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Formality of a higher-codimensional Swiss-cheese operad. (English) Zbl 1495.55005

The little disks operads \(\mathrm{D}_n\) are classical objects in algebraic topology that were first introduced to study iterated loop spaces. The elements of \(\mathrm{D}_n(k)\) consist of configurations of \(k\) disks inside the unit \(n\)-disk \({\mathbb{D}}^n\). An important property of \(\mathrm{D}_n\) is its formality. The Swiss-cheese operads \(\mathrm{SC}_n\), \(n\geq 2\), are relative versions of the little disks operads. Unlike the little disks operads, the Swiss-cheese operads are not formal.
The author introduces higher-codimensional variants of the Swiss-cheese operads, \(\mathrm{CD}_{mn}\), where \(n-2\geq m\geq 1\), called the complementarily constrained disks operads. On the level of homology, these operads encode actions of \(\mathrm{D}_n\)-algebras on \(\mathrm{D}_m\)-algebras by central derivations. The elements of \(\mathrm{CD}_{mn}(k,l)\) are given by configurations of \(k\) disks centered on \({\mathbb{D}}^m\subset {\mathbb{D}}^n\) and \(l\) disks entirely contained in the complement \({\mathbb{D}}^n\setminus {\mathbb{D}}^m\).
The main result of the paper states that \(\mathrm{CD}_{mn}\) is formal over \({\mathbb{R}}\). The proof is inspired by the proofs of the formality of the little disks operad by M. Kontsevich [Lett. Math. Phys. 48, No. 1, 35–72 (1999; Zbl 0945.18008)] and by P. Lambrechts and I. Volić [Formality of the little \(N\)-disks operad. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1308.55006)]. The motivation for the article comes from the study of the configuration spaces of the complement \(N\setminus M\), where \(N\) is a closed \(n\)-manifold and \(M\) is a closed \(m\)-submanifold of \(N\) of codimension \(\geq 2\).

MSC:

55P48 Loop space machines and operads in algebraic topology
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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