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

TL;DR

This work investigates the topology of the moduli quotient $\mathcal{M}_{0,n+1}/\Sigma_n$, showing it is not a topological manifold for $n\ge 4$ while remaining simply connected for all $n$. It introduces a cactus-based combinatorial model that yields a CW-decomposition and establishes a homotopy equivalence to the configuration space quotient $C_n(\mathbb{C})/S^1$, providing a concrete computational framework. The authors develop a torsion-focused program, deriving bounds on torsion, leveraging equivariant cohomology and Mayer–Vietoris sequences to compute $H_*(\mathcal{M}_{0,n+1}/\Sigma_n;\mathbb{F}_p)$ in many cases and determining exact integral homology for small $n$ (contractible for $n\le 5$ and nontrivial at $n=6$). The methods connect moduli-space topology with configuration-space techniques, offering detailed torsion analyses and explicit small-$n$ computations with potential implications for orbifold topology and related moduli problems.

Abstract

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.

Figures (8)

• Figure 1: On the left there is an element $x\in\mathcal{C}_4$, on the right its associated cactus $c(x)$. The basepoint of the circle $S^1$ is depicted in red and corresponds to a basepoint on the cactus $c(x)$ (which we depict as a red spine).
• Figure 2: On the left there are some cacti (without basepoint), on the right the corresponding dual graphs.
• Figure 3: On top where is a full description of $\mathcal{C}_2\cong S^1$: there are two zero cells $(2,1)$ (on the left) and $(1,2)$ (on the right). The $1$-cells are $(2,1,2)$ (on the top) and $(1,2,1)$ (on the bottom). Below we see the cell $(2,3,2,1,2)\cong \Delta^0\times\Delta^2\times\Delta^0$ of $\mathcal{C}_3$ and the codimension one cells in its boundary.
• Figure 4: This picture shows the CW-complex $\mathcal{C}_3/S^1\simeq \mathcal{M}_{0,4}$. There are two zero cells and three edges.
• Figure 5: On the left there are some cells of $\mathcal{C}_4/S^1$. On the right is represented the stabilizer of the cell respect to the $\Sigma_4$ action by relabelling the lobes. In the first row we see a $0$-dimensional cell, whose stabilizer is the cyclic group of order four generated by $(1234)\in\Sigma_4$. In the second row there is a $1$-cell, which has trivial stabilizer. The last two cells are two dimensional with stabilizer respectively a cyclic group of order two and three.

Theorems & Definitions (56)

• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Definition
• Definition
• Definition
• Proposition 2.5