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

TL;DR

This paper investigates the topological properties of the moduli space of genus zero Riemann surfaces with marked points modulo symmetric group action, revealing non-manifold structure for larger n, simple connectivity, and specific homology characteristics.

Contribution

It provides new topological and homological insights into the space _{0,n+1}/_n, including non-manifold behavior, simple connectivity, and explicit homology computations.

Findings

_{0,n+1}/_n is not a topological manifold for n.

It is simply connected for all n.

_{0,p+1}/_p has no p-torsion in homology.

Abstract

This paper contains some results about the topology of $\M_{0,n+1}/\Sigma_n$, where $\M_{0,n+1}$ is the moduli space of genus zero Riemann surfaces with marked points. We show that $\M_{0,n+1}/\Sigma_n$ 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 $\M_{0,p+1}/\Sigma_p$ has no $p$ torsion, where $p$ is a prime. Lastly we compute $H_*(\M_{0,n+1}/\Sigma_n;\Z)$ for small values of $n$, proving that $\M_{0,n+1}/\Sigma_n$ is contractible for $n\leq 5$ while $\M_{0,7}/\Sigma_6$ is not.