Free products of topological groups which are 𝑘_{𝜔}-spaces

Abstract

Let G and H be topological groups and G ∗ H G \ast H their free product topologized in the manner due to Graev. The topological space G ∗ H G \ast H is studied, largely by means of its compact subsets. It is established that if G and H are k ω {k_\omega } -spaces (respectively: countable CW-complexes) then so is G ∗ H G \ast H . These results extend to countably infinite free products. If G and H are k ω {k_\omega } -spaces, G ∗ H 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 π ( G ∗ H ) \pi (G \ast H) . If G 1 {G_1} and H 1 {H_1} are quotients (continuous open homomorphic images) of G and H, then G 1 ∗ H 1 {G_1} \ast {H_1} is a quotient of G ∗ H G \ast H .