SωSω and S2S2 on free topological groups

In this paper, we investigate copies of SωSω and S2S2 on free topological groups. By applying these results, we show that, for a paracompact space with a point-countable k-network, X   is discrete or compact if F5(X)F5(X) is Fréchet–Urysohn, which generalizes Yamada's theorem (Yamada [26]). We also give a negative answer to Yamada's conjecture (Yamada [26]): If X   is a metrizable space, then F4(X)F4(X) is Fréchet–Urysohn if and only if the set of all non-isolated points of X   is compact; and a partial answer to Arhangel'skii's conjecture: Sω1Sω1 cannot be embedded into a sequential topological group. Finally, we prove that, for a k⁎k⁎-metrizable μ-space X  , the free topological group F(X)F(X) is a k  -space if and only if F5(X)F5(X) is a k-space. Some questions about free topological groups are posed.