Formality Theorem for Hochschild Cochains via Transfer

2013

Algebr. Geom. Topol.

We prove that the homology of the Swiss-cheese operad is a Koszul operad. As a consequence, we obtain that the spectral sequence associated to the stratification of the compactification of points on the upper half plane collapses at the second stage, proving a conjecture by A. Voronov in [17]. However, we prove that the operad obtained at the second stage differs from the homology of the Swiss-cheese operad. IntroductionAn operad O is called differentiable when it is defined on the symmetric monoidal category of differentiable manifolds. If each O(n) is a manifold with corners whose connected boundary components are cartesian products of O(k) (with k < n) and the operad structure is given by the inclusion map of the boundary strata, then the operad is called stratified. To every stratified operad is associated a dg-operad given by the spectral sequences induced by a natural filtration on its singular chain complex: the boundary strata filtration. Since that filtration is given by the codimension of the boundary components, it is finite and hence converges to the homology H * (O) at some finite stage. One would naturally wonder wether the operad structure induced on homology by the spectral sequence coincides with the one induced by the topological operad structure.In the present paper we study the spectral sequence of a stratified operad, given by Kontsevich's compactification [14], which is homotopically equivalent to the Swiss-cheese operad SC. We show that H * (SC) is an inhomogeneous Koszul operad. We also show that the operad structure induced on H * (SC) by the spectral sequence of its stratified version differs from the structure given by the topological operad structure of SC. The difference between the two operad structures in H * (SC) is explored in terms of the Koszul duality theory for inhomogeneous quadratic operads developed by Tonks and Vallette in [5].Among the related algebraic structures are Kajiura and Stasheff's OCHA [11,10], Leibniz pairs [3] and extensions of those considered by Dolgushev [2]. The relation between OCHAS, Leibniz pairs and the Swiss-cheese operad has been carefully studied by the authors in [9], where the zeroth homology of the Swiss-cheese operad SC were related to the first row of the spectral sequence associated to the Kontsevich compactification. One of the purpose of [9] was to prove an SC analogue of the following fact concerning the little disks operad D 2 :Proposition. The zero-th homology of the operad D 2 is Koszul dual to a suspension of the top homology.In the case of little discs, the zero-th homology is the operad Com and the top homology is a desuspension of the operad Lie. In fact, the above Proposition is a consequence of a theorem, proved by Getzler and Jones, according to which the Gerstenhaber operad H * (D 2 ) is, up to suspension, a self-dual Koszul operad.

supporting

confidence: 51%

mentioning

confidence: 99%

On the spectral sequence of the Swiss-cheese operad

Hoefel

2013

Algebr. Geom. Topol.

“…The left hand side of the quasi-isomorphism is minimal and algebras over it are OCHA as pointed out by Dolgushev in [3] (see section 7.1). The result, however, is not totally satisfying because H 0 (SC)…”

Section: Introductionmentioning

confidence: 93%

“…But he did not prove that the operad is Koszul. In [3], Dolgushev showed that OCHA correspond to algebras over Ω(D), where D is a cooperad that will be specified later, but without proving that D is Koszul. The aim of our paper is to prove that the operads involved are Koszul, justifying completely that SHLP and OCHA are "strong homotopy" algebras over an operad.…”

Section: Introductionmentioning

confidence: 99%

Open-Closed Homotopy Algebras and Strong Homotopy Leibniz Pairs Through Koszul Operad Theory

Hoefel

2012

Lett Math Phys

“…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 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 .…”

Section: Introductionmentioning

confidence: 99%

Non-formality of the Swiss-cheese operad

2015

J Topology

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...