The projective dimension of a compact abelian group

Abstract

Abstract.Let A" be a compact group, FGX the (Graev) free topological group generated by X, and K the kernel of the canonical quotient morphism from FGX to X. Then A" is a (Graev) free topological group. A corollary to the abelian analogue of this theorem is that the projective dimension of a compact abelian group, relative to the class of all continuous epimorphisms admitting sections, is exactly one. Introduction.Let 'S (respectively, s/) denote the category of topological (abelian) groups and continuous group homomorphisms. We take as known the definitions and elementary properties of topological groups as presented in, for example, [2, pp. 219-250]. We adopt the convention that all groups are topological, using the modifier "algebraic" for those groups without topologies. Moreover, all our topologies satisfy the Hausdorff separation axiom.We assume also some familiarity with the notions of free (topological) groups due to A. A. Markov [8] and M. I. Graev [5]. Given a completely regular space X, we denote by FX (ZX) the Markov free (free abelian) group over X, and by FGX (ZGX) the Graev free (free abelian) group over X. Recall that if A' is a compact space, then FGX is the colimit of the expanding sequence {Fm} of compact spaces, where Fm is the set of all words in FGX whose reduced length relative to X\{e} is less than or equal to m (e is the identity of FGX). An analogous result holds for FX, ZGX, and ZX.Recall that a group P in 'S (s/) is projective relative to a class S of epimorphisms in 'S (stf) if, given any diagram P I

Free Topological Groups and the Projective Dimension of a Locally Compact Abelian Group

“…This result follows immediately from Corollary 3 and the fact (Proposition 2.1 of [11]) that no locally compact abelian group is projective unless it is a discrete free abelian group.…”

“…Apply Theorem 3 with Xn = X and Yn= Y for all n. We now generalize Proposition 1.6 of [11]. Theorem 4.…”

“…In [11] Nummela introduced a concept of projective dimension for abelian topological groups. For our purposes it suffices to note that an abelian topological group is of projective dimension one if and only if it is not projective and it is a quotient group of a free abelian topological group by a free abelian topological group.…”

“…It was observed [6] that Graev's proof worked not only for free topological groups but also for groups with the maximum topology relative to a given compact set, and several applications to free products of topological groups were obtained [8], [9], [11], [12].…”

“…Recently Nummela [11] showed that the projective dimension of a compact abelian group, relative to the class of epimorphisms admitting sections, is exactly one. This was done by proving that a compact abelian group is a quotient group of a free abelian topological group by a free abelian topological group.…”