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We prove that the induced differentiable structure on $S^5$ is not the smooth one and describe the smooth and the singular points. We also consider the action of $T^4$ on $CP^5$ induced by the composition of the second symmetric power $T^4\\subset T^6$ and the standard action of $T^6$ on $CP^5$ and prove that the orbit space $CP^5/T^4$ is homeomorphic to the join $CP^2\\ast S^2$. 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