RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY PRO-GROUPS OF CONTINUA

All basic notions of shape theory and pro-categories used in this paper can be found in [D-S]. In the recent paper [F] S. Ferry has characterized continua having the shape of an LCn-space (n ~ 0, the case n = 0 is due to J Krasinkiewicz [K]) as those possessing2 the following property: (*) for some point x E X the k-th homotopy pro-group pro-TIk(X,x) is stable for k < n and satisfies the Mittag-Leffler condition for k = n + 1. Also the author proved in [0] that a continuum X is3 an LCn+l-divisor (n ~ 0) iff X is nearly I-movable and satisfies the following condition: (**) the k-th homology pro-group pro-Hk(X) is stable for k < n and satisfies the Mittag-Leffler condition for k = n + 1. The interplay between conditions (*) and (**) was investigated in [D2]. Here is the main result of [D2]. Theorem 1. Let X = (X,pm+l) be an inverse sequenae- m m of pointed aonneated CW aompZexes and Zet n> o. If pro-TIk(~) is stabZe for k < n and satisfies the Mittag Leffler aondition for k n + 1, then pro-Hk(~) is stable for k < n and satisfies the Mittag-LeffZer aondition fo