Torus Action on Quaternionic Projective Plane and Related Spaces

Abstract

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number $$\frac{1}{2}\dim X-\dim T$$ is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $${\mathbb {H}}P^2/T^3\cong S^5$$ and $$S^6/T^2\cong S^4$$ , for the homogeneous spaces $${\mathbb {H}}P^2={{\,\mathrm{Sp}\,}}(3)/({{\,\mathrm{Sp}\,}}(2)\times {{\,\mathrm{Sp}\,}}(1))$$ and $$S^6=G_2/{{\,\mathrm{SU}\,}}(3)$$ . Here, the maximal tori of the corresponding Lie groups $${{\,\mathrm{Sp}\,}}(3)$$ and $$G_2$$ act on the homogeneous spaces from the left. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of eight-dimensional manifolds with the action of $$T^3$$ . This class generalizes $${\mathbb {H}}P^2$$ . We prove that their orbit spaces are homeomorphic to $$S^5$$ as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold.