Ryan Budney
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.
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We extend Tamarkin's formality of the little disk operad to the framed little disk operad.
We describe new graphical models of the framed little disks operads which exhibit large symmetry dg Lie algebras.
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。
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…
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。
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.
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.
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…
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。
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.
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.
Cyclic operads through modules
本文提出通过模计算循环operad映射空间的方法，并应用于计算循环Batalin-Vilkovisky（Hopf余）operad的同伦自同构空间。
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…
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。
The framed little 2-discs operad is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes. We show that this cyclic operad is formal, meaning that its chains and its homology (the Batalin-Vilkovisky operad) are quasi-isomorphic cyclic operads. To prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction, and we show that this complex resolves BV as a cyclic operad.
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 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 …
A hypergraph model for the cyclic BV operad and its applications
本文提出一个带有显式同伦映射的超图操作素模型，用于证明其同调等于BV操作素，并利用该模型构造新的同伦映射，从而证明FFM_2的循环形式性。
Cyclic model for the dg dual of the BV operad
本文为 BV 运算子的 dg 对偶构造了新的小循环模型 QBV，并证明其 Feynman 变换与截断 Feynman 变换在互补权和次数上拟同构。
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。
First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a cyclically invariant Maurer-Cartan element in an associated Lie algebra. Using this result we introduce and study the operad of cyclically invariant operations, with instances arising in cyclic cohomology and $S^1$ equivariant homology. We compute the homology of the cyclically invariant operations; the result being the homology operad of $\mathcal{M}_{0,n+1}$, the uncompactified moduli spac…
Equivariant formality of the little disks operad
本文证明了小n-圆盘操作元在有理数上关于SO(n)和O(n)是等变形式的，等价于有向和无向框架小圆盘操作元作为∞-操作元是有理形式的。
It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings.
Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2-discs operad.
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.
We give a direct combinatorial proof that the modular envelope of the cyclic operad Ass is the modular operad of (the homeomorphism classes of) 2D compact surfaces with boundary with marked points.
We propose a $λ$-calculus-style formal language, called the $μ$-syntax, as a lightweight representation of the structure of cyclic operads. We illustrate the rewriting methods behind the formalism by giving a complete step-by-step proof of the equivalence between the unbiased and biased definitions of cyclic operads.
Dwyer--Kan homotopy theory for cyclic operads
该研究提出对称幺半基范畴上着色循环 operad 的通用定义，借助其到着色 operad 的遗忘函子的左右伴随显式公式，将单纯集富集着色 operad 范畴的 Cisinski--Moerdijk 模型结构提升到对应循环 operad 范畴。
We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\widehat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\widehat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\widehat{\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincaré duali…
Models for cyclic infinity operads
本文在循环树状集与空间上构建了循环拟算畴与完备循环树状 Segal 空间的模型结构，证明其与单纯循环算畴 Quillen 等价，并推广至平面情形以支持 Walde 纲领。
We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\hat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\hat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\hat{\mathcal O}$ in terms of derivations and module maps. An application to Poincaré duality on the chain level of a suitable topological space is given.
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 …
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…
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…
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…
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。
Cyclic duality between BV algebras and BV modules
本文证明满足cosimplicial相容条件的循环operad其上同调具有BV代数结构，并给出BV模的循环对偶成为BV代数的条件，以Hopf代数理论中的同调群为例加以说明。
We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $Ξ$ of trees, which carries a tight relationship to the Moerdijk-Weiss category of rooted trees $Ω$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $Ξ$ which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.
Realization of saturated transfer systems on cyclic groups of order $p^nq^m$ by linear isometries $N_\infty$-operads
本文证明了Rubin饱和猜想的一个新情形，即阶为 $p^nq^m$ 的有限循环群上的 $G$-传递系统可通过线性等距 $N_\infty$-operads 实现。
In this paper we apply homotopical localization to the framework of differential graded algebras over an operad. We get plus construction by performing nullification with respect to an universal acyclic algebra. This plus construction for Lie and Leibniz algebras up to homotopy yields two new homology theories for associative algebras. They are closely related to cyclic and Hochschild homologies. We also compute their first homology groups.
