When can a formality quasi-isomorphism over rationals be constructed recursively?

TL;DR

This paper demonstrates that for a broad class of differential graded operads over rationals, a formality quasi-isomorphism can be constructed recursively without explicit knowledge of the connecting zig-zag of quasi-isomorphisms, simplifying the process.

Contribution

It introduces a recursive method to construct formality quasi-isomorphisms for dg operads over rationals, avoiding the need for explicit zig-zag details.

Findings

Recursive construction involves solving finite dimensional linear systems.

Method applies to a large class of dg operads over rationals.

Construction does not require explicit zig-zag knowledge.

Abstract

Let $O$ be a differential graded (possibly colored) operad defined over rationals. Let us assume that there exists a zig-zag of quasi-isomorphisms connecting $O \otimes K$ to its cohomology, where $K$ is any field extension of rationals. We show that for a large class of such dg operads, a formality quasi-isomorphism for $O$ exists and can be constructed recursively. Every step of our recursive procedure involves a solution of a finite dimensional linear system and it requires no explicit knowledge about the zig-zag of quasi-isomorphisms connecting $O \otimes K$ to its cohomology.