We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutativ... more We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-point correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism. As another application, we consider the problem of quantization in the large N limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories.
HAL (Le Centre pour la Communication Scientifique Directe), Mar 4, 2020
The main purpose of this article is to develop an explicit derived deformation theory of algebrai... more The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of algebras (associative, commutative, Poisson...) or bialgebras (associative and coassociative, Lie, Frobenius...), that is algebraic structures parametrized by props. A central aspect is that we define and study moduli spaces of deformations of algebraic structures up to quasi-isomorphisms (and not only up to isomorphims or ∞-isotopies). To do so, we implement methods coming from derived algebraic geometry, by encapsulating these deformation theories as classifying (pre)stacks with good infinitesimal properties and derived formal groups. In particular, we prove that the Lie algebra describing the deformation theory of an object in a given ∞-category of dg algebras can be obtained equivalently as the tangent complex of loops on a derived quotient of this moduli space by the homotopy automorphims of this object. Moreover, we provide explicit formulae for such derived deformation problems of algebraic structures up to quasi-isomorphisms and relate them in a precise way to other standard deformation problems of algebraic structures. This relation is given by a fiber sequence of the associated dg-Lie algebras of their deformation complexes. Our results provide simultaneously a vast generalization of standard deformation theory of algebraic structures which is suitable (and needed) to set up algebraic deformation theory both at the ∞categorical level and at a higher level of generality than algebras over operads. In addition, we study a general criterion to compare formal moduli problems of different algebraic structures and apply our formalism to En-algebras and bialgebras. 1 DERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES Contents Introduction 0.1. Motivations 0.2. Main results 0.3. Further applications and perspectives Notations and conventions 1. Recollections 1.1. Symmetric monoidal categories over a base category 1.2. Props, properads and their algebras 1.3. Algebras and coalgebras over operads 1.4. Homotopy algebras 2. Derived deformation theory of algebraic structures 2.1. A brief preliminary about cdgas 2.2. Relative categories versus ∞-categories 2.3. Formal moduli problems and (homotopy) Lie algebras 2.4. Moduli spaces of algebraic structures and their formal moduli problems 3. Derived formal groups of algebraic structures and associated formal moduli problems 3.1. Generalities on derived formal groups 3.2. Derived prestack group and their tangent L ∞-algebras 3.3. Prestacks of algebras and derived groups of homotopy automorphisms 3.4. The fiber sequence of deformation theories 3.5. Equivalent deformation theories for equivalent (pre)stacks of algebras 4. The tangent Lie algebra of homotopy automorphims 4.1. Homotopy representations of L ∞-algebras and a relevant application 4.2. ∞-actions in infinitesimally cohesive presheaves 4.3. The Lie algebra of homotopy automorphisms as a semi-direct product 4.4. The operad of differentials 4.5. Computing the tangent Lie algebra of homotopy automorphims 5. Examples 5.1. Deformations of E n-algebras 5.2. Deformation complexes of P ois n-algebras 5.3. Bialgebras 6. Concluding remarks and perspectives 6.1. Algebras over operads in vector spaces 6.2. Differential graded algebras over operads 6.3. Algebras over properads References
HAL (Le Centre pour la Communication Scientifique Directe), Feb 21, 2018
A first goal of this paper is to precisely relate the homotopy theories of bialgebras and E 2-alg... more A first goal of this paper is to precisely relate the homotopy theories of bialgebras and E 2-algebras. For this, we construct a conservative and fully faithful ∞-functor from pointed conilpotent homotopy bialgebras to augmented E 2-algebras which consists in an appropriate "cobar" construction. Then we prove that the (derived) formal moduli problem of homotopy bialgebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of E 2-algebra structures on this "cobar" construction. We show consequently that the E 3-algebra structure on the higher Hochschild complex of this cobar construction, given by the solution to the higher Deligne conjecture, controls the deformation theory of this bialgebra. This implies the existence of an E 3-structure on the deformation complex of a dg bialgebra, solving a long-standing conjecture of Gerstenhaber-Schack. On this basis we solve a long-standing conjecture of Kontsevich, by proving the E 3-formality of the deformation complex of the symmetric bialgebra. This provides as a corollary a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which extends to homotopy dg Lie bialgebras and is independent from the choice of an associator. Along the way, we establish new general results of independent interest about the deformation theory of algebraic structures, which shed a new light on various deformation complexes and cohomology theories studied in the literature. Contents 35 5. Bialgebras versus E 2-algebras 48 6. Identification of Deformation complexes with higher Hochschild, Gerstenhaber-Schack and Tamarkin complexes of P ois n-algebras 54 7. The E 3-formality Theorem 67 8. Etingof-Kazdhan deformation quantization 75 References 77
