$\mathbb Z_2$-Homology of the Orbit Spaces $G_{n,2}/T^n$

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 Terzić (2022), where $U_n = \Delta _{n,2}\times \mathcal F_n$ for a hypersimplex $\Delta _{n,2}$ and a universal space of parameters $\mathcal F_n$ defined in the works of Buchstaber and Terzić (2019, 2022). It was proved by Buchstaber and Terzić (2023) 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 of Keel (1992) and Ceyhan (2009) on the homology groups of $\mathcal M_{0,n}$ and express them in terms of the stratification of $\mathcal F_n$ incorporated in the model $(U_n,p_n)$. As a result we provide an inductive, with respect to $n$, description of cycles in $X_n$. We also obtain explicit formulas for the $\mathbb Z_2$-homology groups of $X_5$ and $X_6$. The results for $X_5$ recover by a different method the results of Buchstaber and Terzić (2023) and Süss (2020). The results for $X_6$ seem to be new.