On the topology of $\mathcal{M}_{0,n+1}/Σ_n$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911244624068608 |
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| author | Rossi, Tommaso |
| author_facet | Rossi, Tommaso |
| contents | This paper contains some results about the topology of $\M_{0,n+1}/Σ_n$, where $\M_{0,n+1}$ is the moduli space of genus zero Riemann surfaces with marked points. We show that $\M_{0,n+1}/Σ_n$ is not a topological manifold for $n\geq 4$, and it is simply connected for any $n\in\N$. We also present some homology computations: for example we show that $\M_{0,p+1}/Σ_p$ has no $p$ torsion, where $p$ is a prime. Lastly we compute $H_*(\M_{0,n+1}/Σ_n;\Z)$ for small values of $n$, proving that $\M_{0,n+1}/Σ_n$ is contractible for $n\leq 5$ while $\M_{0,7}/Σ_6$ is not. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the topology of $\mathcal{M}_{0,n+1}/Σ_n$ Rossi, Tommaso Algebraic Topology Algebraic Geometry This paper contains some results about the topology of $\M_{0,n+1}/Σ_n$, where $\M_{0,n+1}$ is the moduli space of genus zero Riemann surfaces with marked points. We show that $\M_{0,n+1}/Σ_n$ is not a topological manifold for $n\geq 4$, and it is simply connected for any $n\in\N$. We also present some homology computations: for example we show that $\M_{0,p+1}/Σ_p$ has no $p$ torsion, where $p$ is a prime. Lastly we compute $H_*(\M_{0,n+1}/Σ_n;\Z)$ for small values of $n$, proving that $\M_{0,n+1}/Σ_n$ is contractible for $n\leq 5$ while $\M_{0,7}/Σ_6$ is not. |
| title | On the topology of $\mathcal{M}_{0,n+1}/Σ_n$ |
| topic | Algebraic Topology Algebraic Geometry |
| url | https://arxiv.org/abs/2404.15494 |