We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are comp... more We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups. Résumé. Dans cet article, onétend le formalisme des intégrales itérées de Chen aux complexes de Hochschild supérieurs. Ces derniers sont des complexes de (co)chaînes modelés sur un espace (simplicial) de la même manière que le complexe de Hochschild classique est modelé sur le cercle. On en déduit des modèles algébriques pour les espaces fonctionnels que l'on utilise pouŕ etudier le produit surfacique. Ce produit, défini sur l'homologie des espaces de fonctions continues de surfaces (de genre quelconque) dans une variété, est un analogue du produit de Chas-Sullivan sur les espaces de lacets en topologie des cordes. En particulier, on en déduit que le produit surfacique est un invariant homotopique. On démontreégalement un théorème du type Hochschild-Kostant-Rosenberg pour les complexes de Hochschild modelés sur les surfaces qui permet d'obtenir des formules explicites pour le produit surfacique des sphères de dimension impaires ainsi que pour les groupes de Lie.
The Helly number of a family of sets with empty intersection is the size of its largest inclusion... more The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G ⊆ F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d Γ + 1), where d Γ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular d R d = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known.
Persistent homology has been recently studied with the tools of sheaf theory in the derived setti... more Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18] after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D b Rc (k R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of D b Rc (k R) and provide some explicit examples of computation of the convolution distance.
Persistent homology has been recently studied with the tools of sheaf theory in the derived setti... more Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18] after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D b Rc (k R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of D b Rc (k R) and provide some explicit examples of computation of the convolution distance.
Proceedings of the American Mathematical Society, 2006
Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and let g 1 be the space of multivecto... more Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and let g 1 be the space of multivector fields on R n. In this paper we prove that given any G ∞-structure (i.e. Gerstenhaber algebra up to homotopy structure) on g 2 , and any C ∞-morphism ϕ (i.e. morphism of a commutative, associative algebra up to homotopy) between g 1 and g 2 , there exists a G ∞-morphism Φ between g 1 and g 2 that restricts to ϕ. We also show that any L ∞-morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G ∞-morphism, using Tamarkin's method for any G ∞-structure on g 2. We also show that any two of such G ∞-morphisms are homotopic.
We introduce a Chas-Sullivan type string product on the homology groups of the inertia stack of a... more We introduce a Chas-Sullivan type string product on the homology groups of the inertia stack of an oriented dierential
We introduce a string coproduct structure on the homology groups of the inertia stack ⁄X and prov... more We introduce a string coproduct structure on the homology groups of the inertia stack ⁄X and prove that H†(⁄X) with the string product and coproduct becomes a (not necessarily unital or counital) Frobenius algebra. As an example, we explicitly describe the Frobenius algebra structure in the case when X is [⁄=G] for a connected compact Lie group G. To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathema... more This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathematics and aims to present the theory of higher Hochschild (co)homology and its application to higher string topology. There is an emphasis on explicit combinatorial models provided by simplicial sets to describe derived structures carried or described by Higher Hochschild (co)homology functors. It contains detailed proofs of results stated in a previous note as well as some new results. One of the main result is a proof that string topology for higher spheres inherits a Hodge filtration compatible with an (homotopy) En+1-algebra structure on the chains for d-connected Poincaré duality spaces. We also prove that the En-centralizer of maps of commutative (dg-)algebras are equipped with a Hodge decomposition and a compatible structure of framed En-algebras. We also study Hodge decompositions suspensions and products by spheres, both as derived functors and combinatorially.
Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale... more Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and g 1 be the space of multivector fi... more Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and g 1 be the space of multivector fields on R n. In this paper we prove that given any G ∞-structure (i.e. Gerstenhaber algebra up to homotopy structure) on g 2 , and any C ∞-morphism ϕ (i.e. morphism of commutative, associative algebra up to homotopy) between g 1 and g 2 , there exists a G ∞morphism Φ between g 1 and g 2 that restricts to ϕ. We also show that any L ∞-morphism (i.e. morphism of Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G ∞-morphism, using Tamarkin's method for any G ∞-structure on g 2. We also show that any two of such G ∞-morphisms are homotopic. 0-Introduction Let M be a differential manifold and g 2 = (C • (A, A), b) be the Hochschild cochain complex on A = C ∞ (M). The classical Hochschild-Kostant-Rosenberg theorem states that the cohomology of g 2 is the graded Lie algebra g 1 = Γ(M, ∧ • T M) of multivector fields on M. There is also a graded Lie algebra structure on g 2 given by the Gerstenhaber bracket. In particular g 1 and g 2 are also Lie algebras up to homotopy (L ∞-algebra for short). In the case M = R n , using different methods, Kontsevich ([Ko1] and [Ko2]) and Tamarkin ([Ta]) have proved the existence of Lie homomorphisms "up to homotopy" (L ∞-morphisms) from g 1 to g 2. Kontsevich's proof uses graph complex and is related to multizeta functions whereas Tamarkin's construction uses the existence of Drinfeld's associators. In fact Tamarkin's L ∞-morphism comes from the restriction of a Gerstenhaber algebra up to homotopy homomorphism (G ∞-morphism) from g 1 to g 2. The G ∞-algebra structure on g 1 is induced by its classical Gerstenhaber algebra structure and a far less trivial G ∞-structure on g 2 was proved to exist by Tamarkin [Ta] and relies on a Drinfeld's associator. Tamarkin's G ∞-morphism also restricts into a commutative, associative up to homotopy morphism (C ∞-morphism for short). The C ∞-structure on g 2 (given by
Let G be a topological group acting on a space X. We construct a family of Steenrod's ∪ i-product... more Let G be a topological group acting on a space X. We construct a family of Steenrod's ∪ i-product [Ann. of Math. (2) 48 (1947) 290] on the Bredon-Illman cochain complex of X [Quart. J. Math. Oxford Ser. (2) 47 (1996) 199]. As corollaries, we get the existence of Steenrod squares on Bredon-Illman cohomology with appropriate coefficients as well as the triviality of the Gerstenhaber bracket induced by the braces at the cochain level [G. Mukherjee, N. Pandey, Homotopy G-algebra structure on Bredon-Illman cochain complex, Preprint].
We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutativ... more We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-point correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism. As another application, we consider the problem of quantization in the large N limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 19, 2019
Bats-toi, signe et persiste"-France Gall A. We define and study several new interleaving distance... more Bats-toi, signe et persiste"-France Gall A. We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In particular, we prove that there exists a persistent A ∞-structure associated to data sets and and we define the associated distance. We prove the stability of these new distances for Čech or Vietoris Rips complexes with respect
Persistent homology has been recently studied with the tools of sheaf theory in the derived setti... more Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18a] after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D b Rc (k R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of D b Rc (k R) and provide some explicit examples of computation of the convolution distance.
In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulma... more In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module A − → 1. The cohomology of the Lie 2-groups corresponding to the universal crossed modules G − → Aut (G) and G − → Aut + (G) is the abutment of a spectral sequence involving the cohomology of GL(n, Z) and SL(n, Z). When the dimension of the center of G is less than 3, we compute explicitly these cohomology groups.
