geqq 1)$ is called the n-th homotopy tower of the pair (X, $\alpha$ ), where $\alpha(t_{j})=*J$ with $\alpha([t_{j}, \infty))\subseteqq U_{f}$ and the bonding maps are induced by the inclusions and the basepoint change isomorphisms. (1.0.5) $A$ space $X$ is said to be properly k-connected if $\mathcal{F}(X)=\{*\}$ and $\Pi_{r}(X, \alpha)\cong 0$ in $(tow- \mathcal{G}_{t}, \mathcal{G}_{i})(0\leqq r\leqq k)$ . Similarly for proper pairs (X, $A$ ) with $\mathcal{F}(X)=\mathcal{F}(A)=\{*\}$ . It is worth pointing out that although $\alpha$ and $\alpha^{\prime}$ represent the same Freudenthal end, the towers $\Pi_{n}(X, \alpha)$ and $\Pi_{n}(X, \alpha^{\prime})$ need not to be isomorphic (see [23; p. 13]). Nevertheless, if $X$ is properly l-connected there is no dependence on the ray $\alpha$ . In fact, any $p$ -homotopy $H:\alpha\cong\alpha^{\prime}$ induces a pro-isomorphism $H_{\$}$ ; $\Pi_{n}(X, \alpha^{\prime})\rightarrow\Pi_{n}(X, \alpha)$ . In addition, if $H:f\cong g$ is a phomotopy there is a commutative diagram $\Pi_{x}(X\underline{\alpha)\overline{g^{*}f_{*}\sim}},’\prod_{\Pi_{n}}n(Y,f\circ\alpha)(Y,g\circ\alpha)\downarrow G_{l}$ where $G=H\circ(\alpha\times id)$ . The towers $\Pi_{n}(X, A, \alpha)(n\geqq 2)$ are also defined for pairs (X, $A$ ) in $\mathcal{P}^{J}$ . In Moore spaces in proper homotopy 309 addition, if $A=J$ we have the identification $\Pi_{n}(X, J;\alpha)=\Pi_{n}(X, \alpha)(n\geqq 1)$ . We recall that E. Brown in [7] gives a functor $P:tow- \mathcal{G}_{s}\rightarrow \mathcal{G}_{t}$ which carries the tower $\Pi_{n}(X, \alpha)$ to the Brown-Grossman group $\pi_{n}^{\infty}(X, \alpha)$ . In a similar way we can define a functor $P_{0}$ : $(tow- \mathcal{G}_{f}, \mathcal{G}_{i})\rightarrow \mathcal{G}_{f}$ which maps $\Pi_{n}(X, \alpha)$ to the global Brown-Grossman group $\pi_{n}(X, \alpha)$ (see [17]). The n-th homology tower of $X$ can be defined as the tower $(n\geqq 0)$ $H_{n}(X)=\{H_{n}(X)-H_{n}(U_{I})-H_{n}(U_{2})-\cdots\}$ where the bonding maps are induced by the inclusions. The chain complex of towers of $X,$ $C_{*}(X)$ is $\{\partial:C_{n}(X)\rightarrow C_{n-1}(X)\}$ where $C_{n}(X)=\{C_{n}(X)\leftarrow C_{n}(U_{1})-C_{n}(U_{2})-\cdots\}$ . Now a proper cohomology theory, $H^{n}$ , with coefficients in a (tow-db, $\cup tb$ ) $-$ object $S$ is defined as the homology of the complex $...\leftarrow C^{n}(X)\leftarrow C^{n-1}(X)\leftarrow\cdots$ where $C^{n}(X)=(tow_{c}- Ab, Ab)(C_{n}(X), S)$ (see [14] for details). Also relative versions of these functors for pairs (X, $A$ ) in $\mathcal{P}$ are defined. Notice that the above functors (including $\Pi_{n}$ ) are well defined up to (tow- $\mathcal{G}_{i},$ $\mathcal{G}_{t}$ )-isomorphisms. Fundamental results on homotopy groups like the Blakers-Massey Theorem, the Freudenthal Theorem or the Hurewicz Theorem can be translated to proper homotopy by using the following proposition 1.1. $p_{ROPOSlT10N}.-([2;1.1])$ Let (X, $A$ ) be a connected strongly locally finite CW-pair with only one Freudenthal end and assume that (X, $A$ ) is properly k-connected. Then there exists a strongly locally finite CW-pair (X’, $A^{\prime}$ ) such that i) $X$ (respectively $A$ ) is a strong deformation p-retract subcomplex of $X^{\prime}$ (respectively $A^{\prime}$ ). ii) $(X^{\prime})^{k}\subseteqq A^{\prime}$ . In particular if $A=J,$ $X$ has the same homotopy type as a CW-complex $X^{\prime}$ with $(X^{\prime})^{k}=J$ . We recall that for any pair in $\mathcal{P}$ and any proper map $\rho:X\rightarrow J$ , there is a natural homeomorphism $J\cong q(A)$ , where $q:X\rightarrow X/pA(\rho)$ is the quotient (see (1.0.1)). Therefore, for any ray $\alpha:J\rightarrow X$ the map $q$ induces morphisms of towers 310 R. AYALA, E. DOMfNGUEZ, A. MARQUEZ and A. QUINTERO $q_{*};$ $\Pi_{r}(X, A, \alpha)-\Pi_{r}(X/pA(\rho), q^{0}\alpha)$ 1.2. THEOREM.