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