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

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 $C{P}^5$ induced by the composition of the second symmetric power $T^4\subset T^6$ and the standard action of $T^6$ on $C{P}^5$ and prove that the orbit space $C{P}^5/T^4$ is homeomorphic to the join $C{P}^2\ast S^2$. The Plücker embedding $G_{4,2}\subset C{P}^5$ is equivariant for these actions and induces embedding $C{P}^1\ast S^2\subset C{P}^2\ast S^2$ for the standard embedding $C{P}^1\subset C{P}^2$. 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 $R{P}^5$ 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 $R{P}^5/Z_2^4$ is homeomorphic to the join $R{P}^2\ast S^2$.

Some typos corrected and few background information added, now 42 pages, to appear in Moscow Mathematical Journal

Cited by in corpus (8)

• Topology of complexity one quotients
• Universal spaces of parameters for complex Grassmann manifolds $G_{q+1,2}$
• Torus actions of complexity one in non-general position
• Real soliton lattices of KP-II and desingularization of spectral curves: the Gr^{TP}(2,4) case
• Orbit spaces of torus actions on Hessenberg varieties
• Toric topology of the Grassmannian of planes in ${\mathbb{C}}^5$ and the del Pezzo surface of degree $5$
• Geometry of central extensions of nilpotent Lie algebras
• The smooth torus orbit closures in the Grassmannians