Free products of topological groups which are $k_{\omega }$-spaces

Abstract:

Let G and H be topological groups and $G \ast H$ their free product topologized in the manner due to Graev. The topological space $G \ast H$ is studied, largely by means of its compact subsets. It is established that if G and H are ${k_\omega }$-spaces (respectively: countable CW-complexes) then so is $G \ast H$. These results extend to countably infinite free products. If G and H are ${k_\omega }$-spaces, $G \ast H$ is neither locally compact nor metrizable, provided G is nondiscrete and H is nontrivial. Incomplete results are obtained about the fundamental group $\pi (G \ast H)$. If ${G_1}$ and ${H_1}$ are quotients (continuous open homomorphic images) of G and H, then ${G_1} \ast {H_1}$ is a quotient of $G \ast H$.