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## Torus Action on Quaternionic Projective Plane and Related Spaces

For an effective action of a compact torus *T* on a smooth compact manifold *X* with nonempty finite set of fixed points, the number 12dimX−dimT12dimX−dimT 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 HP2/T3≅S5HP2/T3≅S5 and S6/T2≅S4S6/T2≅S4, for the homogeneous spaces HP2=Sp(3)/(Sp(2)×Sp(1))HP2=Sp(3)/(Sp(2)×Sp(1)) and S6=G2/SU(3)S6=G2/SU(3). Here, the maximal tori of the corresponding Lie groups Sp(3)Sp(3) and G2G2 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 T3T3. This class generalizes HP2HP2. We prove that their orbit spaces are homeomorphic to S5S5 as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold.