Formality of the chain operad of framed little disks

Formality criteria for algebras over operads

We study some formality criteria for differential graded algebras over differential graded operads. This unifies and generalizes other known approaches like the ones by Manetti and Kaledin. In particular, we construct general operadic Kaledin classes and show that they provide obstructions to formality. Moreover, we show that an algebra A is formal if and only if its operadic cohomology spectral sequence degenerates at $E_2$.

Formality of the framed little 2-discs operad and semidirect products

We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the operadic semidirect product with the circle group. The idea of our proof is to show that Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions and so one can essentially take the operadic semidirect product with the homology of S^1 everywhere to obtain a chain of quasi-isomorphi…

Non-formality of the Swiss-Cheese operad

In this note, we prove that the Swiss-cheese operad is not formal. We also give a criteria in terms of Massey operadic product for the non-formality of a topological operad.

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

We show that the little discs operad $D_2$ is not formal over $\mathbb{F}_2$ as a planar (or non-symmetric) operad. We compute explicitly the homological obstruction using as chain model the cells of the spineless cacti operad

A(nother) model for the framed little disks operad

We describe new graphical models of the framed little disks operads which exhibit large symmetry dg Lie algebras.

Symmetric multiplicative formality of the Kontsevich operad

In his famous paper entitled "Operads and motives in deformation quantization", Maxim Kontsevich constructed (in order to prove the formality of the little d-disks operad) a topological operad, which is called in the literature the Kontsevich operad, and which is denoted Kd in this paper. This operad has a nice structure: it is a multiplicative symmetric operad, that is, it comes with a morphism from the symmetric associative operad. There are many results in the literature concerning the formality of Kd. It is well known (by Kontsevich) that Kd is formal over reals as a symmetric operad. It i…

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.

The framed discs operad is cyclic

The operad of framed little discs is shown to be equivalent to a cyclic operad. This answers a conjecture of Salvatore in the affirmative, posed at the workshop `Knots and Operads in Roma,' at Universita di Roma ``La Sapienza'' in July of 2006.

框架operad的自同构

Automorphisms of framed operads

本文研究了框架operad的自同构，通过建立原operad与半直积operad的同伦自同构群之间的联系，成功计算了框架小圆盘operad的自同构。

Moduli Spaces and Formal Operads

Let overline{M}_{g,n} be the moduli space of stable algebraic curves of genus g with n marked points. With the operations which relate the different moduli spaces identifying marked points, the family (overline{M}_{g,n})_{g,n} is a modular operad of projective smooth Deligne-Mumford stacks, overline{M}. In this paper we prove that the modular operad of singular chains C_*(overline{M};Q) is formal; so it is weakly equivalent to the modular operad of its homology H_*(overline{M};Q). As a consequence, the "up to homotopy" algebras of these two operads are the same. To obtain this result we prove …

Multiplicative formality of operads and Sinha's spectral sequence for long knots

Lambrechts, Turchin and Volić proved the Bousfield-Kan type rational homology spectral sequence associated to the $d$-th Kontsevich operad collapses at $E^2$-page if $d\geq 4$. The key of their proof is formality of the operad. In this paper, we simplify their proof using a model category of operads. As byproducts we obtain two new consequences. One is collapse of the spectral sequence in the case of $d=3$ (and the coefficients being rational numbers). The other says there is no non-trivial extension for the Gerstenhaber algebra structure on the spectral sequence.

带框小 $n$-圆盘算胚的实模型

Real models for the framed little $n$-disks operads

本文研究正交群在小 $n$-圆盘算胚上的作用，为带框小 $n$-圆盘算胚提供了实数域上的小模型，并由此得出其在 $n$ 为偶数时是形式的，对所有 $n$ 均为余形式的。

Framed discs operads and the equivariant recognition principle

The framed n-discs operad fD_n is studied as semidirect product of SO(n) and the little n-discs operad. Our equivariant recognition principle says that a grouplike space acted on by fD_n is equivalent to the n-fold loop space on a SO(n)-space. Examples of fD_2-spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of fD_n. Koszul duality for semidirect product operads of chain complexes is defined and applied to compute the double loop space homology as BV-algebra.

Formality of a higher-codimensional Swiss-Cheese operad

We study bicolored configurations of points in the Euclidean $n$-space that are constrained to remain either inside or outside a fixed Euclidean $m$-subspace, with $n - m \ge 2$. We define a higher-codimensional variant of the Swiss-Cheese operad, called the complementarily constrained disks operad $\mathsf{CD}_{mn}$, associated to such configurations. The operad $\mathsf{CD}_{mn}$ is weakly equivalent to the operad of locally constant factorization algebras on the stratified space $\{\mathbb{R}^{m} \subset \mathbb{R}^{n}\}$. We prove that this operad is formal over $\mathbb{R}$.

