Equivariant formality of corank-one isotropy actions and products of rational spheres

TL;DR

This paper characterizes when certain Lie group actions are equivariantly formal, extending classifications of homogeneous spaces with rational homotopy types resembling products of spheres, which has implications in geometric topology.

Contribution

It provides a complete characterization of rank-one isotropy actions that are equivariantly formal and extends the classification of homogeneous quotients with specific rational homotopy types.

Findings

Characterization of pairs (G,K) with rank difference 1 and equivariantly formal actions

Correction and extension of existing classifications of homogeneous quotients

Identification of conditions for rational homotopy types of products of spheres

Abstract

We completely characterize the pairs of connected Lie groups $G > K$ such that $\mathrm{rank}(G) - \mathrm{rank}(K) = 1$ and the left action of $K$ on $G/K$ is equivariantly formal. The analysis requires us to correct and extend an existing partial classification of homogeneous quotients $G/K$ with the rational homotopy type of a product of an odd- and an even-dimensional sphere.