Non-formality of Voronov’s swiss-cheese operads. (English) Zbl 07936839

Idrissi, Najib; Vasconcellos Vieira, Renato

Q. J. Math. 75, No. 1, 63-95 (2024).

Summary: The Swiss-Cheese operads, which encode actions of algebras over the little \(n\)-cubes operad on algebras over the little \((n-1)\)-cubes operad, comes in several variants. We prove that the variant in which open operations must have at least one open input is not formal in characteristic zero. This is slightly stronger than earlier results of Livernet and Willwacher. The obstruction to formality that we find lies in arity \((2, 2^n)\), rather than \((2, 0)\) (Livernet) or \((4, 0)\) (Willwacher).

MSC:

18M60	Operads (general)
55P48	Loop space machines and operads in algebraic topology
55P62	Rational homotopy theory