On the spectral sequence of the Swiss-cheese operad

Hoefel, Eduardo; Livernet, Muriel

doi:10.2140/agt.2013.13.2039

Cited by 4 publications

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“…For d = 2, Dolgushev studies in [3] the first sheet of the homology spectral sequence for the Fulton-MacPherson version of the Swiss-cheese operad and proves that it is not formal. However, this does not imply the non-formality of SC 2 , because we proved with E. Hoefel in [7] that the homology spectral sequence, though collapsing at page 2, does not converge as an operad, to the homology operad of SC 2 . Lambrechts and Volic study the inclusion of the little n-discs operad into the little m-discs operad and prove in [10], that the inclusion is formal for m > 2n.…”

Section: Introductionmentioning

confidence: 83%

“…There is another version of the Swiss-cheese operad given by Kontsevich in [8]. The homology of these two topological colored operad has been studied by Hoefel and the author in [6] and [7]. In this note, we consider the Kontsevich version of the Swiss-cheese operad denoted by SC as in [7].…”

Section: The Two-dimensional Swiss-cheese Operadmentioning

confidence: 99%

“…The homology of these two topological colored operad has been studied by Hoefel and the author in [6] and [7]. In this note, we consider the Kontsevich version of the Swiss-cheese operad denoted by SC as in [7]. It is a two-colored topological operad, whose colors are the closed color c and the open color o.…”

Section: The Two-dimensional Swiss-cheese Operadmentioning

confidence: 99%

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Non-formality of the Swiss-cheese operad

Livernet

2015

J Topology

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In this note, we prove that the Swiss-cheese operad is not formal. We also give a criterion in terms of Massey operadic product for the non-formality of a topological operad. IntroductionThe question of formality finds its roots in rational homotopy theory. In [12], Sullivan proves that the rational homotopy type of a space is entirely determined by a commutative differential graded algebra. The space is formal if this algebra is quasi-isomorphic to its homology, that is, if the rational homotopy type of the space is entirely determined by its cohomology ring. This notion has been applied with great success in [2], where the authors prove that the real homotopy type of a simply connected compact Kähler manifold is entirely determined by its cohomology ring. The question of non-formality has its own interest, in particular in the study of symplectic manifolds. In such cases, one can ask whether a symplectic manifold admits a Kähler structure or not. A symplectic manifold which is not formal is certainly not a Kähler manifold (see, for example, [4]).More generally, formality is closely related to deformation of structures. And structures are ruled by operads. One of the most famous topological operads is the little d-disks operad of Boardman and Vogt [1], recognizing iterated loop spaces (see [11]). An operad is formal if its singular chain complex is an operad quasi-isomorphic to its homology operad. Algebras over the homology operad of the little d-disks operad are d-Gerstenhaber algebras. In [8], Kontsevich lays the groundwork for the formality of the little d-disks operad and its applications (see the introduction of Giansiracusa and Salvatore in [5] for a history of the proof of the formality of the little d-disks operad). The Swiss-cheese operad appears in the work of Voronov in [14] and Kontsevich in [8] as a tool to handle action of a d-Gerstenhaber algebra on a (d − 1)-Gerstenhaber algebras and their deformations.In this note, we prove that the Swiss-cheese operad SC d is not formal for any d 2. Let us comment some related results. For d = 2, Dolgushev studies in [3] the first sheet of the homology spectral sequence for the Fulton-MacPherson version of the Swiss-cheese operad and proves that it is not formal. However, this does not imply the non-formality of SC 2 , because we proved with Hoefel in [7] that the homology spectral sequence, though collapsing at page 2, does not converge as an operad, to the homology operad of SC 2 . Lambrechts and Volic study the inclusion of the little n-disks operad into the little m-disks operad and prove in [10], that the inclusion is formal for m > 2n. Turchin and Willwacher show in [13] that this inclusion is not formal if m = n + 1. Their proof is based on Kontsevich's graph complex. There might be a link between the deformations of the aforementioned inclusion and the deformations of the Swiss-cheese operad, but this is not clear. If there is, the techniques involved in our paper are much simpler because they only use basic algebraic topology. Indeed, we prove the non...