–Let (X, $A$ ) be a strongly locally finite CW-pair such that (X, $A$ ) is properly n-connected and $A$ is properly m-connected $(n, m\geqq 1)$ . Then the morphism $q*defined$ above is an isomorphism if $2\leqq r\leqq m+n$ , and an epimorphism if $r=m+n+1$ . PROOF. Firstly, we shall prove the theorem when $X^{n}\subseteqq A$ and $A^{m}=J$ . Let $i:J\rightarrow A$ be the inclusion. According to (1.0.2), we can find a proper retraction $r:X\rightarrow J$ of $i$ . Let $\{U_{j}^{\prime}\}$ be a system of $\infty$ -neighbourhoods of $X$ consisting of subcomplexes. Without loss of generality we can assume that $r(U_{j}^{\prime})$ $\subseteqq[t_{j}, \infty)$ . Let $ U_{j}=U_{j}^{\prime}\cup[t_{f}, \infty$ ). It is clear that $\{U_{j}\}$ is a new system of $\infty-$ neighbourhoods with $ U_{j}^{1}=[t_{j}, \infty$ ), $(U_{j}, U_{j}\cap A)$ is n-connected and $A_{j}=U_{j}\cap A$ is m-connected for any $j\geqq 0$ . Then, if $X/pA$ is the proper quotient constructed with the retraction $r$ , it is easily checked that $\{U_{j}/pAj\}$ is a system of $\infty-$ neighbourhoods of $X/pA$ , where $U_{f}/pAj$ is constructed by using the restriction $r|U_{j}$ : $ U_{j}\rightarrow[t_{j}, \infty$ ). Since proper quotlents has the same ordinary homotopy type as ordinary quotients, we can levelwise apply the ordinary Blakers-Massey Theorem [24; 6.22] to get isomorphisms $q_{j*\ddagger}\pi_{r}(U_{j}, A_{j})\rightarrow\pi_{r}(U_{j}/pAf)$ if $r\leqq n+m$ and epimorphisms if $r=m+n+1$ . Now the result follows when $\alpha=i$ and $\rho=r$ . Moreover, we can use the naturality of the base ray change isomorphisms and the homotopical invariance of proper quotients (see (1.0.3), and (1.0.5)) to prove the result for (X, $A$ ) as above and arbitrary $\alpha$ and $\rho$ . The general case can be reduced to the previous case by using Theorem 1.1. Using Theorem 1.2 and a proof similar to the ordinary case ([24; 6.23]) we obtain 1.3. THEOREM.–Let $X$ be a properly n-connected strongly locally finite CWcomplex. Then there is a natural suspension $(tow- \mathcal{G}_{t}, \mathcal{G}_{t})$ -morphism $\Sigma_{*}:$ $\Pi_{k}(X)-\Pi_{k+1}(\Sigma_{p}X)$ which is an isomorphism if $k\leqq 2n$ and an epimorphism if $k=2n+1$ . 1.4. REMARK.–The results stating that $\Sigma$ and $q$ induce isomorphisms between the corresponding homology towers can be proved in a straightforward way without using Proposition 1.1. Moore spaces in proper homotopy 311 Proposition 1.1 and the above remark give an easy proof of the following theorem. 1.5. THEOREM. $-$ ([21; II.4.2.7]) Let $X$ be a properly n-connected strongly locally finite CW-complex. Then the natural Hurewicz morphism $h;\Pi_{k}(X)\rightarrow H_{k}(X)$ is an isomorphism if $k=n+1$ and an epimorphism if $k=n+2$ . Furthermore Theorem 1.2 and Remark 1.4 provide a relative version of Theorem 1.5 for a proper pair $(X, A)$ of strongly locally finite CW-complexes with $A$ properly l-connected and $X$ properly n-connected. Finally Theorem 1.5, the Brown-Grossman functor $P_{0}$ in (1.0.5), and [7; $p$ , $43J$ lead to 1.6. THEOREM.–Given a p-map $f:X\rightarrow Y$ where $X$ and $Y$ are properly 1connected finite dimensional, locally finite CW-complexes such that $f_{*};$ $H_{r}(X)\rightarrow H_{r}(\}^{7})$ is an isomorphism for each $r$ , then the map $f$ is a p-homotopy equivalence. 2. Proper Moore Spaces. We start with some notions in ( $tow_{c}- Ab,$ (Ab). 2.1. DEFINITION.–A free tower in (tow-.A6, A6) is a tower $F(\mathcal{L})=\dagger F(L_{0})-F(L_{1})-\cdots\}$ where the following four conditions hold; i) $\mathcal{L}$ is a filtration $\mathcal{L}\equiv L_{0}\supseteqq L_{1}\supseteqq\cdots$ with $L_{0}$ a countable set. ii) $\bigcap_{j=1}^{\infty}L_{j}=\emptyset$ . iii) The differences $L_{k}\backslash L_{k+1}$ are finite. iv) $F(L_{i})$ is the free group generated by $L_{i}$ and the bonding morphism are induced by the inclusions. Given the towers $F(\mathcal{L})$ and $F(\mathcal{L}^{\prime})$ , it can easily be checked that any bijection $L_{0}\cong L_{0}^{\prime}$ induces an isomorphism $F(\mathcal{L})\cong F(\mathcal{L}^{\prime})$ in (towA6, $\mathcal{A}b$ ). So the isomorphism class of $F(\mathcal{L})$ is determined by the cardinality of $L_{0}$ . 2.2. REMARKS. $-a$ ) Free towers are projective objects $\ln$ (tow-Ab, $\mathcal{A}b$ ) (see [14]). b) Given a strongly locally finite CW-complex $X$ , and a system of $\infty$ -neighbourhoods $\{U_{j}\}$ consisting of subcomplexes, the tower of cellular n-chains of $X$ , $C_{n}(X)=\{C_{n}(X)-C_{n}(U_{1})-C_{n}(U_{2})-\cdots\}$ is obviously a free tower. Also the tower of cellular n-cycles 312 R. AYALA, E. DOMiNGUEZ, A. MARQUEZ and A. QUINTERO $Z_{n}(X)=\{Z_{n}(X)-Z_{n}(U_{1})\leftarrow Z_{n}(U_{2})\leftarrow\cdots\}$ is a free tower. More generally, the kernel of any morphism between two free towers is always a free tower (see [14; 5.1]). 