Voronov 的 Swiss-Cheese 操作子的非形式性

Non-formality of Voronov's Swiss-Cheese operads

本文证明 Voronov 的一类 Swiss-Cheese 操作子在特征零下非形式，并将形式性障碍定位在元数 $(2,2^n)$，强于此前结果。

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…

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…

Formality of little disks and algebraic geometry

We construct a canonical chain of formality quasiisomorphisms for the operad of chains on framed little disks and the operad of chains on little disks. The construction is done in terms of logarithmic algebraic geometry and is remarkable for being rational (and indeed definable integrally) in de Rham cohomology.

Weyl n-algebras and the Swiss cheese operad

We apply Weyl $n$-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.

带挠系数的平展上同调、纯度与形式化

Etale cohomology, purity and formality with torsion coefficients

本研究借助伽罗瓦群在平展上同调的作用，证明带挠系数的dg-Operad与dg-代数的有限次数形式化结论，可应用于小圆盘算子奇异链等构造，形式化次数由系数域基数决定。

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.

等变带框小圆盘 Operad 是可加的

Equivariant Framed Little Disk Operads are Additive

本文提出等变带框小圆盘 operad，并证明其满足可加性，从而把 Dunn-Brinkmeier 关于小立方体 operad 的定理推广到群作用与带框情形。

$\u213e_n$-代数的形式性与球面的上链

Formality of $\mathbb{E}_n$-algebras and cochains on spheres

研究增广𝔼ₙ-代数的环路与悬垂函子，证明n-球面的上链代数作为𝔼ₙ-代数是形式的，且谱中操作子到单子的自由函子在含稳定𝔼ₙ-操作子的子范畴上完全忠实。

Homotopy derivations of the framed little discs operads

We study the homotopy derivations of the framed little discs operads, which correspond to the homotopy derivations of the BV2n operads. By extending a result by Willwacher about the homotopy derivations of the en operads we show that the homotopy derivations of the BV2n operads may be described through the cohomology of a suitable graph complex. We will present an explicit quasi-isomorphic map.

Surface Drinfeld Torsors I : Higher Genus Associators

We develop a higher genus version of Drinfeld associators by means of operad theory. We start by introducing a framed version of rational associators and Grothendieck-Teichmüller groups and show that their definition is independent of the framing data. Next, we define a framed version of the universal KZ connection and we use it to show that over the complex numbers, the rational framed Drinfeld torsor is not empty. Next, we concentrate on the higher genus version of this story. We define an operad module of framed parenthesized higher genus braidings in prounipotent groupoids and we define it…

框架化小圆盘操作子的每个自同态都是自同构

Every Endomorphism of the Framed Little Disk Operad is an Automorphism

本文利用单李群自映射分类，将 Horel-Krannich-Kupers 关于小圆盘操作子自同态均为自同构的结果推广到框架化情形，并进一步考察了瑞士奶酪操作子及若干半直积情形。

Mixed Hodge structures and formality of symmetric monoidal functors

We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex schemes with non-pure weight filtration, we relate the singular chains functor…

Moduli of framed formal curves

We introduce framed formal curves, which are formal algebraic curves with boundary components parametrized by the punctured formal disk. We study the moduli space of nodal framed formal curves, which we endow with a logarithmic structure. We show that this moduli space is a smooth formal logarithmic stack. The remarkable property of our construction is that framed formal curves admit a natural operation of "gluing along the boundary" which works well in families and preserves smoothness (both in a formal and in a logarithmic sense), and this induces gluing maps on the level of moduli. Using mo…

The framed little 2-discs operad and diffeomorphisms of handlebodies

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of K. Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ``open string'' modular operad made …

循环操作符与结的Grothendieck-Teichmüller对称性

Grothendieck-Teichmüller Symmetries of Cyclic Operads and Tangles

本文研究了循环操作符与结的Grothendieck-Teichmüller对称性，通过操作符模型揭示了 profinite 结的算术对称性，并给出了形式性的另一种证明。

Calculus of functors, operad formality, and rational homology of embedding spaces

Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice the embedding dimension of M, the Taylor tower in the sense of orthogonal calculus (henceforward called ``the orthogonal tower'') of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence…

余维一长纽结的Sinha谱序列与小2-圆盘算子的非形式性

Sinha's spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad

本文计算了余维一长纽结空间Sinha谱序列在特征2和3下的若干微分，证明特定元素的$d_2$和$d_3$微分非零，并据此推出小2-圆盘链算子在相应特征下的平面非形式性，同时发展了基于构型空间与胖对角线对偶的Thom空间计算方法。

Operads and cosimplicial objects: an introduction

This paper is an introduction to a series of papers in which we give combinatorial models for certain important operads (including A-infinity and E-infinity operads, the little n-cubes operads, and the framed little disks operad) and combinatorial conditions for them to act on a given space or chain complex. The paper does not assume any prior knowledge of operads--Sections 2, 6 and 9, which can be read independently, are an introduction to the theory of operads.

When can a formality quasi-isomorphism over rationals be constructed recursively?

Let $O$ be a differential graded (possibly colored) operad defined over rationals. Let us assume that there exists a zig-zag of quasi-isomorphisms connecting $O \otimes K$ to its cohomology, where $K$ is any field extension of rationals. We show that for a large class of such dg operads, a formality quasi-isomorphism for $O$ exists and can be constructed recursively. Every step of our recursive procedure involves a solution of a finite dimensional linear system and it requires no explicit knowledge about the zig-zag of quasi-isomorphisms connecting $O \otimes K$ to its cohomology.

