Topology and geometry of the canonical action of $T^4$ on the complex Grassmannian $G_{4,2}$ and the complex projective space $\mathbb CP^5$

We consider the canonical action of the compact torus $T^4$ on the complex Grassmann manifold $G_{4,2}$ and prove that the orbit space $G_{4,2}/T^4$ is homeomorphic to the sphere $S^5$. We prove that the induced map from $G_{4,2}$ to the sphere $S^5$ is not smooth and describe its smooth and singular points. We also consider the action of $T^4$ on $\mathbb CP^5$ induced by the composition of the second symmetric power representation of $T^4$ in $T^6$ and the standard action of $T^6$ on $\mathbb CP^5$ and prove that the orbit space $\mathbb CP^5/T^4$ is homeomorphic to the join $\mathbb CP^2\ast S^2$. The Plücker embedding $G_{4,2}\subset\mathbb CP^5$ is equivariant for these actions and induces the embedding $\mathbb CP^1\ast S^2\subset\mathbb CP^2\ast S^2$ for the standard embedding $\mathbb CP^1\subset\mathbb CP^2$.
All our constructions are compatible with the involution given by the complex conjugation and give the corresponding results for the real Grassmannian $G_{4,2}(\mathbb R)$ and the real projective space $\mathbb RP^5$ for the action of the group $\mathbb Z_2^4$. We prove that the orbit space $G_{4,2}(\mathbb R)/\mathbb Z_2^4$ is homeomorphic to the sphere $S^4$ and that the orbit space $\mathbb RP^5/\mathbb Z_2^4$ is homeomorphic to the join $\mathbb RP^2\ast S^2$.