$ \mathbb{Z}_{2} $- homology of the orbit spaces $ G_{n,2}/ T^{n} $
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| Format: | Preprint |
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2024
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| _version_ | 1866917696170360832 |
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| author | Ivanović, Vladimir Terzić, Svjetlana |
| author_facet | Ivanović, Vladimir Terzić, Svjetlana |
| contents | 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 $U_n = Δ_{n,2}\times \mathcal{F}_{n}$ for a hypersimplex $Δ_{n,2}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_11625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $ \mathbb{Z}_{2} $- homology of the orbit spaces $ G_{n,2}/ T^{n} $ Ivanović, Vladimir Terzić, Svjetlana Algebraic Geometry Algebraic Topology 57S25, 57N65, 53D20, 14M25, 52B11, 14B05 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 $U_n = Δ_{n,2}\times \mathcal{F}_{n}$ for a hypersimplex $Δ_{n,2}$ 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. |
| title | $ \mathbb{Z}_{2} $- homology of the orbit spaces $ G_{n,2}/ T^{n} $ |
| topic | Algebraic Geometry Algebraic Topology 57S25, 57N65, 53D20, 14M25, 52B11, 14B05 |
| url | https://arxiv.org/abs/2406.11625 |