2.3. DEFINITION.–A tower $S$ is said to be geometrically admissible if there exists an exact sequence in (tow-Ab, $\mathcal{A}b$ ) $j_{s}$ $j_{2}$ $j_{1}$ $0-F(\mathcal{L}_{3})-F(\mathcal{L}_{2})-F(\mathcal{L}_{1})-S-0$ When $F(\mathcal{L}_{3})$ is trivial we say that $S$ has geometrical projective dimension (g.p. $d.$ ) 1. 0therwise, we write $g$ . $p.d$ . $S=2$ . 2.4. REMARK.–For any strongly locally finite CW-complex $X$ the n-th homology tower of $X$ is geometrically admissible since the exact sequence of towers $0-Z_{n+1}(X)-C_{n+1}(X)-Z_{n}(X)-H_{n}(X)-0$ is a free resolution. However, the short exact sequence $0\rightarrow{\rm Im}\partial_{n+1}\rightarrow Z_{n}(X)\rightarrow H_{n}(X)\rightarrow 0$ is not always a free resolution. Indeed, let $X$ be the CW-complex obtained from the cylinder $ S^{n}\times[0, \infty$ ) by attaching an $(n+1)$ -cell at $S^{n}\times\{j\}$ by a map $f:S^{n}\rightarrow S^{n}$ of degree $2^{j}(j\geqq 1)$ . Then one can check that ${\rm Im}\partial_{n+1}$ is not free since $\partial_{n+1}$ has no right inverse. 2.5. REMARK.–In [9] Dymov introduced the notion of copresentation of a tower $S$ . More explicitly, following [9] we say that $s=\{G_{0}\leftarrow G_{1}\leftarrow\cdots\}$ admits a copresentation if there exist a levelwise epimorphism $\phi:F(\mathcal{L})\rightarrow S$ with $F(\mathcal{L})$ a free tower and subsets $R_{f}\subseteqq F(L_{f})$ with $R_{j}-R_{J+1}$ finite and $\cap\infty R_{j}=\emptyset$ , such $j\Leftarrow 1$ that $Ker\phi_{j}$ is the subgroup $\langle R_{j}\rangle$ generated by $R_{j}$ . In general, the tower $\langle R\rangle$ $\equiv\langle R_{0}\rangle\leftarrow\langle R_{1}\rangle\leftarrow\cdots$ needs not to be projective. It is easy to check that a tower $S$ is geometrically admissible if and only if admits a copresentation (up to isomorphism). In fact, if $\langle 9\rangle\rightarrow fF(\mathcal{L})\rightarrow S\rightarrow 0$ is a Dymov copresentation, and $F(R)$ denotes the free tower consisting of the free groups $F(R_{i})$ , there is a levelwise epimorphism $ F(R)\rightarrow k\langle R\rangle$ and an exact sequence $F(R)^{fk}\rightarrow\langle R\rangle\rightarrow F(\mathcal{L})$ . By Remark 2.2(b), $Ker(f\circ k)$ is a free tower, and so $S$ is geometrically admissible. Conversely, for any free resolution $j_{3}$ $j_{2}$ $j_{1}$ $0\rightarrow F(\mathcal{L}_{3})-F(\mathcal{L}_{2})-F(\mathcal{L}_{1})-S-0$ Moore spaces in proper homotopy 313 let $\{\varphi_{i} : F(L_{n(i)}^{2})\rightarrow F(L_{t}^{1})\}$ be a levelwise representative of $j_{2}$ . Then $s^{J}=$ {Coker $\varphi_{i}$ } is isomorphic to $S$ by exactness and admits the following copresentation. Let $\phi:F(\mathcal{L}_{1})\rightarrow S^{\prime}$ be the natural levelwise quotient morphism. We take $R_{i}=\varphi_{i}(L_{n(i)}^{2})\subseteqq F(L_{i}^{1})$ . Now, it is clear that $R_{i}-R_{i+1}\subseteqq\varphi_{i}(L_{n(i)}^{2}-L_{nti+1}^{2})$ is finite. If $\tilde{S}^{n}$ ( $\tilde{B}^{n}$ respectively), is the space obtained by attaching finitely many copies (possibly no copy) of $S^{n}$ ( $B^{n}$ respectively) $(n\geqq 2)$ at each $ m\in N\subseteqq[0, \infty$ ), one checks that $\Pi_{n}(\tilde{S}^{n})(\Pi_{n}(\tilde{B}^{n},\tilde{S}^{n-1})$ respectively) can be identified in a natural way with some $F(\mathcal{L})$ , with $L_{0}\subseteqq N$ . Moreover the following result holds: 2.6. LEMMA.–Given a space $X$ in $\mathcal{P}^{J}$ , there is a natural bijection $\rho:[\tilde{S}^{n}, X]_{p}^{J}$ $\cong(towA6, Ab)(F(\mathcal{L})\rightarrow\Pi_{n}(X))(n\geqq 2)$ , given by $\rho([f])=f*\cdot$ Moreover, if $Z$ is the mapping cone $X\bigcup_{f}\tilde{B}^{n+1}$ in the cofibration category $\mathcal{P}^{J}$ , then $f*can$ be also regarded as the boundary operator $d_{n+1}$ ; $\Pi_{n+1}(Z, X)\rightarrow\Pi_{n}(X)$ . PROOF.