This paper is a continuations of the project initiated in [BGNX]. We construct string operations ... more This paper is a continuations of the project initiated in [BGNX]. We construct string operations on the S 1-equivariant homology of the (free) loop space LX of an oriented differentiable stack X and show that H S 1 * +dim X−2 (LX) is a graded Lie algebra. In the particular case where X is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a general machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack and further geometric if X is geometric. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on. Contents
We use factorization homology and higher Hochschild (co)chains to study various problems in algeb... more We use factorization homology and higher Hochschild (co)chains to study various problems in algebraic topology and homotopical algebra, notably brane topology, centralizers of En-algebras maps and iterated bar constructions. In particular, we obtain an E n+1-algebra model on the shifted integral chains C •+m (M ap(S n , M)) of the mapping space of the n-sphere into an m-dimensional orientable closed manifold M. We construct and use E∞-Poincaré duality to identify the higher Hochschild cochains, modeled over the n-sphere, with the chains on the above mapping space, and then relate the Hochschild cochains to the deformation complex of the E∞-algebra C * (M), thought of as an En-algebra. We invoke (and prove) the higher Deligne conjecture to furnish En-Hochschild cohomology, and all that is naturally equivalent to it, with an E n+1-algebra structure and further prove that this construction recovers the sphere product. In fact, our approach to the Deligne conjecture is based on an explicit description of the En-centralizers of a map of E∞algebras f : A → B by relating it to the algebraic structure on Hochschild cochains modeled over spheres, which is of independent interest and explicit. More generally, we give a factorization algebra model/description of the centralizer of any En-algebra map and a solution of Deligne conjecture. We also apply similar ideas to the iterated bar construction. We obtain factorization algebra models for (iterated) bar construction of augmented Em-algebras together with their En-coalgebras and E m−n-algebra structures, and discuss some of its features. For E∞-algebras we obtain a higher Hochschild chain model, which is an En-coalgebra. In particular, considering the E∞-algebra structure of an n-connected topological space Y , we obtain a higher Hochschild cochain model of the natural En-algebra structure of the chains of the iterated loop space C * (Ω n Y).
We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutativ... more We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-point correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism. As another application, we consider the problem of quantization in the large N limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories.
HAL (Le Centre pour la Communication Scientifique Directe), Mar 4, 2020
The main purpose of this article is to develop an explicit derived deformation theory of algebrai... more The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of algebras (associative, commutative, Poisson...) or bialgebras (associative and coassociative, Lie, Frobenius...), that is algebraic structures parametrized by props. A central aspect is that we define and study moduli spaces of deformations of algebraic structures up to quasi-isomorphisms (and not only up to isomorphims or ∞-isotopies). To do so, we implement methods coming from derived algebraic geometry, by encapsulating these deformation theories as classifying (pre)stacks with good infinitesimal properties and derived formal groups. In particular, we prove that the Lie algebra describing the deformation theory of an object in a given ∞-category of dg algebras can be obtained equivalently as the tangent complex of loops on a derived quotient of this moduli space by the homotopy automorphims of this object. Moreover, we provide explicit formulae for such derived deformation problems of algebraic structures up to quasi-isomorphisms and relate them in a precise way to other standard deformation problems of algebraic structures. This relation is given by a fiber sequence of the associated dg-Lie algebras of their deformation complexes. Our results provide simultaneously a vast generalization of standard deformation theory of algebraic structures which is suitable (and needed) to set up algebraic deformation theory both at the ∞categorical level and at a higher level of generality than algebras over operads. In addition, we study a general criterion to compare formal moduli problems of different algebraic structures and apply our formalism to En-algebras and bialgebras. 