On the topology of ${\mathcal{M}}_{0,n+1}\mathrm{/}{\mathrm{\Sigma }}_n$

This paper contains some results about the topology of ${\text{M}}_{0,n+1}\mathrm{/}{\mathrm{\Sigma }}_n$, where ${\text{M}}_{0,n+1}$ is the moduli space of genus zero Riemann surfaces with marked points. We show that ${\text{M}}_{0,n+1}\mathrm{/}{\mathrm{\Sigma }}_n$ is not a topological manifold for $n\ge 4$, and it is simply connected for any $n\in \mathbb{N}$. We also present some homology computations: for example we show that ${\text{M}}_{0,p+1}\mathrm{/}{\mathrm{\Sigma }}_p$ has no $p$ torsion, where $p$ is a prime. Lastly we compute $H_{\ast }({\text{M}}_{0,n+1}\mathrm{/}{\mathrm{\Sigma }}_n;\mathbb{Z})$ for small values of $n$, proving that ${\text{M}}_{0,n+1}\mathrm{/}{\mathrm{\Sigma }}_n$ is contractible for $n\le 5$ while ${\text{M}}_{0,7}\mathrm{/}{\mathrm{\Sigma }}_6$ is not.