Non-formality of Voronov's Swiss-Cheese operads

Authors

N. I., Renato Vasconcellos Vieira.

Publication

In: Q. J. Math. 75.1 (2024), pp. 63-95.

Online on

Online on March 29, 2023.

Accepted on

Accepted on October 19, 2023.

Updated on

Updated on April 16, 2024.

Full text

Full text.

10.1093/qmath/haad041

10.1093/qmath/haad041.

arXiv:2303.16979

arXiv:2303.16979.

hal-04053257

hal-04053257.

MR4732946

MR4732946.

Zbl07936839

Zbl07936839.

Source

Source.

Abstract

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).