Publications
Résumé: On démontre un théorème de formalité d'algèbres de Gerstenhaber à homotopie près pour le complexe de Hochschild d'une quantification d'une algèbre de Poisson. English detailed abstractAbstract. Let M be a differential manifold. Using different methods, Kontsevich and Tamarkin have proved a formality theorem, which states the existence of a Lie homomorphism “up to homotopy” between the Lie algebra of Hochschild cochains on C∞(M), the smooth functions on M, and its cohomology (Γ(M, ΛTM), [−,−]), the Schouten algebra of polyvector fields. Suppose M is a Poisson manifold equipped with a Poisson tensor π; then one can deduce from this theorem the existence of a star product ⋆ on C∞(M). In this paper we prove that the formality theorem can be extended to a Lie (and even Gerstenhaber) homomorphism “up to homotopy” between the Lie (resp. Gerstenhaber “up to homotopy”) algebra of Hochschild cochains on the deformed algebra (C∞(M)[[h]], ⋆) and the Poisson complex (Γ(M, ΛTM)[[h]], [hπ,−]). We first recall Tamarkin’s proof and see how the formality maps can be deduced from Etingof-Kazhdan’s theorem using only homotopies formulas. The formality theorem for Poisson manifolds will then follow. Hide abstract
Résumé: On donne une formule explicite pour le caractère de Chern ch reliant la K-théorie algébrique d'un anneau à son homologie cyclique négative et on calcule l'image de certains symboles par ch English detailed abstractAbstract. We give an explicit formula for the natural map ϒ: H(G) → HC⁻(Z[G]), from the homology of a group to the negative cyclic homology of its group algebra, inducing the universal (Goodwilie-Jones) Chern character from algebraic K-theory to Negative cyclic homology. In degree 2, we deduce an explicit formula from Milnor's K_2(A) K-theory to HC⁻_2(A) (A is any ring). We compute formula for the Chern character of Steinberg, Loday as well as Dennis-Stein symbols. Finally we deduce from the previous results a new proof of the compatibility of the Chern character with products. Hide abstract
Résumé: On étudie les notions d'algèbres de Gerstenhaber à homotopie près et d'homologie des algèbres de Gerstenhaber suivant la théorie des opérades. English detailed abstractAbstract. We give an explicit description of homotopy Gerstenhaber algebras defined as algebras over a minimal model of the operad of Gerstenhaber algebras (using Koszul duality). We also give a description of the natural complex computing the homology of Gerstenhaber algebras and gives a spectral sequence to compute this cohomology. We also explain how to adapt the above constructions to the case of Poisson algebras or more generally ot the case of any "Poisson" n-algebra. Hide abstract
Résumé: On construit des ièmes produits ∪i de Steenrod sur le complexe de Bredon-Illman des espaces G-équivariants et on en déduit l'existence des puissances de Steenrod sur cette cohomologie. English detailed abstractAbstract. Let G be a topological group acting on a space X. We construct a family of Steenrod's ∪i -product on the Bredon- Illman cochain complex of X. 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.
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Résumé: On montre que pour toute structure d'algèbre de Gerstenhaber à homotopie près (G∞) sur les multivecteurs de Rn, toute formalité C∞ se relève en une formalité G∞ et que toute formalité L∞ peut être déformée en une G∞. English detailed abstractAbstract. Let g2 be the Hochschild complex of cochains on the smooth functions on the n-dimensional euclidean space and g1 be the space of multivector fields on the same space . Tamarkin has proved that the dg Lie algebra structure on g2 can be lifted to a G∞-structure (i.e. Gerstenhaber algebra up to homotopy structure) and that there is a G∞-quasi-isomorphism in between g1 and g2 (where g1 is equipped with its usual Gerstenhaber algebra structure), in other words that g2 is formal, inducing a quantization of R^n. In this paper we prove that given any G∞-structure on g2 , and any morphism f: g1 → g2 of associative, commutative up to homotopy algebras between g1 and g2 , there exists a G∞-morphism F between g1 and g2 that restricts to f. We also show that any Lie algebra up to homotopy morphism, in particular the one constructed by Kontsevich, can be deformed into a G∞-morphism, using Tamarkin's method for any G∞ -structure on g2 . We also show that any two of such G∞-morphisms are homotopic in a certain sense.
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Résumé: On munit les groupes d’homologie du champ des lacets libres d’un champ orienté d’un produit et d’un coproduit induisant une structure d’algèbre de Frobenius. De plus, l’homologie en degrés décalés H•(LX) = H•+d(LX) est une algèbre BV. La version détaillée de cet article est incluse dans l'article String Topology for Stacks English abstractAbstract. We explain that the homology groups of the free loop stack of an oriented stack are equipped with a canonical loop product and coproduct, which makes it into a Frobenius algebra. Moreover, the shifted homology H•(LX) = H•+d(LX) admits a BV algebra structure. These results are taken from the Preprint String Topology for Stacks.