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。
Categories of graphs for operadic structures
本文回顾了用于描述广义算盘同伦相干版本的图的范畴，给出态射的新统一定义，并证明了广义算盘范畴间的自由-遗忘伴随关系可在预层层面实现。
Cyclic operads, Koszul complexes, and hairy graph complexes
利用向下Brauer范畴的扭曲线性化，将毛状图复形构造为Koszul复形，并解释其同调；引入第二类Koszul复形，揭示不稳定情形与无穷维情形图同调的关系。
We provide a new combinatorial approach to studying the collection of N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic group. In particular, we show that for G the cyclic group of order p^n the natural order on the collection of N-infinity-operads stands in bijection with the poset structure of the (n+1)-associahedron. We further provide a lower bound for the number of possible N-infinity-operads for any finite cyclic group G.
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。
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…
Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group
本文计算了多个循环（余）operad 费曼变换的权 2 及权 top$-2$ 上同调，利用 Giansiracusa 结果得到柄体群的权 top$-2$ 上同调，并与曲线模空间相关结果比较，还为柄体群权 top 上同调与 Kontsevich 图上同调等价提供新证明。
The Getzler-Gauss-Manin connection and Kontsevich-Soibelman operations on the periodic cyclic homology
本文研究了周期性循环同调中源于Kontsevich-Soibelman操作的等变操作，并将其与等变Gromov-Witten理论联系起来，同时探讨了其在辛拓扑中的应用。
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。
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。
In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Ho…
The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and Feynman transform. Specifically, for each twisted modular operad $\mathcal{P}$ (taking values in dg-vector spaces over a field $k$ of characteristic 0), there is a certain sheaf $\mathcal{F}$ associated with it on the moduli space of stable metric graphs such that the Verdier dual sheaf $D\mathcal{F}$ is associated with the Feynman transform $F\mathcal{P}$ of $\mathcal{P}$. In…
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。
We introduce twisted arrow categories of operads and of algebras over operads. Up to equivalence of categories, the simplex category $Δ$, Segal's category $Γ$, Connes cyclic category $Λ$, Moerdijk-Weiss dendroidal category $Ω$, and categories similar to graphical categories of Hackney-Robertson-Yau are twisted arrow categories of symmetric or cyclic operads. Twisted arrow categories of operads admit Segal presheaves and 2-Segal presheaves, or decomposition spaces. Twisted arrow category of an operad $P$ is the $(\infty, 1)$-localization of the corresponding category $Ω/P$ by the boundary prese…
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.
We exhibit the simplex category $Δ$ and Segal's category $Γ$ as $\infty$-categorical localizations of the dendroidal categories $Ω_π$ and $Ω$ introduced by Moerdijk and Weiss. As an application we obtain an equivalence of $\infty$-categories between invertible $\infty$-operads and the $2$-Segal spaces of Dyckerhoff and Kapranov. Finally, we describe a cyclic version of the dendroidal category and explain how it $\infty$-localizes to Connes's cyclic category $Λ$.
We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.
Cyclic Symmetries of Chord Diagrams
本文通过直接证明，将Proalgebraic graded Grothendieck-Teichmüller群与基于Furusho对五边形方程的循环括号弦图prounipotent循环操作符的自同构群证明同构，并描述了该群对带状弦图范畴的作用。
Transfer systems are combinatorial objects which classify $N_\infty$ operads up to homotopy. By results of A. Blumberg and M. Hill, every transfer system associated to a linear isometries operad is also saturated (closed under a particular two-out-of-three property). We investigate saturated and linear isometric transfer systems with equivariance group $C_{p^mq^n}$, the cyclic group of order $p^mq^n$ for $p,q$ distinct primes and $m,n\ge 0$. We give a complete enumeration of saturated transfer systems for $C_{p^mq^n}$. We also prove J. Rubin's saturation conjecture for $C_{pq^n}$; this says th…
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…
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