Related papers: Cyclic operad formality for compactified moduli sp…
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
We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new…
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…
It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero. In this paper we…
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…
We characterize cyclic algebras over the associative and the framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a coherent way, i.e. up to coherent isomorphism. When the symmetric…
In this work we study the homotopy type of multipath complexes of bidirectional path graphs and polygons, motivated by works of Vre\'cica and \v{Z}ivaljevi\'c on cycle-free chessboard complexes (that is, multipath complexes of complete…
S. Axelrod and I.M. Singer constructed a compactification of the configuration space of distinct points in a Riemannian manifold V. A similar compactification for the moduli space of configurations of distinct points in the plane (mod the…
The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of…
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…
A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial…
We construct a new (cyclic) operad of `mosaics' defined by polygons with marked diagonals. Its underlying (aspherical) spaces are the sets of real points of the moduli space of punctured Riemann spheres, which are naturally tiled by…
The real locus of the moduli space of stable genus-zero curves with marked points, $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$, is known to be a smooth manifold and is the Eilenberg-MacLane spaces for the so-called pure Cactus groups.…
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…
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)…
We prove that the tree-like Deligne-Mumford operad is a homotopical model for the trivialization of the circle in the higher-genus framed little discs operad. Our proof is based on a geometric argument involving nodal annuli. We use as a…
Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of…
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions…
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…
In the classical Batalin--Vilkovisky formalism, the BV operator $\Delta$ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is…