${\mathbb{Z}}_2$- homology of the orbit spaces $G_{n,2}/T^n$

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 Terzić (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 Terzić (2019), (2020). It is proved by Buchstaber and Terzić (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 Terzić (2021) and Süss (2020). The results for $X_6$ we consider to be new.

38 pages