The A-infinity operad and the moduli space of curves

TL;DR

This paper demonstrates that the modular envelope of the A-infinity cyclic operad corresponds to the moduli space of Riemann surfaces with boundary, providing a new proof of its homotopy equivalence with a ribbon graph cell complex.

Contribution

It establishes that the modular operad constructed from moduli spaces is the modular envelope of the A-infinity cyclic operad, linking operad theory with moduli space topology.

Findings

The modular operad from moduli spaces is the modular envelope of the A-infinity cyclic operad.

Provides a new proof of the homotopy equivalence between ribbon graph complexes and moduli spaces.

Connects operad structures with the topology of Riemann surface moduli spaces.

Abstract

The modular envelope of a cyclic operad is the smallest modular operad containing it. A modular operad is constructed from moduli spaces of Riemann surfaces with boundary; this modular operad is shown to be the modular envelope of the A-infinity cyclic operad. This gives a new proof of the result of Harer-Mumford-Thurston-Penner-Kontsevich that a cell complex built from ribbon graphs is homotopy equivalent to the moduli space of curves.