Disks Operad Research Articles

A Generalization of Operads Based on Subgraph Contractions

Abstract We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\operatorname{\overline{{\mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.

On the algebras over equivariant little disks

We describe the structure present in algebras over the little disks operads for various representations of a finite group G, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for G=C2, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.

On the formality of the little disks operad in positive characteristic

Using a variant of the Boardman–Vogt tensor product, we construct an action of the Grothendieck–Teichmüller group on the completion of the little n-disks operad E n . This action is used to establish a partial formality theorem for E n with mod p coefficients and to give a new proof of the formality theorem in characteristic zero.

The intrinsic formality of $E_n$-operads

We establish that E_n -operads satisfy a rational intrinsic formality theorem for n\geq 3 . We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in spaces. We consider, in this context, the dual cooperad of the n -Poisson operad \mathsf{Pois}_n^c , which represents the cohomology of the operad of little n -discs \mathsf{D}_n . We assume n\geq 3 . We explicitly prove that a Hopf cooperad in cochain graded dg-modules \mathsf{K} is weakly-equivalent (quasi-isomorphic) to \mathsf{Pois}_n^c as a Hopf cooperad as soon as we have an isomorphism at the cohomology level H^*(\mathsf{K})\simeq\mathsf{Pois}_n^c when 4\nmid n . We just need the extra assumption that \mathsf{K} is equipped with an involutive isomorphism mimicking the action of a hyperplane reflection on the little n -discs operad in order to extend this formality statement in the case 4\mid n . We deduce from these results that any operad in simplicial sets \mathsf{P} which satisfies the relation H^*(\mathsf{P},\mathbb{Q})\simeq\mathsf{Pois}_n^c in rational cohomology (and an analogue of our extra involution requirement in the case 4\mid n ) is rationally weakly equivalent to an operad in simplicial sets \mathtt L \mathtt G_{\bullet}(\mathsf{Pois}_n^c) which we determine from the n -Poisson cooperad \mathsf{Pois}_n^c . We also prove that the morphisms \iota: \mathsf{D}_m\rightarrow\mathsf{D}_n , which link the little discs operads together, are rationally formal as soon as n-m\geq 2 . These results enable us to retrieve the (real) formality theorems of Kontsevich by a new approach, and to sort out the question of the existence of formality quasi-isomorphisms defined over the rationals (and not only over the reals) in the case of the little discs operads of dimension n\geq 3 .

A topological origin of quantum symmetric pairs

It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are {mathbb {Z}}_2-orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Examples of these are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra.

Gerstenhaber algebra and Deligne’s conjecture on the Tate–Hochschild cohomology

Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we also obtain that the singular Hochschild cochain complex has a $B$-infinity algebra structure and its cohomology ring is a Gerstenhaber algebra. Inspired by the original definition of Tate cohomology for finite groups, we define a generalized Tate-Hochschild complex with the Hochschild chains in negative degrees and the Hochschild cochains in non-negative degrees. There is a natural embedding of this complex into the singular Hochschild cochain complex. In the case of a self-injective algebra, this embedding becomes a quasi-isomorphism. In particular, for a symmetric algebra, this allows us to show that the Tate-Hochschild cohomology ring, equipped with the Gerstenhaber algebra structure, is a Batalin-Vilkovisky algebra.

The Lambrechts\u2013Stanley model of configuration spaces

We prove the validity over {mathbb {R}} of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of Lambrechts–Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold. We moreover prove, if the dimension of the manifold is at least 4, that our model is compatible with the action of the Fulton–MacPherson operad (weakly equivalent to the little disks operad) when the manifold is framed. We use this more precise result to get a complex computing factorization homology of framed manifolds. Our proofs use the same ideas as Kontsevich’s proof of the formality of the little disks operads.

Planar non-formality of the little discs operad in characteristic two

AbstractWe show that the little discs operad D2 is not formal over F2 as a planar (or non-symmetric) operad. We compute explicitly the homological obstruction using as chain model the cells of the spineless cacti operad.

Non-Formality of the Odd Dimensional Framed Little Balls Operads

We prove that the chain operad of the framed little balls (or disks) operad is not formal as a non-symmetric operad over the rationals if the dimension of their balls is odd and greater than 4.