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Résumé: A la manière de Pirashvili, on peut associer une cohomologie de Hochschild supérieure associée aux sphères de dimension d, définie pour toute algèbre commutative A et module M. Lorsque M = A, cette cohomologie est munie d’un produit gradué commutatif, d’un crochet de Lie de degré d et d’opérations d’Adams. Ces structures sont compatibles entre elles et sont reliées à la topologie des Branes. Une bonne partie des constructions techniques à ce papier sont détaillées dans mes articles avec Tradler et Zeinalian. Une version détaillée et améliorée des résultats de cet article concernant les structures d'algèbres sur les opérades des petits disques et les applications à la topologie des Branes est incluse dans mon mémoire d'Habilitation et le preprint sur la "Brane Topology and Centralizers"(voir ci-dessous). Les résultats sur les opérations d'Adams seront probablement détaillés (et étendus ?) ailleurs lorsque j'en ai l'énergie... English abstractAbstract. Following ideas of Pirashvili, we define higher order Hochschild cohomology over spheres Sd defined for any commutative algebra A and module M. When M = A, we prove that this cohomology is equipped with graded commutative algebra and degree d Lie algebra structures as well as with Adams operations. All operations are compatible in a suitable sense. These structures are related to Brane topology. Many results in these note have been included and expanded in my joint work with Tradler-Zeinalian as well as mémoire d'Habilitation.
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Résumé: On généralise les constructions des opérations d'Adams et la décomposition de Hodge de la (co)homologie de Hochschild des algèbres commutatives à leur analogue à homotopie près (les C∞-algèbres). On applique cette étude en topologie des cordes, pour obtenir une décomposition de Hodge compatible à une structure BV sur l'homologie de l'espace des lacets libres d'un espace à dualité de Poincaré. On étudie également la (co)homologie de Harrison et on obtient une généralisation homotopique de la suite exacte de Jacobi-Zariski. En particulier, on étudie des notions relatives des C∞-algèbres (c'est à dire lorsque l'anneau de base est remplacé par une C∞-algèbre) et de leurs (co)homologies. English detailed abstractAbstract. For commutative algebras (and not merely associative ones) Gerstenhaber-Schak [GS1] and Loday [Lo1] have shown that the Hochschild (co)homology groups (with value in symmetric bimodules) are equipped with an additional structure: the Adams (also called λ)-operations, which induce a Hodge decomposition. In this paper we study Hochschild (co)homology of commutative and associative up to homotopy algebras with coefficient in a (new) homotopy analogue of symmetric bimodules. We prove that Hochschild (co)homology is equipped with λ-operations and Hodge decomposition generalizing the results in [GS1] and [Lo1] for strict algebras. The main application is concerned with string topology: we obtain a Hodge decomposition compatible with a non-trivial BV-structure on the homology H•(LX) of the free loop space of a triangulated Poincaré-duality space. Harrison (co)homology of commutative and associative up to homotopy algebras (C∞-algebra for short) can be defined similarly and is related to the weight 1 piece of the Hodge decomposition. We study (an homotopy generalization of) Jacobi-Zariski exact sequence for this theory in characteristic zero. In particular, we define (co)homology of relative A∞-algebras, i.e., A∞-algebras with a C∞-algebra playing the role of the ground ring. We also give a relation between the Hodge decomposition and homotopy Poisson-algebras cohomology.
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Résumé: On étudie la cohomologie des algébroides de Courant et on construit une suite spectrale qui la relie à la cohomologie naive des algébroides de Courant. On en déduit que ces cohomologies coincident pour les algébroides transitifs, et, pour le cas plus général des algébroides à base scindée, on montre que la cohomologie naive et la connaissance d'un homomorphisme de transgression permet de calculer complétement la cohomologie de l'algébroide de Courant. English detailed abstractAbstract. In this paper we study the cohomology H•(E) of a Courant algebroid E. This cohomology was defined by Roytenberg using a symplectic realization of the Courant algebroid. A less involved construction is the naive cohomology Hnaive(E) defined by Stiénon and Xu. We prove that if E is a transitive algebroid, both cohomology coincides as was conjectured by Stiénon and Xu. This result is proved using a spectral sequence converging to H•(E). The second page of the spectral sequence contains the naive cohomology and allows computation of the cohomology of E for more general Courant algebroids than transitive ones. Indeed, if E is with split base, we prove that there exists a natural transgression homomorphism T3 (with image in the degree 3 part of Hnaive(E)) which, together with the knoweldge of Hnaive(E), gives the full cohomology of E. In many cases, the transgression homomorphism can be determined explicitly. For instance, for generalized exact Courant algebroids, we give an explicit formula for T3 depending only on the Severa characteristic class of E.
