Are there any kinds of self maps on the loop structure whose induced homomorphic images are the Lie brackets in tensor algebra? We will give an answer to this question by defining a self map of $\Omega \Sigma K(\mathbb{Z},2d)$, and then by computing efficiently some self maps. We also study the topological rationalization properties of the suspension of the Eilenberg-MacLane spaces. These results will be playing a powerful role in the computation of the same $n$-type problems and giving us an information about the rational homotopy...

For the n-dimensional Hawaiian earring ${ℍ}_n,$ n ≥ 2, ${\pi }_n({ℍ}_n,o)\simeq {\mathbb{Z}}^{\omega }$ and ${\pi }_i({ℍ}_n,o)$ is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CX ∨ CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then $H_n\left(X\vee Y\right)\simeq H_n\left(X\right)\oplus H_n\left(Y\right)\oplus H_n\left(CX\vee CY\right)$ for n ≥ 1.