Hyper-commutative algebras and cyclic cohomology

This paper introduces a chain model for the Deligne-Mumford operad formed by homotopically trivializing the circle in a chain model for the framed little disks. We then show that under degeneration of the Hochschild to cyclic cohomology spectral sequence, a known action of the framed little disks on Hochschild cochains lifts to an action of this new chain model. We thus establish homotopy hyper-commutative algebra structures on both Hochschild and cyclic cochain complexes, and we interpret the gravity brackets on cyclic cohomology as obstructions to degeneration of this spectral sequence. Our …

On several varieties of cacti and their relations

Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and Fiedorowicz' braided operad structure are given by the same ex…

Knots, operads and double loop spaces

We show that the space of long knots in an euclidean space of dimension larger than three is a double loop space, proving a conjecture by Sinha. We construct also a double loop space structure on framed long knots, and show that the map forgetting the framing is not a double loop map in odd dimension. However there is always such a map in the reverse direction expressing the double loop space of framed long knots as a semidirect product. A similar compatible decomposition holds for the homotopy fiber of the inclusion of long knots into immersions. We show also via string topology that the spac…

Deligne-Mumford 算元作为圆周作用的平凡化

The Deligne-Mumford operad as a trivialization of the circle action

本文证明树状 Deligne-Mumford 算元是高亏格带标架小圆盘算元中圆周平凡化的同伦模型，通过节点环带的几何论证和拓扑模问题形式化框架实现。

Monoidal functors, acyclic models and chain operads

We prove that for a topological operad $P$ the operad of oriented cubical chains, $C^{ord}_\ast(P)$, and the operad of singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{ord}_\ast(P;\mathbb{Q})$ is formal if and only if $S_\ast(P;\mathbb{Q})$ is formal, thus linking together some formality results spread in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give different variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets defined by $R$-simplicial differentia…

地面域的范畴计数不变量

Categorical enumerative invariants of the ground field

本文引入了$S^1$-framed模操作的Feynman紧化，证明其同调操作与Deligne-Mumford操作同构，并通过Sen-Zwiebach的弦顶点得到地面域的范畴计数不变量与点的Gromov-Witten不变量一致。

Relations in the modulo 2 homology of framed disks algebras

We study the homology structure of spaces having an action of the \emph{framed disks} operad. In particular, we compute the relations between Kudo-Araki operations and generalized Batalin-Vilkovisky operators. As an application, we complete the computations of the modulo 2 homology of $Ω^2 S^3$ as a Batalin-Vilkovisky algebra, and give some evidence for a classical conjecture about the modulo 2 Hurewicz homomorphism of the infinite loop space of the sphere spectrum.

偶数维小圆盘操作代数的对数几何模型

Log Geometric Models for Little Disks Operads in Even Dimensions

本文利用对数几何为偶数维小圆盘操作代数构造模型，并证明无框架情形可用带虚拟态射的对数概形实现。

A proof of a cyclic version of Deligne's conjecture via Cacti

In this note, we show that the normalized Hochschild co--chains of an associative algebra with a non--degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in this sense they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber's bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the l…

配置空间的一点紧化与小圆盘操作的自对偶性

One point compactifications of configuration spaces and the self duality of the little disks operad

本文通过配置空间层次的庞特里亚金-托姆构造，为小圆盘操作构建了自对偶映射，并证明了带框架流形上的兼容性。

立方体、仙人掌与标架长纽结

Cubes, cacti, and framed long knots

该研究定义了射影无刺仙人掌operad对标架长纽结空间泰勒塔各层的作用，证明其与Budney小2-立方体作用的空间相容性，改进了相关Vassiliev不变量猜想的前期结果，还重新证明了Sakai发现的一类Browder括号类的非平凡性。

Cyclic A_\infty Structures and Deligne's Conjecture

First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyc…

图配置空间、合同腺与形式性

Graphical configuration spaces, Contractads and Formality

本文研究图配置空间的合同腺结构，证明其在特定条件下形式性，并探讨了循环图的共形式性障碍及二维合同腺的细胞结构。

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 "de Rham chains" 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 w…

Toric varieties of Loday's associahedra and noncommutative cohomological field theories

We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-disks, framed little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, co…

循环配边小圆盘代数、Grothendieck-Verdier对偶与柄体群表示

Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations

该研究刻画对称幺半双范畴中配边小圆盘operad的循环代数，证明其与两类Grothendieck-Verdier范畴等价，进而得到量子拓扑中柄体群表示推广与无半单假设的Grothendieck-Verdier对偶两项应用。

对数态射、切向基点和小盘

Logarithmic morphisms, tangential basepoints, and little disks

本文通过虚拟态射理论，将小盘操作符的形性证明与切向基点理论结合，构建了整数上的log schemes操作符，并揭示了辫群与模空间的关系。