Formality of a higher-codimensional Swiss-Cheese operad

TL;DR

This paper introduces a higher-codimensional Swiss-Cheese operad variant, proves its formality over the real numbers, and relates it to factorization algebras on stratified spaces, advancing understanding of configuration spaces.

Contribution

It defines the complementarily constrained disks operad for higher codimension configurations and proves its formality, linking it to factorization algebras on stratified Euclidean spaces.

Findings

The operad $ extsf{CD}_{mn}$ is weakly equivalent to locally constant factorization algebras.

The operad $ extsf{CD}_{mn}$ is formal over $ eal$.

Provides new insights into the topology of higher-codimensional configuration spaces.

Abstract

We study bicolored configurations of points in the Euclidean $n$-space that are constrained to remain either inside or outside a fixed Euclidean $m$-subspace, with $n - m \ge 2$. We define a higher-codimensional variant of the Swiss-Cheese operad, called the complementarily constrained disks operad $\mathsf{CD}_{mn}$, associated to such configurations. The operad $\mathsf{CD}_{mn}$ is weakly equivalent to the operad of locally constant factorization algebras on the stratified space $\{\mathbb{R}^{m} \subset \mathbb{R}^{n}\}$. We prove that this operad is formal over $\mathbb{R}$.