–Take $\varphi:F(\mathcal{L})\rightarrow\Pi_{n}(X)$ . After identifylng $\Pi_{n}(\tilde{S}^{n})$ with $F(\mathcal{L})$ , let $\pi_{n}(\tilde{S}^{n})-\pi_{n}(\tilde{S}_{k(1)}^{n})-\pi_{n}(\tilde{S}_{k(2)}^{n}-\cdots$ $\downarrow$ $\downarrow$ $\downarrow$ $\pi_{n}(X)-\pi_{n}(U_{1})-\pi_{n}(U_{2})-\cdots$ be a levelwise representative of $\varphi$ , where $\tilde{S}_{k(t)}^{n}$ is obtained by deleting from $\tilde{S}^{n}$ the copies, $S_{j}^{n}$ , of $S^{n}$ placed at the points $1\leqq\int<k(t),$ $ k(1)<k(2)<\cdots$ . We define $f|S_{j}^{n}$ as a representative of $\varphi_{h(j)}[l_{f}]$ where $l_{j}$ : $S_{j}^{n}\subseteqq\tilde{S}_{k(t)}^{n},$ $ k(t)\leqq$ $j<k(t+1)$ . One easily checks that $ f_{*}=\varphi$ . This proves that $\rho$ is onto. The injectivity follows in a similar way. Finally, we have the diagram $\Pi_{n+1}(Z, X)\Pi_{n+1}(\tilde{B}^{n+1}\underline{p_{\cong^{*}}}\tilde{S}^{n})$ $\Pi_{n}\downarrow_{(X)}$ $-\Pi_{n}(\tilde{S}^{n})=F(\mathcal{L})\downarrow\cong$ where $p:\tilde{B}^{n+1}\rightarrow Z$ is the canonical p-map and $p_{*}$ is an isomorphism by Theorem 1.2 since $\tilde{B}^{n+1}/p\tilde{S}^{n}\cong\tilde{S}^{n+1}\cong Z/pX$ . For the sake of simplicity, we shall use the single notation $\tilde{S}^{n}(\tilde{B}^{n})$ for all the “strings” of spheres (balls) described above. The particular objects $\tilde{S}^{n}(\tilde{B}^{n})$ we are using in the future will be clear from the context. Similarly for $F(\mathcal{L})$ . 320 R. AYALA, E. DOMiNGUEZ, A. MARQUEZ and A. QUINTERO is the p-map defined by the characteristic maps of the $(n+1)$ -cells of $X$ . We easily check that $f_{n+1*}=\varphi_{n+1}$ . When one tries to go further an obstruction appears as follows. In the diagram $\Pi_{n+2}(X^{n+2}, X^{n+1})$ $\Pi_{n+1}(X^{n+1})$ $\partial_{n+2}-\Pi_{n+1}(X^{n+1}, X^{n})$ $\varphi_{n+2}\downarrow$ (1) $\downarrow f_{n+1*}$ $\downarrow f_{n+1*}=\varphi_{n+1}$ $\Pi_{n+2}(Y^{n+2}, Y^{n+1})$ $\Pi_{n+1}(Y^{n+1})-\Pi_{n+1}(Y^{n+1} \partial_{n+2}^{\prime}Y^{n})$ the square (1) needs not be commutative. One defines an element in $(tow-\Lambda b, \Lambda b)$ $(F(\mathcal{L}_{s});\Pi_{n+1}(Y^{n+1}))$ by the difference $\beta(f_{n+1})=f_{n+12}\partial_{n+2}-\partial_{n+2}^{\prime}\varphi_{n+2}$ Since the other square is commutative we have, by definition of $Kerj_{*}$ in (tow- $\llcorner Ab,$ $\mathcal{A}b$ ) that $\beta$ can be regarded as a morphism $\beta(f_{n+1}):F(\mathcal{L}_{\theta})\rightarrow Kerj_{*}=\Gamma_{n+1}Y$ . Obviously $\beta$ is a cocycle and defines a class $c(\varphi)\in H^{n+2}(X, \Gamma_{?l+1}Y)$ . The next lemma shows that $c(\varphi)$ is a well defined obstruction. 3.3. LEMMA. $-1$ ) $c(\varphi)$ does not depend on the morphisms $\varphi_{i}(i=n, n+1, n+2)$ . 2) $c(\varphi)$ is an obstruction to realizing $\varphi$ . PROOF. 1) Let $\{\varphi_{i}^{\prime}\}$ be another morphism such that the diagram $(^{*})$ commutes and let $f_{n+1}^{\prime}$ : $X^{n+1}\rightarrow Y^{n+1}$ be a $p$ -map realizing $\varphi_{n+1}^{\prime}$ . It is a well-known fact from Homological Algebra in abelian categories that $\{\varphi_{i}\}$ and $\{\varphi_{i}^{\prime}\}$ are homotopic chain morphisms. Thus, there exist morphisms $\{\alpha_{i} : C_{i}(X)\rightarrow C_{i+1}(Y)\}$ $(i=n, n+1, n+2)$ such that the following equalities hold a) $\varphi_{n}^{\prime}-\varphi_{n}=d_{n+1}^{\prime}\circ\alpha_{n}$ ; b) $\varphi_{n+1}^{\prime}-\varphi_{n+1}=\alpha_{n}\circ d_{n+1}+d_{n+2}^{\prime}\circ\alpha_{n+1}$ ; and c) $\varphi_{n+2}^{\prime}-\varphi_{n+2}=d_{n+3}^{\prime}\circ\alpha_{n+2}+\alpha_{n+1}\circ d_{n+2}$ . By c) and the definition of $\beta(f_{n+1})$ we have (I) $\beta(f_{n+1}^{\prime})-\beta(f_{n+1})=(f_{n+1*}^{\prime}-f_{n+1^{*}}-\partial_{n+2}^{\prime}\circ\alpha_{n+1}\circ j_{*})\circ\partial_{n+2}$ where $j_{*}:$ $\Pi_{n+1}(Y^{n+1})\rightarrow C_{n+1}(Y)$ . Now, a) provides a $p$ -homotopy $H:X^{n}\times I$ $\rightarrow Y^{n+1}$ between $f_{n}$ and $f_{n}^{\prime}=f_{n+1}^{\prime}|_{X^{n}}$ . As in ordinary homotopy theory, $H$ yields a “difference” morphism $\Delta=d(f_{n+1}^{\prime}, H, f_{n+1});C_{n+1}(X)-\Pi_{n+1}(Y^{n+1})$ (see [15]). Moreover, $H$ can be chosen in such a way that $\int*\circ\Delta=f_{n+1*}^{\prime}-f_{n+15}$ $-\alpha_{n}\circ d_{n+1}$ . Take $\beta=\Delta-\partial_{n+2}^{\prime}\circ\alpha_{n+1}$ . By b), $j_{*}(\beta)=0$ . And the projectiveness of $C_{n+1}(X)$ allows us to regard $\beta$ as an element in (tow-.A6, $\cup tb$ ) $(C_{n+1}(X);\Gamma_{n+1}(Y))$ since $Kerj_{*}=\Gamma_{n+1}(Y)$ . Moore spaces in proper homotopy 321 Finally, one can readily check from the definitions that $f_{n+1}^{\prime}\cong pf_{n+1}+$ $(\beta+\partial_{n+2}^{\prime}\circ\alpha_{n+1})$ and then, by Lemma 3.1 the right side in the equality (I) is $\beta^{Q}j_{*}o\partial_{n+2}=\beta\circ d_{n+2}=\delta\beta$ . This proves $[\beta(f_{n+1})]=[\beta(f\text{{\it \’{n}}}_{+1})]\in H^{n+2}(X;\Gamma_{n+1}(Y))$ . 