arXiv subjects
Publications and source records attributed to Thomas Willwacher.
We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined $L_\infty$-algebras of diagrams.
We discuss a graph complex formed by directed acyclic graphs with external legs. This complex comes in particular with a map to the ribbon graph complex computing the (compactly supported) cohomology of the moduli space of points $\mathcal M_{g,n}$, extending an earlier result of Merkulov-Willwacher. It is furthermore quasi-isomorphic to the hairy graph complex computing the weight 0 part of the compactly supported cohomology of $\mathcal{M}_{g,n}$ according to Chan-Galatius-Payne. Hence we can naturally connect the works Chan-Galatius-Payne and of Merkulov-Willwacher and the ribbon graph complex and obtain a fairly satisfying picture of how all the pieces and various graph complexes fit together, at least in weight zero.
We compute small rational models for configuration spaces of points on oriented surfaces, as right modules over the framed little disks operad. We do this by splitting these surfaces in unions of several handles. We first describe rational models for the configuration spaces of these handles as algebras in the category of right modules over the framed little disks operad. We then express the configuration spaces of the surface as an "iterated Hochschild complex" of these algebras with coefficients in the module given by configurations in a sphere with holes. Physically, our results may be interpreted as saying that the partition function of the Poisson-$σ$-model on closed surfaces has no quantum corrections, i.e., no terms coming from Feynman diagrams of positive loop order.
We define a simplicial enrichment on the category of differential graded Hopf cooperads (the category of dg Hopf cooperads for short). We prove that our simplicial enrichment satisfies, in part, the axioms of a simplicial model category structure on the category of dg Hopf cooperads. We use this simplicial model structure to define a model of mapping spaces in the category of dg Hopf cooperads and to upgrade results of the literature about the homotopy automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid structures. The rational homotopy theory of operads implies that the homotopy automorphism spaces of dg Hopf cooperads can be regarded as models for the homotopy automorphism spaces of the rationalization of operads in topological spaces (or in simplicial sets). We prove, as a main application, that the spaces of Maurer--Cartan forms on the Kontsevich graph complex Lie algebras are homotopy equivalent, in the category of simplicial monoids, to the homotopy automorphism spaces of the rationalization of the operads of little discs.
We study two colored operads of configurations of little $n$-disks in a unit $n$-disk, with the centers of the small disks of one color restricted to an $m$-plane, $m<n$. We compute the rational homotopy type of these \emph{extended Swiss Cheese operads} and show how they are connected to the rational homotopy types of the inclusion maps from the little $m$-disks to the little $n$-disks operad.
Given a closed manifold $M$. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincaré duality model of the manifold, and involves the configuration space of two points on $M$. We moreover, construct an $IBL_\infty$-structure on (a model of) cyclic chains on the cochain algebra of $M$, such that the natural comparison map to the $S^1$-equivariant loop space homology intertwines the Lie bialgebra structure on homology. The construction of the coproduct/cobracket depends on the perturbative partition function of a Chern-Simons type topological field theory. Furthermore, we give a construction for these string topology operations on the absolute loop space (not relative to constant loops) in case that $M$ carries a non-vanishing vector field and obtain a similar description. Finally, we show that the cobracket is sensitive to the manifold structure of $M$ beyond its homotopy type. More precisely, the action of ${\rm Diff}(M)$ does not (in general) factor through ${\rm aut}(M)$.
The mapping spaces of the $r$-truncated versions of the $E_n$ operads appear as the $r$-th stage of the Taylor tower for long embedding spaces. It has been shown that their rational homotopy groups can be expressed through graph homology in the limit $r\to \infty$. For finite $r$ only a part of the graph homology appears in the homotopy groups of the Taylor tower at this stage, possibly along with some additional unstable homotopy groups. In this paper we study the convergence properties in some more detail. In particular, we provide bounds for the stage $r$ at which the various graph homology classes start appearing, and we provide degree bounds for the unstable (i.e., vanishing as $r\to \infty$) terms in the homotopy groups of the tower.
We describe new graphical models of the framed little disks operads which exhibit large symmetry dg Lie algebras.
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 $LG_{\bullet}(\mathsf{Pois}_n^c)$ which we determine from the $n$-Poisson cooperad $\mathsf{Pois}_n^c$. We also prove that the morphisms $ι: \mathsf{D}_m\rightarrow\mathsf{D}_n$, 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 $n\geq 3$.
We study the subcategory of topological operads $P$ such that $P(0) = *$ (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of $k$-truncated unitary operads within the model category of all $k$-truncated operads, for any fixed arity bound $k\geq 1$, where a $k$-truncated operad denotes an operad that is defined up to arity $k$.
We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ and $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n)$ are simply connected and rationally equivalent. As application we determine the rational homotopy type of the deloopings of spaces of long embeddings. Some of the results hold also for mapping spaces $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$, $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n)$, $n-m\geq 2$, of the truncated little discs operads, which allows one to determine rationally the delooping of the Goodwillie-Weiss tower for the spaces of long embeddings.
We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the $E_n$ operads. We show that twisting with certain Maurer-Cartan elements trivializes their Lie algebra structure. The result can be used to understand the rational homotopy type of many connected components of the mapping spaces of $E_n$ operads.
We settle several questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. An important new technical ingredient introduced in this paper is an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on the graded commutative tensor algebra S(V) given in terms of polydifferential operators. Applying this functor to the prop LieB of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in in 1-1 correspondence with prop morphisms from the minimal resolution AssB_infty of the prop of associative bialgebras to the polydifferential prop DLieB_infty satisfying certain boundary conditions. We prove that the set of such formality morphisms (having an extra property of being Lie connected) is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator there is an associated Lie_infty quasi-isomorphism between the Lie_infty algebras controlling, respectively, deformations of the standard bialgebra structure in S(V) and deformations of any given Lie bialgebra structure in V. We study the deformation complex of an arbitrary universal formality morphism and show that it is quasi-isomorphic (up to one class corresponding to the standard rescaling automorphism of the properad LieB) to the full oriented graph complex studied earlier in \cite{Wi2}. This result gives a complete classification of the set of gauge equivalence classes of universal Lie connected formality maps --- it is a torsor over the Grothendieck-Teichmüller group GRT and can hence can be identified with the set of Drinfeld associators.
We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron). In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras. The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras.
We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many hairy graph cohomology classes out of non-hairy classes by a mechanism which we call the waterfall mechanism. By this mechanism we can construct many previously unknown classes and provide a first glimpse at the tentative global structure of the hairy graph cohomology.
Motivated by the universal obstruction to the deformation quantization of Poisson structures in infinite dimensions we introduce the notion of quantizable odd Lie bialgebra. The main result of the paper is a construction of a highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so called quantizable Poisson structures.
We show that the Hochschild-Pirashvili homology on any suspension admits the so called Hodge splitting. For a map between suspensions $f\colon ΣY\to ΣZ$, the induced map in the Hochschild-Pirashvili homology preserves this splitting if $f$ is a suspension. If $f$ is not a suspension, we show that the splitting is preserved only as a filtration. As a special case, we obtain that the Hochschild-Pirashvili homology on wedges of circles produces new representations of $Out(F_n)$ that do not factor in general through $GL(n,Z)$. The obtained representations are naturally filtered in such a way that the action on the graded quotients does factor through $GL(n,Z)$.
We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersion) of R^m in R^n as "decorated" graph complexes associated to certain representations of the outer automorphism groups of free groups. This interpretation gives rise to a natural spectral sequence, which allows us to shed some light on the structure of the hairy graph cohomology. We also explain briefly the connection to the deformation theory of the little discs operads and some conclusions that this brings.