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

Free products of topological groups with a closed subgroup amalgamated

“…Topological amalgamated free products 415 | The proof of our theorem has similarities to Ordman's proof [10] that the free j k-group on a t 2 k-space exists and is a t 2 k-group, and more especially to the i proof of Brown and Hardy [1] that the universal topological groupoid on a k u -groupoid exists and is k w .…”

“…In case (a), G\% in fact lies \*W n _ x *W n _ x , so that r/;> = T n G; ; 4 = ^n GJ;^, which is closed in GJ ^ by the inductive assumption. In case (b), we claim that Tj^ = U o (o X t)~x(r n -i)> where t is the identity on From Proposition 4.25 of [7] (or Proposition A.I of [1]), we may immediately deduce the following result, using the fact that W is a k u -space.…”

“…A topological group is a kj-group if as a topological space it is k u . The appendix of [1] contains a useful list of the properties of k u -spaces.…”

“…[1], [7]) which uses the facts that p: W -» G and pXp: WxW^>GxG are both quotient maps of k u -spaces ( [1], [7]). It also follows routinely that G has the universal property required of it by the definition.…”

Homotopy theory, and change of base for groupoids and multiple groupoids

“…Since these vertex groups are isomorphic to subgroups of the original groupoid, classical subgroup theorems may be obtained by this method, in some cases giving stronger versions [39]. These methods also allow for topological versions of the main subgroup theorems [7]. Section 6 refers to other methods of proving the required lifting of universal morphisms without using the solution of the word problem.…”