Factorization homology and calculus $\textit{à la}$ Kontsevich Soibelman

We use factorization homology over manifolds with boundaries in order to construct operations on Hochschild cohomology and Hochschild homology. These operations are parametrized by a colored operad involving disks on the surface of a cylinder defined by Kontsevich and Soibelman. The formalism of the proof extends without difficulties to a higher dimensional situation. More precisely, we can replace associative algebras by algebras over the little disks operad of any dimensions, Hochschild homology by factorization (also called topological chiral) homology and Hochschild cohomology by higher Hochschild cohomology. Our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum giving interesting operations on topological Hochschild homology and cohomology.

A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented $C^\infty$-manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of Rham of free loops with marked points. A notion of de Rham chains, which is a certain hybrid of the notions of singular chains and differential forms, is a key ingredient in our construction. Combined with a generalization of cyclic Deligne's conjecture, this dg operad produces a chain model of the free loop space which admits an action of a chain model of the framed little disks operad, recovering the string topology BV algebra structure on the homology level.

The homotopy braces formality morphism

We extend M. Kontsevich's formality morphism to a homotopy braces morphism and to a homotopy Gerstenhaber morphism. We show that this morphism is homotopic to D. Tamarkin's formality morphism, obtained using formality of the little disks operad, if in the latter construction one uses the Alekseev-Torossian associator. Similar statements can also be shown in the "chains" case, i.e., on Hochschild homology instead of cohomology. This settles two well known and long standing problems in deformation quantization and unifies the several known graphical constructions of formality morphisms and homotopies by Kontsevich, Shoikhet, Calaque, Rossi, Alm, Cattaneo, Felder and the author.

Cross-effects and the classification of Taylor towers

Let F be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of F have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of F are a right module over a certain `pro-operad'. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivial operad. We show that the Taylor tower of the functor F can be reconstructed from this structure on the derivatives.

Homology of non-$k$-overlapping discs

In this paper we describe the homology and cohomology of some natural bimodules over the little discs operad, whose components are configurations of non-k-overlapping discs. At the end we briefly explain how this algebraic structure intervenes in the study of spaces of non-k-equal immersions.

Crossed interval groups and operations on the Hochschild cohomology

We prove that the operad \mathcal{B} of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that \mathcal{B} is a certain crossed interval extension of an operad \mathcal{T} whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.

On the rational homology of high-dimensional analogues of spaces of long knots

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of $\mathbb{R}^m$ into $\mathbb{R}^n$. We view the space of embeddings as the value of a certain functor at $\mathbb{R}^m$, and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show that the formality of the little disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when $2m+1<n$, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.

Deloopings of the spaces of long embeddings

The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary. As a corollary, we show that the homotopy fiber is weakly equivalent to a space on which the framed little disks operad acts possibly nontrivially, and hence its rational homology is a (higher) BV-algebra in a stable range of dimensions.

Homotopically trivializing the circle in the framed little disks

This paper confirms the following suggestion of Kontsevich. In the appropriate derived sense, an action of the framed little disks operad and a trivialization of the circle action is the same information as an action of the Deligne-Mumford-Knudsen operad. This improves an earlier result of the author and Bruno Vallette.

Minimal models, GT-action and formality of the little disk operad

We give a new proof of formality of the operad of little disks. The proof makes use of an operadic version of a simple formality criterion for commutative differential graded algebras due to Sullivan. We see that formality is a direct consequence of the fact that the Grothendieck-Teichm\"uller group operates on the chain operad of little disks.

Groups, cacti and framed little discs

Let G be a topological group. Then the based loopspace of G is an algebra over the cacti operad, while the double loopspace of the classifying space of G is an algebra over the framed little discs operad. This paper shows that these two algebras are equivalent, in the sense that they are weakly equivalent E-algebras, where E is an operad weakly equivalent to both framed little discs and cacti. We recover the equivalence between cacti and framed little discs, and Menichi's isomorphism between the BV-algebras obtained by taking the homology of the loopspace of G and of the double loopspace of BG.