$ \mathbb{Z}_{2} $- homology of the orbit spaces $ G_{n,2}/ T^{n} $

TL;DR

This paper computes the $Z_2$-homology groups of the orbit spaces of complex Grassmannians under torus actions, providing explicit formulas for cases $X_5$ and $X_6$, with the latter being a novel result.

Contribution

It offers a new explicit description of the $Z_2$-homology of $X_n$ for $n=5,6$, extending previous results and utilizing stratification of moduli spaces.

Findings

Explicit $Z_2$-homology formulas for $X_5$ and $X_6$

Recovery of known results for $X_5$

First computation of $X_6$ homology groups

Abstract

We study the $\mathbb{Z}_2$-homology groups of the orbit space $X_n = G_{n,2}/T^n$ for the canonical action of the compact torus $T^n$ on a complex Grassmann manifold $G_{n,2}$. Our starting point is the model $(U_n, p_n)$ for $X_n$ constructed by Buchstaber and Terzi\'c (2020), where $U_n = \Delta _{n,2}\times \mathcal{F}_{n}$ for a hypersimplex $\Delta_{n,2}$ and an universal space of parameters $\mathcal{F}_{n}$ defined in Buchstaber and Terzi\'c (2019), (2020). It is proved by Buchstaber and Terzi\'c (2021) that $\mathcal{F}_{n}$ is diffeomorphic to the moduli space $\mathcal{M}_{0,n}$ of stable $n$-pointed genus zero curves. We exploit the results from Keel (1992) and Ceyhan (2009) on homology groups of $\mathcal{M}_{0,n}$ and express them in terms of the stratification of $\mathcal{F}_{n}$ which are incorporated in the model $(U_n, p_n)$. In the result we provide the description of cycles in $X_n$, inductively on $ n. $ We obtain as well explicit formulas for $\mathbb{Z}_2$-homology groups for $X_5$ and $X_6$. The results for $X_5$ recover by different method the results from Buchstaber and Terzi\'c (2021) and S\"uss (2020). The results for $X_6$ we consider to be new.