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Section: Introductionmentioning

confidence: 83%

Section: The Two-dimensional Swiss-cheese Operadmentioning

confidence: 99%

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Non-formality of the Swiss-cheese operad

Livernet

2015

J Topology

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“…The first difference relies in the fact that each space SC 1 (J; x) has contractible path connected components and that SC 1 is a regular operad. The second difference relies in the fact that for n ≥ 2 the operad SC n is not formal as proved in [21], though sc n is Koszul (in the quadratic-linear sense) as proved in [16], whereas SC 1 is formal and sc 1 is Koszul quadratic.…”

Section: Deformation Complex Of Sc 1 -Algebras and Its Variantsmentioning

confidence: 99%

“…with ♭ and ♮ are the bijections defined in (16) and (19) and D c is the bijection Υ : Nest(S(n)) → (T ) ≤M 1,n from Proposition 4.2. 6.…”

Section: The Poset T and The Blow-up Componentsmentioning

confidence: 99%

On the deformation complex of homotopy affine actions

Hoefel

Livernet

Quesney

2019

Advances in Mathematics

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An affine action of an associative algebra A on a vector space V is an algebra morphism A → V ⋊ End(V ), where V is a vector space and V ⋊ End(V ) is the algebra of affine transformations of V . The one dimensional version of the Swiss-Cheese operad, denoted sc 1 , is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by (sc 1 ) ∞ . Algebras over this minimal model are called Homotopy Affine Actions, they consist of an A ∞ -morphism A → V ⋊ End(V ), where A is an A ∞ -algebra. In this paper we prove a relative version of Deligne's conjecture. In other words, we show that the deformation complex of a homotopy affine action has the structure of an algebra over an SC 2 operad. That structure is naturally compatible with the E 2 structure on the deformation complex of the A ∞ -algebra. 1complex. An As ∞ -version of Deligne's conjecture states that the deformation complex of any E 1 -algebra is an E 2 -algebra. This was done by Kontsevich and Soibelman in [20], where they build an explicit operad M acting on the deformation complex of an As ∞ -algebra (see also [6] for a very nice proof of this result). This operad will correspond to the closed part of ours. We refer also to Kauffman and Schwell in [18] for a topological proof of this fact.The 2-dimensional Swiss cheese operad SC vor 2 has been introduced by S. Voronov in [28] as a topological 2-colored operad, in order to understand spaces of configuration of points used by M. Kontsevich in deformation quantization and by B. Zwiebach in [29] in Open-Closed string field theory. The two colors are commonly called closed and open. It is shown by A. Voronov that its homology, sc vor 2 , governs Gerstenhaber algebras (the closed structure) acting on associative algebras (the open structure).In [14], the first author studied another version of the Voronov's Swiss-Cheese operad introduced by M. Kontsevich in [19], that we call here the Swiss-cheese operad and denote SC 2 . This operad admits operations having only closed input and an open output. In particular, there is an operation that transforms a closed variable into an open one, called the whistle. The operad SC vor 2 is a suboperad of SC 2 . This operad is more accurate in compactification and deformation theory though it has the disadvantage of not being formal, as proved in [21]. Given an associative algebra, the pair (HH * (A), A) is an algebra over sc 2 , the homology of SC 2 . The map HH * (A, A) → A sends HH * (A, A) onto HH 0 (A, A), the center of A.As in the classical case, an operad is said to be SC (vor) n if it is weakly equivalent to the operad of chains of the topological operad SC (vor) n . A differential graded vector space is said to be an SC (vor) n -algebra if it has an action of an SC (vor) n operad. In [8], V. Dolgushev, D. Tamarkin and B. Tsygan proved that the pair (CH * (A, A), A) is an SC vor 2 -algebra, where CH * (A, A) denotes the Hochschild cochain complex of the associative algebra A.Main result. In this paper we prove a...

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Moduli problems for operadic algebras

Calaque

Campos

Nuiten

2022

Journal of London Math Soc

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A theorem of Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. We prove a generalization of this correspondence, relating formal moduli problems parametrized by algebras over a Koszul operad to algebras over its Koszul dual operad. In particular, when the Lie algebra associated to a deformation problem is induced from a pre-Lie structure, it corresponds to a permutative formal moduli problem. As another example, we obtain a correspondence between operadic formal moduli problems and augmented operads.

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On the spectral sequence of the Swiss-cheese operad

Cited by 4 publications

References 15 publications

Non-formality of the Swiss-cheese operad

Non-formality of the Swiss-cheese operad

On the deformation complex of homotopy affine actions

Moduli problems for operadic algebras

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