The Dror-Whitehead Theorem in Pro-Homotopy and Shape Theories

Abstract

Many analogues of the classical Whitehead theorem from homotopy theory are now available in pro-homotopy and shape theories. E. Dror has significantly extended the homology version of the Whitehead theorem from the well-known simply connected case to the more general, for instance, nilpotent case. We prove a full analogue of Dror's theorems in pro-homotopy and shape theories. More specifically, suppose f: X -+ Y is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then f induces isomorphisms of the homotopy pro-groups, for instance, when X and Y are simple, nilpotent, complete, or H-objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories. 0. Introduction. The purpose of this paper is to establish an analogue of Dror's generalization of the Whitehead theorem (see [DR]), stated below as the DrorWhitehead theorem, in the context of the pro-homotopy and shape theories. We shall elaborate on these matters in the next few paragraphs. THE DROR-WHITEHEAD THEOREM. Suppose a map f: X -* Y induces isomorphisms of all the homology groups of spaces X and Y with integral coefficients. Then f induces isomorphisms of all the homotopy groups if and only if F.97Tf is an epimorphism, 1'j,Tf is a monomorphism, and rTf is a monomorphism. The functors r1',, r,, and P are defined by considering the action of the fundamental group on the homotopy groups; see [DR] or ?2 of this paper. An important class of spaces to which the Dror-Whitehead theorem applies is nilpotent spaces; see [DR], [(], [BK] or ?3 of this paper. Theorem (4.1.3) of this paper is our extension of the Dror-Whitehead theorem to pro-homotopy and shape theories; also, see Theorems (4.1.1), (4.1.2), (4.1.3), (4.5.1), and Corollary (4.3). Our Corollary (4.3) extends a theorem of Raussen [RA] in the same manner as Dror extends the Whitehead theorem. A parallel development of pivotal ingredients of pro-algebra is provided by [SIJ] which extends the work of Stallings [ST] and Dror [DR] concerning algebra; and our entire program is a natural extension of Dror's work. As a concluding remark, we may add that many analogues of the various versions of the classical Whitehead theorem have been studied in pro-homotopy Received by the editors September 3, 1980 and, in revised form, December 8, 1980. 1980 Mathemnatics Subject Classification. Primary 55P55, 55Q07; Secondary 54C56, 55N05.