Pre-Lie Pairs and Triviality of the Lie Bracket on the Twisted Hairy Graph Complexes

Abstract

We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the little disks operads. We show that twisting with certain Maurer–Cartan elements trivializes their Lie algebra structure. The result can be used to understand the rational homotopy type of many connected components of these mapping spaces.

Homotopical rigidity of the pre-Lie operad

“…Our arguments use filtration arguments that appear quite often when working with graph complexes [Wil15,Wil17], so in addition to furnishing the proofs of several new results, this paper hopefully might serve a pedagogical purpose, giving the reader an insight into several useful tricks that are normally hidden deep inside very condensed papers.…”

Weight 11 compactly supported cohomology of moduli spaces of curves

“…The terms of valence 2 cancel by an argument similar to that at the end of the proof of Lemma 5.5. For (20) note that ({γ 1 , γ 2 }) 1 is a linear combination of graphs obtained by connecting γ 1 and γ 1 by one edge, and marking one vertex of γ 1 or γ 2 as external. The terms in which the external vertex is in γ 2 are the same as γ • 1 • γ 1 2 , and the terms in which the external vertex is in γ 1 are the same as (−1)…”

“…where the sum is over all ways of inserting γ 2 into a vertex of γ 1 . We shall only need to use the following property of •, which is a special case of Proposition 5.3 (proved in [20]):…”