On the topology of M0,n+1/Σₙ and M̄0,n+1/Σₙ

This paper contains some results about the topology of M0,n+1/Σₙ and M̄0,n+1/Σₙ, where M0,n+1 is the moduli space of genus zero Riemann surfaces with marked points and M̄0,n+1 is its Deligne-Mumford compactification. We show that M0,n+1/Σₙ and M̄0,n+1/Σₙ are not topological manifolds for n≥ 4, and they are simply connected for any n. We also present some homology computations: for example we show that the homology of M0,n+1/Σₙ is all torsion and that M0,p+1/Σₚ has no p torsion, where p is a prime. Lastly we compute H_*(M0,n+1/Σₙ;Z) for small values of n, proving that M0,n+1/Σₙ is contractible for n≤ 5 while M0,7/Σ₆ is not.