2) If $c(\varphi)=0$ , let $w\in(towAb, \llcorner Ab)(C_{n+1}(X), \Gamma_{n+1}(Y))$ be such that $\beta(f_{n+1})=$ $\delta w=w\circ d_{n+2}=w\circ j_{*}\circ\partial_{n+2}$ . Take $\overline{f}_{n+1}=f_{n+1}+w$ . By Lemma 3.1, $\beta(\overline{f}_{n+1})=$ $(f_{n+1*}-w\circ j_{*})\partial_{n+2}-\partial_{n+2}^{\prime}\circ\varphi_{n+2}=0$ , and $\overline{f}_{n+1}$ extends to a p-map $f_{n+2}$ : $X\rightarrow Y$ with $f_{n+2*}=\varphi:H_{n}(X)\rightarrow H_{n}(Y)$ . 3.4. REMARKS. $-a$ ) The obstruction $c(\varphi)$ was already considered by J. H. C. Whithead in ordinary homotopy (see [25; \S 6]) and it can be defined within the general settlng of cofibratlon categories (see [5; VII. 1.13]). b) For any $n\geqq 3$ there are two non properly equivalent Moore spaces of type $(S;n)$ . The examples are given in Appendix A. As an immediate consequence of Theorem 3.2 we have 3.5. COROLLARY.–If $S$ is a tower $ w\iota$ th g.p.d. $S=1$ then there exists (up to p-homotopy) a unique proper Moore space of type $(S, n)(n\geqq 2)$ . More generally, we can state 3.6. COROLLARY. –If $S$ is a geometrically admz ssible tower with $Ext^{2}(S;\Gamma_{n}(S))$ $=0$ , then there exists a unique proper Moore space of type $(S, n)(n\geqq 2)$ . Here $\Gamma_{n}(S)$ denotes the tower obtained from $S$ by applying levelwise the algebraic Whitehead $\Gamma_{n}$ -functor (see $\lceil_{-}26$ ; Ch. II] or [5; IX. 4]). It is known that $\Gamma_{n}=-\otimes Z_{2}$ when $n\geqq 3$ . So, (3.6) yields 3.7. COROLLARY. –If $S$ is a geometrically admissible tower with $Ext^{2}(S;S\otimes Z_{2})$ $=0$ then there exists a unique proper Moore space of type $(S, n)$ for all $n\geqq 3$ . PROOF $0F(3.6)$ . Let $Y$ be a proper Moore space of type $(S, n)$ constructed as in the proof of Theorem 2.9. Let $ Y\supseteqq U_{1}\supseteqq\cdots\supseteqq U_{n}\cdots$ be a system of $\infty-$ neighbourhoods such that each $U_{j}$ is a subcomplex. Moreover, each $U_{j}$ is $(n-1)$ -connected by construction. Thus by [19; VIII. 2.4] for $n\geqq 3$ and [26; III. 14] for $n=2$ we have $\Gamma_{n+1}U_{j}\cong\Gamma_{n}(H_{n}(U_{j}))$ and therefore $\Gamma_{n+1}Y\cong\Gamma_{n}(S)$ . If $X$ is another proper Moore space of type $(S;n)$ we can realize id: $S\rightarrow S$ by a $p$ -map $f:X\rightarrow Y$ by Theorem 3.2 since $H^{n+2}(X;\Gamma_{n+1}Y)=Ext^{2}(S;\Gamma_{n+1}Y)=0$ . By Theorem 1.6 $f$ is actually a p-homotopy equivalence. 322 R. AYALA, E. $DoMINGUEZ$ , A. M\’ARQUEZ and A. QUINTERO 3.8. REMARK.–The tower $s=\{Z_{2}\leftarrow Z_{4}p_{1}\leftarrow Z_{8}\leftarrow p_{2}\ldots\}$ where $p_{i}(1)=1$ , has geometrical projective dimension 2 since $S$ is the n-th homology tower of the CWcomplex given in Remark 2.4. Nevertheless $\Gamma_{n}S$ is the constant tower $Z_{2}$ when $n\geqq 3$ , and $\Gamma_{2}S$ is isomorphic to $S$ since $\Gamma_{2}Z_{2n}=Z_{4n}$ according to [26; II.(B)]. Then, one can check as in Appendix A that $Ext^{2}(S;\Gamma_{n}S)=0$ . So, there is a unique proper Moore space of type $(S;n)(n\geqq 3)$ by Corollary 3.7. The same result holds for $n=2$ . Another sufficient algebraic condition on $S$ for the uniqueness of proper Moore spaces of type $(S;n)n\geqq 3)$ is the following. 3.9. $p_{ROPOSITION}.$ –Let $S$ be a geometrically admissible tower such that $Tor^{1}(S;Z_{2})=0$ . Then there is a unique Moore space of type $(S;n)(n\geqq 3),$ unique up to p-homotopy. Before starting the proof of Proposition 3.9 we shall fix notation and prove a lemma whose proof $1S$ similar to the proof of [14; Lemma 2]. Let $(towZ_{2}, Z_{2})$ denote the abelian category defined in the same way as ( $tow- \mathcal{A}b$ , A6) by using $Z_{2}$ -vector spaces instead of abelian groups. Given a free tower $F(\mathcal{L})$ , let $Z_{2}(L_{i})$ denote $F(L_{i})\otimes Z_{2}$ . Then 3.10. LEMMA.