1 DERIVED DEFORMATION THEORY OF ALGEBRAIC STRUCTURES Contents Introduction 0.1. Motivations 0.2. Main results 0.3. Further applications and perspectives Notations and conventions 1. Recollections 1.1. Symmetric monoidal categories over a base category 1.2. Props, properads and their algebras 1.3. Algebras and coalgebras over operads 1.4. Homotopy algebras 2. Derived deformation theory of algebraic structures 2.1. A brief preliminary about cdgas 2.2. Relative categories versus ∞-categories 2.3. Formal moduli problems and (homotopy) Lie algebras 2.4. Moduli spaces of algebraic structures and their formal moduli problems 3. Derived formal groups of algebraic structures and associated formal moduli problems 3.1. Generalities on derived formal groups 3.2. Derived prestack group and their tangent L ∞-algebras 3.3. Prestacks of algebras and derived groups of homotopy automorphisms 3.4. The fiber sequence of deformation theories 3.5. Equivalent deformation theories for equivalent (pre)stacks of algebras 4. The tangent Lie algebra of homotopy automorphims 4.1. Homotopy representations of L ∞-algebras and a relevant application 4.2. ∞-actions in infinitesimally cohesive presheaves 4.3. The Lie algebra of homotopy automorphisms as a semi-direct product 4.4. The operad of differentials 4.5. Computing the tangent Lie algebra of homotopy automorphims 5. Examples 5.1. Deformations of E n-algebras 5.2. Deformation complexes of P ois n-algebras 5.3. Bialgebras 6. Concluding remarks and perspectives 6.1. Algebras over operads in vector spaces 6.2. Differential graded algebras over operads 6.3. Algebras over properads References
HAL (Le Centre pour la Communication Scientifique Directe), Feb 21, 2018
A first goal of this paper is to precisely relate the homotopy theories of bialgebras and E 2-alg... more A first goal of this paper is to precisely relate the homotopy theories of bialgebras and E 2-algebras. For this, we construct a conservative and fully faithful ∞-functor from pointed conilpotent homotopy bialgebras to augmented E 2-algebras which consists in an appropriate "cobar" construction. Then we prove that the (derived) formal moduli problem of homotopy bialgebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of E 2-algebra structures on this "cobar" construction. We show consequently that the E 3-algebra structure on the higher Hochschild complex of this cobar construction, given by the solution to the higher Deligne conjecture, controls the deformation theory of this bialgebra. This implies the existence of an E 3-structure on the deformation complex of a dg bialgebra, solving a long-standing conjecture of Gerstenhaber-Schack. On this basis we solve a long-standing conjecture of Kontsevich, by proving the E 3-formality of the deformation complex of the symmetric bialgebra. This provides as a corollary a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which extends to homotopy dg Lie bialgebras and is independent from the choice of an associator. Along the way, we establish new general results of independent interest about the deformation theory of algebraic structures, which shed a new light on various deformation complexes and cohomology theories studied in the literature. Contents 35 5. Bialgebras versus E 2-algebras 48 6. Identification of Deformation complexes with higher Hochschild, Gerstenhaber-Schack and Tamarkin complexes of P ois n-algebras 54 7. The E 3-formality Theorem 67 8. Etingof-Kazdhan deformation quantization 75 References 77
We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are comp... more We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups. Résumé. Dans cet article, onétend le formalisme des intégrales itérées de Chen aux complexes de Hochschild supérieurs. Ces derniers sont des complexes de (co)chaînes modelés sur un espace (simplicial) de la même manière que le complexe de Hochschild classique est modelé sur le cercle. On en déduit des modèles algébriques pour les espaces fonctionnels que l'on utilise pouŕ etudier le produit surfacique. Ce produit, défini sur l'homologie des espaces de fonctions continues de surfaces (de genre quelconque) dans une variété, est un analogue du produit de Chas-Sullivan sur les espaces de lacets en topologie des cordes. En particulier, on en déduit que le produit surfacique est un invariant homotopique. On démontreégalement un théorème du type Hochschild-Kostant-Rosenberg pour les complexes de Hochschild modelés sur les surfaces qui permet d'obtenir des formules explicites pour le produit surfacique des sphères de dimension impaires ainsi que pour les groupes de Lie.
The Helly number of a family of sets with empty intersection is the size of its largest inclusion... more The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G ⊆ F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d Γ + 1), where d Γ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular d R d = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known.
Persistent homology has been recently studied with the tools of sheaf theory in the derived setti... more Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18] after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D b Rc (k R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of D b Rc (k R) and provide some explicit examples of computation of the convolution distance.
