# Torus action on quaternionic projective plane and related spaces

## TL;DR

This paper investigates torus actions of complexity one on quaternionic projective spaces and related manifolds, describing their orbit spaces and establishing topological equivalences with spheres, thus extending classical results in toric topology.

## Contribution

It provides explicit descriptions of orbit spaces for specific quaternionic and homogeneous spaces under torus actions, including new topological classifications and generalizations.

## Key findings

- alculated orbit spaces as spheres for quaternionic projective spaces
- Established homeomorphisms between orbit spaces and spheres
- Linked results to Kuiper--Massey theorem and its extensions

## Abstract

For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated 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 $\mathbb{H}P^2=Sp(3)/(Sp(2)\times Sp(1))$ and $S^6=G_2/SU(3)$. Here the maximal tori of the corresponding Lie groups $Sp(3)$ and $G_2$ act on the homogeneous spaces by the left multiplication. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of 8-dimensional manifolds with the action of $T^3$, generalizing $\mathbb{H}P^2$. We prove that their orbit spaces are homeomorphic to $S^5$ as well. We link this result to Kuiper--Massey theorem and some of its generalizations.