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).