Subgroups of Free Topological Groups and Free Topological Products of Topological Groups

Introduction Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup theorem when a continuous Schreier transversal exists, and we give such a result in the category of Hausdorff /<Vgroups (Theorem 8). In the same category, we give an open subgroup version of the Kurosh Theorem (Theorem 13). The method of proof in both cases is a topological version of the groupoid method given by Higgins in [8]—that is, we use topological groupoids. The key steps are first of all to construct universal morphisms of topological groupoids, and secondly to prove that the pull-back by a covering morphism of a universal morphism is again universal. For the second step it is essential to know that the pullback of quotient maps of topological groupoids is again a quotient, and to obtain this we work in the category of Hausdorff /^-spaces. The chief technical work is then in constructing universal morphisms in this category—the results on /r^-spaces needed are Propositions A1-A3 which are given in the Appendix. This restriction to ^-spaces means that our theorems specialise only to countable versions of the abstract theorems. More general results can be given by using kgroupoids, but the proofs involve extra technicalities, and so we refer the interested reader to [7]. The authors wish to thank P. J. Higgins and P. Nickolas for helpful comments.