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.
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.
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…
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…
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.
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…
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 …
The paper gives an explicit description of the Weiss embedding tower in terms of spaces of maps of truncated modules over the framed Fulton-MacPherson operad.
We build new algebraic structures, which we call genuine equivariant operads, which can be thought of as a hybrid between equivariant operads and coefficient systems. We then prove an Elmendorf-Piacenza type theorem stating that equivariant operads, with their graph model structure, are equivalent to genuine equivariant operads, with their projective model structure. As an application, we build explicit models for the $N_{\infty}$-operads of Blumberg and Hill.
We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a right-Quillen equivalence from the model category of equivariant simplicial operads with norm maps to the model category structure for equivariant-$\infty$-operads in equivariant dendroidal sets.
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.
We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N-infinity operads, equivariant generalizations of E-infinity operads. Algebras in equivariant spectra over an N-infinity operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N-infinity operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant in…
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 build model structures on the category of equivariant simplicial operads with a fixed set of colors, with weak equivalences determined by families of subgroups. In particular, by specifying to the family of graph subgroups (or, more generally, one of the indexing systems of Blumberg-Hill), we obtain model structures on the category of equivariant simplicial operads with a fixed set of colors, with weak equivalences determined by norm map data.
We construct a generalization of the operadic nerve, providing a translation between the equivariant simplicially enriched operadic world to the parametrized $\infty$-categorical perspective. This naturally factors through genuine equivariant operads, a model for "equivariant operads with norms up to homotopy". We introduce the notion of an op-fibration of genuine equivariant operads, extending Grothendieck op-fibrations, and characterize fibrant operads as the image of genuine equivariant symmetric monoidal categories. Moreover, we show that under the operadic nerve, this image is sent to $G$…
We build model structures on the category of equivariant simplicial operads with weak equivalences determined by families of subgroups, in the context of operads with a varying set of colors (and building on the fixed color model structures in the prequel). In particular, by specifying to the family of graph subgroups (or, more generally, one of the indexing systems of Blumberg-Hill), we obtain model structures on the category of equivariant simplicial operads whose weak equivalences are determined by norm map data.
We introduce the analogues of the notions of complete Segal space and of Segal category in the context of equivariant operads with norm maps, and build model categories with these as the fibrant objects. We then show that these model categories are Quillen equivalent to each other and to the model category for $G$-$\infty$-operads built in a previous paper. Moreover, we establish variants of these results for the Blumberg-Hill indexing systems. In an appendix, we discuss Reedy categories in the equivariant context.
In this short note we explain in detail the construction of a $O(n)$-equivariant isomorphism of topological operads $F_n \cong WF_n$ , where $F_n$ is the Fulton Mac Pherson operad and $W$ is the Boardman-Vogt construction
In this paper, we prove a conjecture of Blumberg and Hill regarding the existence of $N_\infty$-operads associated to given sequences $\mathcal{F} = (\mathcal{F}_n)_{n \in \mathbb{N}}$ of families of subgroups of $G\times Σ_n$. For every such sequence, we construct a model structure on the category of $G$-operads, and we use these model structures to define $E_\infty^{\mathcal{F}}$-operads, generalizing the notion of an $N_\infty$-operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for …
From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operad C_2 (which is an S^1 operad) we obtain variations on Getzler's gravity operad, which we show governs the Chas-Sullivan string bracket.
Equipping a non-equivariant topological E_\infty operad with the trivial G-action gives an operad in G-spaces. The algebra structure encoded by this operad in G-spectra is characterised homotopically by having no non-trivial multiplicative norms. Algebras over this operad are called naive-commutative ring G-spectra. In this paper we let G be a finite group and we show that commutative algebras in the algebraic model for rational G-spectra model the rational naive-commutative ring G-spectra.
We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When $G$ is a finite abelian group, we prove that a $G$-indexing system is realized by a Steiner operad if and only if it is generated by cyclic $G$-orbits. When $G$ is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than $3$, we prove that a $G$-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill's horn-filling condition. We also repackage the data in an indexing system as a certain kind o…
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…
We extend the Cisinski-Moerdijk-Weiss theory of $\infty$-operads to the equivariant setting to obtain a notion of $G$-$\infty$-operads that encode "equivariant operads with norm maps" up to homotopy. At the root of this work is the identification of a suitable category of $G$-trees together with a notion of $G$-inner horns capable of encoding the compositions of norm maps. Additionally, we follow Blumberg and Hill by constructing suitable variants associated to each of the indexing systems featured in their work.
Equipping a non-equivariant topological $E_\infty$-operad with the trivial $G$-action gives an operad in $G$-spaces. For a $G$-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring $G$-spectra. In this paper we take $G=SO(2)$ and we show that commutative algebras in the algebraic model for rational $SO(2)$-spectra model rational naïve-commutative ring $SO(2)$-spectra. In particular, this applies to show that the $SO(2)$-equivariant cohomology associated to an elliptic curve $C$ f…
Hurwitz spaces are homotopy quotients of the braid group action on the moduli space of principal bundles over a punctured plane. By considering a certain model for this homotopy quotient we build an aspherical topological operad that we call the little bundles operad. As our main result, we describe this operad as a groupoid-valued operad in terms of generators and relations and prove that the categorical little bundles algebras are precisely Turaev's crossed categories. Moreover, we prove that the evaluation on the circle of a homotopical two-dimensional equivariant topolological field theory…
We develop the rudiments of a theory of parametrized $\infty$-operads, including parametrized generalizations of monoidal envelopes, Day convolution, operadic left Kan extensions, results on limits and colimits of algebras, and the symmetric monoidal Yoneda embedding.
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.
In this paper, we construct incomplete versions of the equivariant stable category; i.e., equivariant stabilization of the category of $G$-spaces with respect to incomplete systems of transfers encoded by an $N_\infty$ operad $\mathcal{O}$. These categories are built from the categories of $\mathcal{O}$-algebras in $G$-spaces. Using this operadic formulation, we establish incomplete versions of the usual structural properties of the equivariant stable category, notably the tom Dieck splitting. Our work is motivated in part by the examples arising from the equivariant units and Picard space fun…
For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it is $N_\infty$ and includes all norms). For G cyclic of order 2 or 3, we determine presentations of the object operad of $Q_G$ and conclude with a discussion of algebras over $Q_G$, which we call biased permutative equivariant categories.
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 ge…
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…
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…