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Résumé: On étudie la cohomologie des 2-groupes de Lie. English detailed abstractAbstract. We study the cohomology of (strict) Lie 2-groups, in particular those arising from gerbes or the string 2-group. In particular we obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module A → 1. For the two universal 2-groups (given by the crossed modules G →Aut (G) and G → Aut+(G)) associated to a Lie group G, we prove that the cohomology H(G → Aut+(G)) and H(G → Aut(G)) are 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. We also compute the cohomology of the Lie 2-group corresponding to a crossed module G→ H for which Ker(i) is compact and Coker(i) is connected, simply connected and compact and apply the result to the string 2-group.
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Résumé: Dans cet article, on étend le formalisme des intégrales itérées de Chen aux complexes de Hochschild supérieurs. On en déduit des modèles algébriques pour les espaces fonctionnels que l’on utilise pour étudier le produit surfacique, un analogue 2-dimensionnel du produit de Chas-Sullivan en topologie des cordes. En particulier, on en déduit que le produit surfacique est un invariant homotopique. On démontre un theorème du type Hochschild-Kostant-Rosenberg pour les complexes de Hochschild modelés sur les surfaces et on en déduit des formules explicites pour le produit surfacique des sphères de dimension impaires et des groupes de Lie. English detailed abstractAbstract. We develop a machinery of Chen iterated integrals for higher Hochschild complexes. Classically Chen iteratd integrals provides an explicit quasi-isomorphism between the Hochschild chains of the de Rham forms on a manifold and the de Rham forms of the free Loop space of the manifold. The higher Hochschild complexes 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 (and which are functorial in both simplicial sets and algebras). We use these to give algebraic models for general mapping spaces, in particular we obtain a model for the algebra of de Rham forms on the mapping space Map(X,M) for a manifold M and space X thanks to a generalization of Chen iterated integrals. In a second part of the paper we define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is a two dimensional analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove a Hochschild-Kostant-Rosenberg type theorems, that is an explicit computation of the surface (co)homology of smooth algebras, and use them to give explicit formulae for the surface product of odd spheres and Lie groups.
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Résumé: On introduit un cadre général pour la topologie des cordes des champs différentiels et topologiques , qui permet de traiter aussi bien les lacets libres que les lacets fantômes. English detailed abstractAbstract. We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. In particular, we give a good notion of a free loop stack, and of a mapping stack Map(Y;X), where Y is a compact space and X a topological stack, which is functorial both in X and Y and behaves well enough with respect to pushouts. We also construct a (kind of generalized)bivariant (in the sense of Fulton and MacPherson) theory for topological stacks: it gives us a flexible theory of Gysin maps which are automatically compatible with pullback, pushforward and products. We introduce a suitable notion of oriented stacks, generalizing oriented manifolds, which are stacks on which we can do string topology. We prove that the homology of the free loop stack of an oriented stack is a BV-algebra and a Frobenius algebra, and the homology of hidden loops is a Frobenius algebra. Here the stack of hidden loops is the inertia stack describing loops which vanish on the coarse space of the stack. Using the formalism of stacks and of our Gysin maps, we show that the homology of the free loop stack defiens a 2-dimensional homological conformal field theories (with closed positive boundaries) and we study the sphere product of oriented stacks as well. Using our general machinery, we construct an intersection pairing for (non necessarily compact) almost complex orbifolds which is in the same relation to the intersection pairing for manifolds as Chen-Ruan orbifold cup-product is to ordinary cup-product of manifolds. We show that the hidden loop product of almost complex orbifold is isomorphic to the orbifold intersection pairing twisted by a canonical class. We gave several examples, including a detailled study of the Frobenius structures of quotient stacks BG=[*/G] of a compact Lie group.
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Résumé: Ce mémoire est basé sur (et décrit) mes travaux concernant la topologie algébrique des espaces fonctionnels et des questions connexes de topologie algébrique des champs différentiables. Ils sont en particulier motivés par les théories des champs quantiques et la topologie des cordes (et ses avatars supérieurs). Il peut, en particulier, servir d'introduction aux travaux sur la topologie des cordes des champs et à ceux sur la (co)homologie de Hochschild supérieure. English abstractAbstract. This "mémoire" is based on and is explaining my work on algebraic topology of mapping spaces as well as some related questions concerning differentiable and topological stacks. This work was greatly motivated and influenced by quantum field theories, (higher) string topology and understanding their algebraic models. In particular, it can be used as an introduction to algebraic topology of stacks and higher Hochschild (co)homology.
