\(\mathbb Z_2\)-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.