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We consider the following topological spaces: $\mathbf{O}=\{z\in ℂ:|z+i|=1\}$, ${\mathbf{O}}_3=\mathbf{O}\cup \{z\in ℂ:z^4\in [0,1],Imz\ge 0\}$, ${\mathbf{O}}_4=\mathbf{O}\cup \{z\in ℂ:z^4\in [0,1]\}$, ${\infty }_1=\mathbf{O}:|z-i|=1\}\cup \{z\in ℂ:z\in [0,1]\}$, ${\infty }_2={\infty }_1\cup \{z\in ℂ:z^2\in [0,1]\}$, et $\mathbf{T}=\{z\in ℂ:z=\cos (3\theta )e^{i\theta },0\le \theta \le 2\pi \}$. Set $E\in \{{\mathbf{O}}_3,{\mathbf{O}}_4,{\infty }_1,{\infty }_2,\mathbf{T}\}$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $Per\left(f\right)$ the set of periods of all periodic points of $f$. The set $K\subset \mathbb{N}$ is the of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset Per\left(f\right)$, then $Per\left(f\right)=\mathbb{N}$. (2) If $S\subset \mathbb{N}$ is a set such that for every $E$ map $f$, $S\subset Per\left(f\right)$ implies $Per\left(f\right)=\mathbb{N}$, then $K\subset S$. In this paper we compute the full periodicity kernel of ${\mathbf{O}}_3,{\mathbf{O}}_4,{\infty }_1,{\infty }_2$ and $\mathbf{T}$.