On topological homotopy groups and relation to Hawaiian groups

By generalizing the whisker topology on the $n$th homotopy group of pointed space $insert ignore into journalissuearticles values(X, x_0);$, denoted by $\pi_n^{wh}insert ignore into journalissuearticles values(X, x_0);$, we show that $\pi_n^{wh}insert ignore into journalissuearticles values(X, x_0);$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $\pi_n^{wh}insert ignore into journalissuearticles values(X,x_0);$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_ninsert ignore into journalissuearticles values(X,x_0);$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_ninsert ignore into journalissuearticles values(X, x_0);$ is equal to the set of all classes of convergent sequences to the identity in $\pi_n^{wh}insert ignore into journalissuearticles values(X, x_0);$. As a consequence, we show that $L_ninsert ignore into journalissuearticles values(X, x_0); \cong L_ninsert ignore into journalissuearticles values(Y, y_0);$ if $\pi_n^{wh}insert ignore into journalissuearticles values(X, x_0); \cong \pi_n^{wh}insert ignore into journalissuearticles values(Y, y_0);$, but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally $n$-simply connected spaces and $n$-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of $n$-loop space coincide. Finally, we show that $n$-SLT paths can transfer $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.