Moore spaces in proper homotopy
Tsukuba Journal of Mathematics
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  . 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
... q 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  , 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).