[Paper Review] Real moduli space of stable rational curves revisted

Anton Khoroshkin, Thomas Willwacher|arXiv (Cornell University)|May 11, 2019

Homotopy and Cohomology in Algebraic Topology35 references4 citations

TL;DR

This paper provides a homotopy-theoretic model of the real moduli space of stable rational curves, $¯{\mathcal{M}}_{0,n+1}(\mathbb{R})$, using a quotient of an associative operad, revealing that the rational cohomology ring is Koszul and that the space is a rational $K(\pi,1)$ but not formal. It establishes the non-formality of the operad and describes the associated Lie algebras of the pure Cacti groups.

ABSTRACT

We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ in terms of a homotopy quotient of an operad of associative algebras. We use this model to find different Hopf models of the algebraic operad of Chains and homologies of $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$. In particular, we show that the operad $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal. The manifolds $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ are known to be Eilenberg-MacLane spaces for the so called pure Cacti groups. As an application of the operadic constructions we prove that for each $n$ the cohomology ring $H(\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}),{\mathbb{Q}})$ is a Koszul algebra and that the manifold $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal but is a rational $K(\pi,1)$ space. We give the description of the Lie algebras associated with the lower central series filtration of the pure Cacti groups.

Motivation & Objective

• To describe the real locus of the moduli space $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ as a homotopy quotient of an associative operad.
• To construct multiple Hopf models for the chain and homology operads of $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$.
• To prove that the operad $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ is not formal.
• To establish that $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ is a rational $K(\pi,1)$ space and that its rational cohomology ring is Koszul.
• To describe the Lie algebras associated with the lower central series of pure Cacti groups.

Proposed method

• Modeling $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ as a homotopy quotient of an operad of associative algebras.
• Using this model to construct explicit Hopf operad structures on chains and homology of the moduli space.
• Applying operadic techniques to detect non-formality via cohomological obstructions.
• Leveraging the known Eilenberg-MacLane property of $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ for pure Cacti groups to analyze rational homotopy type.
• Analyzing the cohomology ring $H^*(\overline{\mathcal{M}}_{0,n+1}(\mathbb{R}), \mathbb{Q})$ to prove it is a Koszul algebra.
• Computing the associated graded Lie algebras of the lower central series of pure Cacti groups using the operadic framework.

Experimental results

Research questions

• RQ1 Is the operad $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ formal over $\mathbb{Q}$, and if not, what obstructions arise?
• RQ2 Can the real moduli space $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ be modeled as a homotopy quotient of an associative operad?
• RQ3 Is the rational cohomology ring of $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ a Koszul algebra?
• RQ4 Does $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ admit the structure of a rational $K(\pi,1)$ space despite non-formality?
• RQ5 What is the structure of the Lie algebras associated with the lower central series of pure Cacti groups?

Key findings

• The operad $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ is not formal over $\mathbb{Q}$, as shown by cohomological obstructions in the operadic structure.
• The rational cohomology ring $H^*(\overline{\mathcal{M}}_{0,n+1}(\mathbb{R}), \mathbb{Q})$ is a Koszul algebra.
• The manifold $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$ is a rational $K(\pi,1)$ space, meaning its rational homotopy type is determined by its fundamental group.
• The pure Cacti groups are realized as the fundamental groups of $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$, and their associated graded Lie algebras are described via the operadic model.
• Multiple Hopf models are constructed for the chain and homology operads of $\overline{\mathcal{M}}_{0,n+1}(\mathbb{R})$, providing new algebraic tools for studying its topology.
• The homotopy quotient model of the real moduli space allows for a systematic study of its rational homotopy theory and operadic structure.