Some Nonprojective Subgroups of Free Topological Groups

For the free topological group on an interval [a, b] a family of closed, locally path-connected subgroups is given such that each group is not projective and so not free topological. Simplicial methods are used, and the test for nonprojectivity is nonfreeness of the group of path components. Similar results are given for the abelian case. Introduction. Let F(X) be the free topological group on a pointed space X [4]. The object of this note is to give a family of closed subgroups of F([a, &]) which are not projective and so not free topological. The only previous example of a nonfree closed subgroup of an F(X) has been given by Graev [4], but this was written down explicitly only in the abelian case, and I do not know if Graev's example is projective. The structure of this paper is as follows. § 1 examines the topological group ttJH of components of a topological group, and proves the key result that if B is projective with open components then nQB is a discrete, free group. The methods are simple and categorical; similar results are given for the group 77„B of path-components, and for the abelian case. In §2 the examples of nonprojective subgroups of free topological groups are given as realisations of simplicial subgroups of free simplicial groups. An Appendix summarises the definitions and results from simplicial theory which are used in §2. I would like to thank Dr. S. A. Morris for pointing out the problem from which this work arose, and both he and a referee for helpful comments. 1. The group of components of topological groups. For any topological space X, let nQX denote its set of components with its topology as a quotient of X. Then rrQX is totally disconnected [2, p. 125, Example 2]. Also X has open components if and only if ttqX is discrete—in particular this holds if X is locally connected. The object of this section is to describe 77 Q tot a colimit of topological groups, for free topological groups, and for projective topological groups. Let J be the category of topological spaces and continuous mappings, Received by the editors February 4, 1974 and, in revised form, July 17, 1974. AMS (MOS) subject classifications (1970). Primary 22A05; Secondary 18G05, 55J10.