# Homotopy invariants for $\overline{\mathcal{M}}_{0,n}$ via Koszul duality

## TL;DR

This paper proves that the integral cohomology rings of moduli spaces of stable rational marked curves are Koszul, computes their rational homotopy Lie algebras, and explores implications for Betti numbers of free loop spaces.

## Contribution

It establishes the Koszul property for these cohomology rings, answering a longstanding open question, and extends the understanding of their rational homotopy structure.

## Key findings

- Integral cohomology rings are Koszul.
- Computed rational homotopy Lie algebras.
- Provided estimates for Betti numbers of free loop spaces.

## Abstract

We show that the integral cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by Berglund, we compute the rational homotopy Lie algebras of those spaces, and obtain some estimates for Betti numbers of their free loop spaces in case of torsion coefficients. We also prove and conjecture some generalisations of our main result.