Independence of homogeneous GKM manifolds and symmetric spaces

TL;DR

This paper investigates the independence properties of torus actions on homogeneous spaces, revealing connections to symmetric spaces and implications for the topology of orbit spaces.

Contribution

It introduces the concept of $j$-independence for torus actions on homogeneous spaces and classifies maximal independence cases, linking them to symmetric spaces of higher rank.

Findings

Maximal independence is 2, 3, or the dimension of the torus.

Cases of 3 or maximal dimension correspond to certain symmetric spaces.

Lower-degree homology groups of the orbit space vanish.

Abstract

Let $G/H$ be a simply connected homogeneous space of maximal rank. Then the maximal torus $T$-action on $G/H$ is a GKM manifold. We call the $T$-action $j$-independent if any $i(\leq j)$ pairwise distinct isotropy weights at a fixed point are linearly independent. Using weighted graphs, we show that the maximal independence of $G/H$ is $2$, $3$ or $n=\dim T$, and that the cases of $3$ or $n=\dim T$ correspond to some symmetric spaces of rank $>2$. As a corollary, using the results of Ayzenberg and Masuda, the lower-degree reduced homology groups (with appropriate coefficients) of the orbit space $T\backslash G/H$ vanish.