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

Abstract

This paper contains some results about the topology of \(\mathcal {M}_{0,n+1}/\Sigma _n\), where \(\mathcal {M}_{0,n+1}\) is the moduli space of genus zero Riemann surfaces with marked points. We show that \(\mathcal {M}_{0,n+1}/\Sigma _n\) 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 \(\mathcal {M}_{0,p+1}/\Sigma _p\) has no p torsion, where p is a prime. Lastly we compute \(H_*(\mathcal {M}_{0,n+1}/\Sigma _n;\mathbb {Z})\) for small values of n, proving that \(\mathcal {M}_{0,n+1}/\Sigma _n\) is contractible for \(n\le 5\) while \(\mathcal {M}_{0,7}/\Sigma _6\) is not.