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

Abstract

We consider the canonical action of the compact torus $T^4$ on the 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 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 $CP^2\ast S^2$. The Pl\"ucker embedding $G_{4,2}\subset CP^5$ is equivariant for these actions and induces embedding $CP^1\ast S^2 \subset CP^2 \ast S^2$ for the standard embedding $CP^1 \subset CP^2$. All our constructions are compatible with the involution given by the complex conjugation and give the corresponding results for the real Grassmannian $G_{4,2}(R)$ and the real projective space $RP^5$ for the action of the group $Z _{2}^{4}$. We prove that the orbit space $G_{4,2}(R)/Z _{2}^{4}$ is homeomorphic to the sphere $S^4$ and that the orbit space $RP^{5}/Z _{2}^{4}$ is homeomorphic to the join $RP^2\ast S^2$.