The Complex of a Group Relative to a Set of Generators: Part II

Abstract

In order to describe the contents of the present paper we begin with some definitions. An extension of a local group Y is a pair (E, 4) where E is a local group and-? is a strong homomorphism of E onto X. Let (E, 4) be an extension of Y. It follows from the definition of strong homomorphism (?31) that the set N = 0_1 is a subgroup of E; N will naturally be called the kernel of (E, 4). A function u on Y to E such that qu(y) = y is a selector of (E, 4). A selector u of (E, 0) is symmetric if u(y'1) = u(y)V'. A selector u(y) is multiplicative if u(yiy2) = u(yI)u(y2) whenever YiY2 is defined. (Since the homomorphism 4 is strong, u(y1)u(y2) is defined whenever YiY2 is defined.) The extension (E, 0) of Y is inessential if it admits a multiplicative selector. Let Y be a local subgroup of a group Q generated by Y. We call an extension (E, 0) of Y extendible over Q if E can be imbedded in a group which is mapped onto Q by a homomorphism whose contraction to E is 0. Assume now (and during the remainder of the introduction) that Q is simply connected relative to Y (?34) and that Y contains no elements of order 2. Consider the abelian group P2(Y) defined in ??13, 34. Although it is determined by Y alone, P2(Y) is isomorphic to the second homotopy group of the complex K(Q, Y) (?35). The main theorem in the present paper is the following: in order that every extension of Y be extendible over Q it is necessary and sufficient that P2(Y) = { 1}. We leave open the question of the relationship between extendibility and the structure of P2(Y) when the latter is non-trivial. Consider the natural homomorphism aQY:G(Y) -4 Q defined in ?34. (G(Y) is a free group since Y has no elements of order 2.) Let C = C(Q, Y) be the preimage of X under aQY ; C is a local subgroup of G(Y). The problem of determining whether an extension of Y is extendible over Q is closely related to the problem of determining whether a given extension of C is inessential. We shall show that every extension of C is inessential if p2(Y) = {1}. Since the proof gives a somewhat misleading appearance of complexity, it will be well to outline the steps. It will be shown that the local subgroup C of G(Y) contains the canonical generators of G(Y) (?32) as well as the group R of defining relations of Q (?34).