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.