–Any tower $\{V_{0}\leftarrow V_{1}\leftarrow V_{2}\cdots\}$ with $V_{i}\subseteqq Z_{2}(L_{i})$ and with bonding morphisms the corresponding restrictions is projective in $(towZ_{2}, Z_{2})$ . PROOF. We may find a basis $T_{i}$ of $V_{i}/V_{i+1}$ such that any element of $T_{i}$ is represented by a linear combination of elements in $L_{i}-L_{i+1}$ . Let $B_{i}$ be the union $\cup\{T_{j} ; j\geqq I\}$ . It is easy to check that $B_{i}$ is a basis for $V_{i}$ . Since $ B_{0}\supseteqq B_{1}\cdots$ and $\cap B_{i}=\emptyset$ it is straightforwardly shown that $\{V_{0}\leftarrow V_{1}\leftarrow V_{2}\cdots\}$ is projective. PROOF OF PROPOSITION 3.9. Let $d_{2}$ $d_{2}$ $d_{0}$ $0-C_{2}-C_{1}-C_{0}-S-0$ be a free resolution of $S$ . Since $S\otimes Z_{2}$ is a tower of groups of order 2 we have an isomorphism (towA6, $Ab$ ) $(C_{i} ; S\otimes Z_{2})\cong(towZ_{2}, Z_{2})(C_{i}\otimes Z_{2} ; S\otimes Z_{2})$ Therefore, $Ext^{2}(S;S\otimes Z_{2})\cong Ext^{1}({\rm Im} d_{1} ; S\otimes Z_{2})\cong Coker((d_{2}\otimes 1)^{*})$ Moore spaces in proper homotopy 323 where $*$ stands for the dual morphism. Now from the exact sequence $0-C_{2}-C_{1}-{\rm Im} d_{1}-0$ we get the exact sequence $0-C_{2}\otimes Z_{2}$ $d_{2}\otimes 1\rightarrow C_{1}\otimes Z_{2}{\rm Im} d_{1}\otimes Z_{2}\underline{d_{1}\otimes 1}\rightarrow 0$ since the tower $Ker(d_{2}\otimes 1)$ is isomorphic to the tower $\{Tor({\rm Im} d_{1}^{i} ; Z_{2})\}$ which is trivial because each component ${\rm Im}[d_{1}^{i} : C_{1}^{k(i)}\rightarrow C_{0}^{i}]$ of the tower ${\rm Im} d_{1}$ is a free abelian group. Thus, Coker $((d_{2}\otimes 1)^{*})\cong Ext_{Z_{2}}^{1}({\rm Im} d_{1}\otimes Z_{2} ; S\otimes Z_{2})$ where the right side is the $Ext1$ functor in the category $(towZ_{2}, Z_{2})$ . On the other hand we have the commutative diagram $0-C_{2}\otimes Z_{2}$ $C_{1}\otimes Z_{2}{\rm Im} d_{1}\otimes Z_{2}$ $d_{2}\otimes 1\underline{d_{1}\otimes 1}-0$ $\Vert$ $\Vert$ $0-c_{2}\otimes z_{2^{-}}^{d_{2}\otimes 1}$ $C_{1}\otimes Z_{2}\rightarrow^{d_{1}\otimes 1}{\rm Im}(d_{1}\otimes 1)-0$ where the upper row is exact as it was proven above. And the lower row is also exact since $Tor^{1}(S;Z_{2})=Ker(d_{1}\otimes 1)/{\rm Im}(d_{2}\otimes 1)=0$ by hypothesis. Thus ${\rm Im} d_{1}\otimes Z_{2}\cong{\rm Im}(d_{1}\otimes 1)$ and hence $Ext_{Z_{2}}^{1}({\rm Im} d_{1}\otimes Z_{2} ; S\otimes Z_{2})\cong Ext_{Z_{2}}^{1}({\rm Im}(d_{1}\otimes 1);S\otimes Z_{2})$ . Now the former term vanishes because ${\rm Im}(d_{1}\otimes 1)$ is projective by Lemma 3.10. This yields $Ext^{2}(S;S\otimes Z_{2})=0$ and the uniqueness follows from Corollary 3.7. FINAL REMARK. The category of trees of abelian groups (see [12]) seems to be the right algebraic framework for a generalization of the results of this paper to spaces with many Freudenthal ends. Appendix A. Two non properly equivalent proper More spaces of type $(S;n),$ $n\geqq 3$ . Let $S$ be the tower $\{\bigoplus_{1}^{\infty}Z_{2}\oplus Z_{2}\rightarrow^{k_{1}\oplus 1}\bigoplus_{2}^{\infty}Z_{2}Z_{2}\}$ $k_{2}\oplus 1$ $\ldots$ in (tow-di, $\mathcal{A}b$ ), with $k_{j}$ standing for the natural inclusion morphism. A free resolution for $S$ is 324 R. AYALA, E. $DoMtNGUEZ$ , A. M\’ARQUEZ and A. QUINTERO $\partial_{2}$ $\partial_{2}$ $0-F(\mathcal{L}_{3})-F(\mathcal{L}_{2})-F(\mathcal{L}_{1})-S-0$ where $\mathcal{L}_{3}$ is $ L_{1}^{3}\supseteqq L_{2}^{3}\supseteqq\cdots$ with $L_{j}^{3}=\{\alpha_{i} ; i\geqq]\}$ . $\mathcal{L}_{2}$ is $ L_{1}^{2}\supseteqq L_{2}^{2}\supseteqq\cdots$ with $L_{j}^{2}=$ $\{\rho_{i}, \mu_{i}, \sigma_{i} ; i\geqq j\}$ and $\mathcal{L}_{1}$ is $ L_{1}^{1}\supseteqq L_{2}^{1}\supseteqq\cdots$ with $L_{j}^{I}=\{\epsilon_{i}, \gamma_{i} ; i\geqq j\}$ . And the morphisms are given by $\partial_{2}(\alpha_{i})=\mu_{i+1}-2\sigma_{i}-\mu_{i}$ ; $\partial_{1}(\rho_{i})=2\epsilon_{i},$ $\theta_{1}(\mu_{i})=2\gamma_{i}$ and $\partial_{1}(\sigma_{i})=$ $\gamma_{i+1}-\gamma_{i}$ . Clearly, $S=S_{1}\oplus S_{2}$ , where $S_{1}$ is the tower $\{\bigoplus_{1}^{\infty}Z_{2}\bigoplus_{2}^{\infty}Z_{2}\}\underline{k_{1}}\underline{k_{2}}\ldots$ and $S_{2}$ is the constant tower $\{Z_{2}=Z_{2}=\cdots\}$ . A. 1. LEMMA. $-Ext^{2}(S;S)\neq 0,$ $Ext^{2}(S_{1} ; S_{1})=Ext^{2}(S_{2} ; S_{2})=0$ . PROOF. By the naturality of $Ext$ ’ and $\oplus$ we have $Ext^{2}(S;S)=\oplus$ $\{Ext^{2}(S_{i} ; S_{j});i, j\leqq 2\}$ . On the other hand, it is easy to check that $g$ . $p$ . $d$ . $S_{1}=1$ , and so $Ext^{2}(S_{1} ; S_{j})$ $=0$ . With the above notations, $S_{2}$ admits the free resolution $\partial_{2}$ $\partial_{1}$ $0\rightarrow F(\{\alpha_{i}\})-F(\{\mu_{i}, \sigma_{i}\})-F(\{\gamma_{i}\})-S_{2}-0$ and by the standard Hom-Ext exact sequence we get $Ext^{2}(S_{2} ; S_{2})=Ext^{1}({\rm Im}\partial_{1} ; S_{2})=0$ . Indeed, for any $\varphi\in(towAb, d6)(F(\{\alpha_{i}\}), S_{2})$ we may define $\overline{\varphi}\in$ ( $towAb,$ Ab) ( $F(\{\mu_{i}, \sigma_{i}\}, S)$ by $\overline{\varphi}(\sigma_{i})=0,\overline{\varphi}(\mu_{1})=0$ and $\overline{\varphi}(\mu_{J})=\Sigma\{\varphi(\alpha_{i});i\leqq j-1\}$ . Then $\overline{\varphi}\circ\partial_{2}$ $=\varphi$ . Finally $Ext^{2}(S_{2}, S_{1})=Ext^{1}({\rm Im}\partial_{1} ; S_{1})\neq 0$ since $\xi:F(\{\alpha_{i}\})\rightarrow S_{1}$ given by $\xi(\alpha_{i})=$ $\epsilon_{i}\infty 1$ defines a non-trivial element. 0therwise, $\xi=\tau\circ\partial_{2}$ for some $\epsilon;F(\{\mu_{i}, \sigma_{i}\})$ $\rightarrow S_{1}$ and $\tau$ yields the equalities $\epsilon_{i}\otimes 1=\tau(\mu_{i+1})-\tau(\mu_{i})(i\geqq 1)$ . As $\tau$ is a promorphism one can inductively prove that $\tau(\mu_{i})\in\bigoplus_{k\geq 1}Z_{2}$ and the sequence $\{\epsilon_{i}\otimes 1\}$ would represent the trivial element $\ln\lim^{1}S_{I}$ and it is a well-known fact that it does not. Lemma A.1 and Corollary 3.7 yield that $R$ ( $S_{1}$ ; n) and $R$ ( $S_{2}$ ; n) are uniquely determined up to p-homotopy $(n\geqq 3)$ . Actually these types are represented by $\tilde{W}$ and $ W\times[0, \infty$ ), where $W^{\approx}$ is obtained by attaching one copy of $W$ at each natural coordinate of $[0, \infty$ ) and $W=S^{n}\bigcup_{2}e^{n+1}$ is the n-sphere with an $(n+1)-$ cell attached by a map of degree 2. Thus, $X=R(S_{1} ; n)_{p}R$ ( $S_{2}$ ; n) is a representative of $R(S;n)$ by Corollary 2.10. A.2. LEMMA. The natural map $[X; X]_{p}^{J}\rightarrow(towAA, \Lambda b)(S;S)$ is onto. Moore spaces in proper homotopy 325 PROOF. By Proposition 2.14 $[R(S_{1} ; n);X]_{p}^{J}\rightarrow[\circ$ is onto. On the other hand, (tow-db, $Ab$ ) $(S_{2} ; S_{1})=\lim S_{1}=0\leftarrow$ and $[R(S_{2} ; n);X]_{p}^{J}\rightarrow$ $[R(S_{2} ; n);R(S_{2} ; n)]_{p}^{J}\cong[W;W]^{J}\rightarrow_{\zeta}A6(Z_{2} ; Z_{2})\cong(towAb, \mathcal{A}b)(S_{2} ; S_{2})$ . where the first bijection is given by the Edwards-Hastings embedding Theorem ([10; 6.27]). Finally the natural bijection $[X; X]_{p}^{J}\cong[R(S_{1} ; n);X]_{p}^{J}\times$ $[R(S_{2} ; n);X]_{p}^{J}$ completes the proof. Now we choose a non-trivial element $\alpha\in H^{n+2}(X;\Gamma_{n+1}X)\cong Ext^{2}(S;S)\neq 0$ . By Remark 2.7 we may assume $X=\tilde{B}^{n+2}\bigcup_{h_{0}}X^{n+1}$ . Now, the commutative diagram $C_{n+2}(X)=\Pi_{n+2}(X, X^{n+1}\Pi_{n+2}(\tilde{B}^{n+1}\vec{h}\cong_{*}S^{n+1})$ $\downarrow d_{n+2}$ $\cong\downarrow d_{n+2}$ $\Pi_{n+1}(X^{n+1})\Pi_{n+1}(S^{n+1})\underline{h_{0*}}$ allows us to identify the boundary operator $d_{n+2}$ with the morphism $h_{0*}(h$ is the characteristic map $h:\tilde{B}^{n+2}\rightarrow X$ ). The isomorphism $d_{n+2}\circ h_{*}^{-1}$ also gives the identification (I) (tow-Ab, $\cup tb$ ) $(C_{n+2}(X);\Pi_{n+1}(X^{n+1}))$ $\cong(towAb, \mathcal{A}b)(\Pi_{n+1}(\tilde{S}^{n+1});\Pi_{n+1}(X^{n+1}))\cong[S^{n+1} ; X^{n+1}]_{p}^{J}$ where the second isomorphism is given by Lemma 2.6. Thus, if $\alpha=[a],$ $a$ can be regarded as a p-map $g;S^{n+1}\rightarrow X^{n}\subseteqq X^{n+I}$ . Let $\overline{h}_{0}$ be a representative of $[h_{0}]+[g]\in[S^{n+1} ; X]_{p}^{J}$ and let $Y$ be the proper cone of $\overline{h}_{0}$ . Since ${\rm Im} g\subseteqq X^{n}$ , the complexes towers of cellular chains of $X$ and $Y$ are the same. But A.3. LEMMA.–The obstruction $c(id)\in H^{n+2}(X;\Gamma_{n+1}Y)$ given in Theorem 3.2 for id: $H_{n}(X)=S\rightarrow S=H_{n}(Y)$ is non-trivial. PROOF. Smce $c(id)$ does not depend on the morphisms $\varphi_{i}$ : $C_{i}(X)\rightarrow C_{i}(Y)$ inducing id: $S\rightarrow S$ (see Lemma $3.3(1)$ ), one can consider $\varphi_{i}=id$ for each $i=n$ , $n+1,$ $n+2$ . So, $c(id)$ is represented by $\beta(id)=d_{n+2}-d_{n+2}^{\prime}$ , where $d_{n+2}$ is given in the above diagram for $X$ . Simllarly $d_{n+2}^{\prime}$ for $Y$ . Bearing in mlnd the identification (I) $\beta(id)$ is regarded as $h_{*}-h_{*}-g_{*}=-g_{*}$ . Then $\beta(id)$ is actually $-a$ and $c(id)=-\alpha\neq 0$ . Finally we get, 326 R. AYALA, E. DOMiNGUEZ, A. M\’ARQUEZ and A. QUINTERO A.4. $p_{ROPOSITION}.-X$ and $Y$ are not p-homotopically equivalent. PROOF. If $h:X\rightarrow Y$ is a p-homotopy equivalence, let $h^{\prime}$ be a $p$ -homotopic inverse of $h$ . The morphism $h_{*}^{\prime}$ : $H_{n}(Y)=S\rightarrow H_{n}(X)=S$ can be realized by a hmap $f:X\rightarrow X$ according to Lemma A.2. Then $h\circ f:X\rightarrow Y$ is a $p$ -map with $(h\circ f)_{*}=id:H_{n}(X)=s\rightarrow s=H_{n}(1^{\nearrow})$ , and this cannot happen by Lemma A.3. References [1] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Maths vol. 100, Springer 1969. [2] R. Ayala, E. Dominguez and A. Quintero, Hurewicz Theorem for homology at infinity, Quart. J. Math. Oxford 44 (1993), 139-153. [3] R. Ayala, E. Dominguez and A. Quintero, A theoretical framework for Proper Homotopy Theory, Math. Proc. Camb. Philos. Soc. 107 (1990), 475-482. [4] R. Ayala, E. Dominguez, A. M\’arquez, A. Quintero and S. Rodriguez, Espaces de Moore dans la cat\’egorie propre, C. R. Acad. Sci. Paris, t. 309, S\’erie I, (1989), 695-698. [5] H. J. Baues, Algebraic Homotopy, Cambridge Studies in Advanced Math. n. 15, Cambridge Univ. Press. 1989. [6] M.G. Brin and T.L. Thickstun, On the proper Steenrod homotopy groups, and proper embeddings of planes into 3-manifolds, Trans. AMS 289 (1985), 737-755. [7] E. M. Brown, Proper homotopy theory in simplicial complexes. Lecture Notes in Maths. vol. 375, pp. 41-46, Springer 1973. [8] Z. Cerin, On various relative proper homotopy groups, Tsukuba J. Math. 4 (1980), 177-202. [9] A. Dymov, On the behaviour at infinity of the fundamental group of a homologically trivial manifold, Izv. Nauk. SSSR 41 (1977), 529-550. [10] D. A. Edwards and H. M. Hastings, Cech and Steenrod homotopy theories with applications to Geometric Topology, Lecture Notes in Maths. vol. 542, Springer 1976. [11] F. T. Farrell, L. R. Taylor and J. B. Wagoner, The Whitehead Theorem in proper category, Compositio Math. 27 (1973), 1-23. [12] F. T. Farrell and J. B. Wagoner, Infinite matrices in Algebraic K-theory and Topology, Comm. Math. Helv. 47 (1972), 474-501. [13] Y. Felix, La dichotomie elliptique-hyperbolique en homotopie rationnelle, Ast\’erisque, 176, Soc. Math. de France, 1989. [14] L. J. Hern\’andez, Proper cohomologies and the proper classification problem, Lecture Notes in Maths. vol. 1298, pp. 171-191, Springer 1987. [15] L. J. Hem\’andez, About the extension problem for proper maps, Topology Appl. 25 (1987), 51-67. [16] L. J. Hern\’andez and T. Porter, Proper pointed maps from $R^{n+1}$ to a $\sigma$ -compact space, Math. Proc. Camb. Phil. Soc. 103 (1988), 457-462. [17] L. J. Hern\’andez and T. Porter, Global anlogues of the Brown-Grossman proper homotopy groups of an end, Math. Proc. Camb. Phil. Soc. 104 (1988), 483-496. [18] P. J. Hilton, Homotopy Theory and duality, Gordon and Breach 1965. [19] P.J. Hilton, An introduction to Homotopy Theory, Cambridge Univ. Press, 1966 [20] I.M. James, General Topology and Homotopy Theory, Springer 1984. Moore spaces $\ln$ proper homotopy 327 [21] S. Mardesic and J. Segal, Shape theory, North Holland 1982. [22] T. Porter, Homotopy groups in strong shape and proper homotopy, Suppl. Rendiconti Circ. Mat. Palermo. Serie II, 4 (1984), 101-111. [23] L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Thesis. Princeton University, 1965. [24] R. Switzer, Algebraic Topology-Homotopy and Homology, Springer 1975. [25] J.H. C. Whitehead, Combinatorial Homotopy, II, Bull. Amer. Math. Soc. 55 (1949), 453-496. [26] J.H. C. Whitehead, A certain exact sequence, Ann. Math. 52 (1950), 51-110. R. Ayala, A. M\’arquez, A. Quintero Facultad de Matem\’atlcas Dpto. Algebra, Computaci\’on, Geometria y Topologia Apdo 1160 41080-SEVILLA (SPAIN) E. Dom\’inguez Dpto $Ingenier^{\prime}1a$ El\’ectrica $e$ Inform\’atica Facultad de Ciencias Ciudad Universitaria 50009-ZARAGOZA (SPAIN)