Persistent homology has been recently studied with the tools of sheaf theory in the derived setti... more Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18] after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D b Rc (k R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of D b Rc (k R) and provide some explicit examples of computation of the convolution distance.
Proceedings of the American Mathematical Society, 2006
Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and let g 1 be the space of multivecto... more Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and let g 1 be the space of multivector fields on R n. In this paper we prove that given any G ∞-structure (i.e. Gerstenhaber algebra up to homotopy structure) on g 2 , and any C ∞-morphism ϕ (i.e. morphism of a commutative, associative algebra up to homotopy) between g 1 and g 2 , there exists a G ∞-morphism Φ between g 1 and g 2 that restricts to ϕ. We also show that any L ∞-morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G ∞-morphism, using Tamarkin's method for any G ∞-structure on g 2. We also show that any two of such G ∞-morphisms are homotopic.
We introduce a Chas-Sullivan type string product on the homology groups of the inertia stack of a... more We introduce a Chas-Sullivan type string product on the homology groups of the inertia stack of an oriented dierential
We introduce a string coproduct structure on the homology groups of the inertia stack ⁄X and prov... more We introduce a string coproduct structure on the homology groups of the inertia stack ⁄X and prove that H†(⁄X) with the string product and coproduct becomes a (not necessarily unital or counital) Frobenius algebra. As an example, we explicitly describe the Frobenius algebra structure in the case when X is [⁄=G] for a connected compact Lie group G. To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathema... more This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathematics and aims to present the theory of higher Hochschild (co)homology and its application to higher string topology. There is an emphasis on explicit combinatorial models provided by simplicial sets to describe derived structures carried or described by Higher Hochschild (co)homology functors. It contains detailed proofs of results stated in a previous note as well as some new results. One of the main result is a proof that string topology for higher spheres inherits a Hodge filtration compatible with an (homotopy) En+1-algebra structure on the chains for d-connected Poincaré duality spaces. We also prove that the En-centralizer of maps of commutative (dg-)algebras are equipped with a Hodge decomposition and a compatible structure of framed En-algebras. We also study Hodge decompositions suspensions and products by spheres, both as derived functors and combinatorially.
Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale... more Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and g 1 be the space of multivector fi... more Let g 2 be the Hochschild complex of cochains on C ∞ (R n) and g 1 be the space of multivector fields on R n. In this paper we prove that given any G ∞-structure (i.e. Gerstenhaber algebra up to homotopy structure) on g 2 , and any C ∞-morphism ϕ (i.e. morphism of commutative, associative algebra up to homotopy) between g 1 and g 2 , there exists a G ∞morphism Φ between g 1 and g 2 that restricts to ϕ. We also show that any L ∞-morphism (i.e. morphism of Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a G ∞-morphism, using Tamarkin's method for any G ∞-structure on g 2. We also show that any two of such G ∞-morphisms are homotopic. 0-Introduction Let M be a differential manifold and g 2 = (C • (A, A), b) be the Hochschild cochain complex on A = C ∞ (M). The classical Hochschild-Kostant-Rosenberg theorem states that the cohomology of g 2 is the graded Lie algebra g 1 = Γ(M, ∧ • T M) of multivector fields on M. There is also a graded Lie algebra structure on g 2 given by the Gerstenhaber bracket. In particular g 1 and g 2 are also Lie algebras up to homotopy (L ∞-algebra for short). In the case M = R n , using different methods, Kontsevich ([Ko1] and [Ko2]) and Tamarkin ([Ta]) have proved the existence of Lie homomorphisms "up to homotopy" (L ∞-morphisms) from g 1 to g 2. Kontsevich's proof uses graph complex and is related to multizeta functions whereas Tamarkin's construction uses the existence of Drinfeld's associators. In fact Tamarkin's L ∞-morphism comes from the restriction of a Gerstenhaber algebra up to homotopy homomorphism (G ∞-morphism) from g 1 to g 2. The G ∞-algebra structure on g 1 is induced by its classical Gerstenhaber algebra structure and a far less trivial G ∞-structure on g 2 was proved to exist by Tamarkin [Ta] and relies on a Drinfeld's associator. Tamarkin's G ∞-morphism also restricts into a commutative, associative up to homotopy morphism (C ∞-morphism for short). The C ∞-structure on g 2 (given by
Let G be a topological group acting on a space X. We construct a family of Steenrod's ∪ i-product... more Let G be a topological group acting on a space X. We construct a family of Steenrod's ∪ i-product [Ann. of Math. (2) 48 (1947) 290] on the Bredon-Illman cochain complex of X [Quart. J. Math. Oxford Ser. (2) 47 (1996) 199]. As corollaries, we get the existence of Steenrod squares on Bredon-Illman cohomology with appropriate coefficients as well as the triviality of the Gerstenhaber bracket induced by the braces at the cochain level [G. Mukherjee, N. Pandey, Homotopy G-algebra structure on Bredon-Illman cochain complex, Preprint].
