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.

Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.