Cyclic operad formality for compactified moduli spaces of genus zero surfaces

TL;DR

This paper proves the formality of a cyclic operad related to moduli spaces of genus zero curves, establishing a quasi-isomorphism between its chains and homology, with implications for algebraic topology.

Contribution

It introduces a new graph complex with a differential combining edge deletion and contraction to resolve the BV operad as a cyclic operad.

Findings

The cyclic operad of moduli spaces is formal.

A new graph complex resolves BV as a cyclic operad.

Chains and homology are quasi-isomorphic cyclic operads.

Abstract

The framed little 2-discs operad is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes. We show that this cyclic operad is formal, meaning that its chains and its homology (the Batalin-Vilkovisky operad) are quasi-isomorphic cyclic operads. To prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction, and we show that this complex resolves BV as a cyclic operad.