Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma

Tourtchine, Victor; Willwacher, Thomas

doi:10.48550/arxiv.1409.0163

Cited by 11 publications

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“…1 Next, let us describe the implications of Theorem 1 for the hairy graph cohomology. To this end, we need to recall one more ingredient: By results of V. Turchin and the second author [23,19] it is known that on each of the complexes HGC n,n and HGC n−1,n there is a deformation of the differential (say D ′ ) such that the cohomology of the deformed complex is equal to the ordinary (non-hairy) graph cohomology:…”

Section: Theoremmentioning

confidence: 99%

Differentials on graph complexes II: hairy graphs

2017

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Abstract. We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many hairy graph cohomology classes out of non-hairy classes by a mechanism which we call the waterfall mechanism. By this mechanism we can construct many previously unknown classes and provide a first glimpse at the tentative global structure of the hairy graph cohomology.

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Section: Theoremmentioning

confidence: 99%

Differentials on graph complexes II: hairy graphs

2017

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“…E.g. H 5 Γ is the part of Γ with 5 hairs. Some differentials do not change some numbers from the above, so the complex splits as a direct product of complexes with fixed that number, e.g.…”

Section: 7mentioning

confidence: 99%

“…The most nontrivial result is that H 0 (GC 2 ) is isomorphic to Grothendiech-Teichmüller Lie algebra grt 1 , shown by Willwacher in [7]. There are also some results connecting hairy and non-hairy complexes, see [6] and [5].…”

Section: Introductionmentioning

confidence: 99%

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Differentials on Graph Complexes III - Deleting a Vertex

Živković¹

2016

Preprint

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“…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. 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.…”

Section: Introductionmentioning

confidence: 99%

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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Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma

Cited by 11 publications

References 0 publications

Differentials on graph complexes II: hairy graphs

Differentials on graph complexes II: hairy graphs

Differentials on Graph Complexes III - Deleting a Vertex

Non-formality of the Swiss-cheese operad

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