We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutativ... more We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-point correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism. As another application, we consider the problem of quantization in the large N limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 19, 2019
Bats-toi, signe et persiste"-France Gall A. We define and study several new interleaving distance... more Bats-toi, signe et persiste"-France Gall A. We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In particular, we prove that there exists a persistent A ∞-structure associated to data sets and and we define the associated distance. We prove the stability of these new distances for Čech or Vietoris Rips complexes with respect
Persistent homology has been recently studied with the tools of sheaf theory in the derived setti... more Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira [KS18a] after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D b Rc (k R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, give a precise description of connected components of D b Rc (k R) and provide some explicit examples of computation of the convolution distance.
In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulma... more In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module A − → 1. The cohomology of the Lie 2-groups corresponding to the universal crossed modules G − → Aut (G) and G − → Aut + (G) is the abutment of a spectral sequence involving the cohomology of GL(n, Z) and SL(n, Z). When the dimension of the center of G is less than 3, we compute explicitly these cohomology groups.
This paper is a continuations of the project initiated in [BGNX]. We construct string operations ... more This paper is a continuations of the project initiated in [BGNX]. We construct string operations on the S 1-equivariant homology of the (free) loop space LX of an oriented differentiable stack X and show that H S 1 * +dim X−2 (LX) is a graded Lie algebra. In the particular case where X is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a general machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack and further geometric if X is geometric. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on. Contents
We use factorization homology and higher Hochschild (co)chains to study various problems in algeb... more We use factorization homology and higher Hochschild (co)chains to study various problems in algebraic topology and homotopical algebra, notably brane topology, centralizers of En-algebras maps and iterated bar constructions. In particular, we obtain an E n+1-algebra model on the shifted integral chains C •+m (M ap(S n , M)) of the mapping space of the n-sphere into an m-dimensional orientable closed manifold M. We construct and use E∞-Poincaré duality to identify the higher Hochschild cochains, modeled over the n-sphere, with the chains on the above mapping space, and then relate the Hochschild cochains to the deformation complex of the E∞-algebra C * (M), thought of as an En-algebra. We invoke (and prove) the higher Deligne conjecture to furnish En-Hochschild cohomology, and all that is naturally equivalent to it, with an E n+1-algebra structure and further prove that this construction recovers the sphere product. In fact, our approach to the Deligne conjecture is based on an explicit description of the En-centralizers of a map of E∞algebras f : A → B by relating it to the algebraic structure on Hochschild cochains modeled over spheres, which is of independent interest and explicit. More generally, we give a factorization algebra model/description of the centralizer of any En-algebra map and a solution of Deligne conjecture. We also apply similar ideas to the iterated bar construction. We obtain factorization algebra models for (iterated) bar construction of augmented Em-algebras together with their En-coalgebras and E m−n-algebra structures, and discuss some of its features. For E∞-algebras we obtain a higher Hochschild chain model, which is an En-coalgebra. In particular, considering the E∞-algebra structure of an n-connected topological space Y , we obtain a higher Hochschild cochain model of the natural En-algebra structure of the chains of the iterated loop space C * (Ω n Y).
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