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Résumé: Le nerf d'une famille d'ensemble est un complexe simplicial qui encode les intersections de ses sous-familles. En topologie et géométrie algorithmique, les nerfs d'un bon recouvrement d'un objet géométrique sont très utilisés car leur type d'homotopie est celui de la réunion du recouvrement (Théorème du nerf ou de Leray). Dans ce papier, on s'intéresse à des recouvrements qui ne sont pas tout à fait bons mais tels que les intersections forment des réunions disjointes de cellules homologiques. On utilise la notion de multinerf pour obtenir une variante du théorème du nerf et on l'applique pour obtenir des Théorèmes à la Helly unifiant notamment des résultats précédents de Amenta, Kalai and Meshulam, et Matousek. English abstractAbstract.
The nerve of a family of sets is a simplicial complex that records the intersection pattern of its subfamilies. Nerves are widely used in computational geometry and topology, because the nerve theorem guarantees that the nerve of a family of geometric objects has the same topology as the union of the objects, if they form a good cover. In this paper, we relax the good cover assumption to the case where each subfamily intersects in a disjoint union of possibly several homology cells, and we prove a generalization of the nerve theorem in this framework, using spectral sequences from algebraic topology. We then deduce a new topological Helly-type theorem that unifies previous results of Amenta, Kalai and Meshulam, and Matousek. This Helly-type theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory. Hide abstract
Résumé: Ce papier (pas encore totalement sous une forme définitive) est consacré aux applications de l'homologie de Hochschild supérieure (telle qu'étudiée dans nos papiers précédents) à la topologie des cordes supérieures et aux espaces de lacets itérés. En particulier on étend l'homologie de Hochschild supérieure aux E∞-algèbres et on étudie également la cohomologie de Hochschild supérieure. On donne alors une description de la structure de En-algèbre du centralisateur de tout morphisme de E∞-algèbre, vu comme morphisme de En-algèbre, en termes de cohomologie de Hochschild modelée sur les sphères, puis on l'applique pour donner une solution à la conjecture de Deligne supérieure. On étudie aussi un relevé E∞ de la dualité de Poincaré et des intégrales itérées de Chen. Ceci nous donne une équivalence entre les chaines (sur un anneau k de caractéristique arbitraire) sur l'espace fonctionnel des n-sphères dans un espace à dualité de Poincaré M (avec un décalage du degré correspondant à la dimension de M), et, par l'étude précédente, une structure En+1 sur les chaines qui induit le produit Sullivan-Voronov. Par ailleurs, les techniques étudiées pour les centralisateurs s'appliquent également aux constructions Bar itérées et donnent des modèles de la structure de En-coalgèbre et de E∞-algèbre des espaces de lacets itérés. Finalement, en utilisant les algèbres de factorisation, on donne aussi une construction du centralisateur de tout morphisme de En-algèbres et on l'applique à l'étude des constructions Bar itérées des En-algèbres. English detailed abstractAbstract. This is the first draft of a paper dedicated to applying the technique of Higher Hochschild chains and factorization algebras to the study of mapping spaces from spheres into manifolds and iterated loop spaces as well. In order to work over a ground ring of arbitrary characteristic (for instance the integers or a finite field), we develop the higher Hochschild cochains for E∞-algebras. Here are the main results of the paper. We obtain an En+1-algebra model on C_{*+m}(Map(S^n, M)), the shifted integral chains on the mapping space of the n-sphere into an m-dimensional orientable manifold M. Our main tool is factorization homology and higher Hochschild (co)chains and we discuss some other applications of these tools of independent interest. 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 show that these Hochschild cochains can be naturally identified with the deformation complex of the E∞-algebra of singular cochains on M thought of as an En-algebra. We then invoke (and prove) the higher Deligne conjecture to furnish the cotangent complex, and all that is naturally equivalent to it, with an En+1-algebra structure and further prove that this construction recovers the sphere product. In fact, our approach to 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. The latter also applies to iterated Bar construction for E∞-algebras together with their En-coalgebra and E∞-algebra structure. In particular, we give a higher Hochschild chain model of the natural En-algebra structure of the chains of the iterated loop space C_*(Ωn Y). Furthermore, for general En-algebras, we apply factorization algebras to give a construction of the centralizers of any map of En-algebras and to discuss several features of the iterated bar construction for En-algebras.
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Résumé: On étudie les complexes de Hochschild supérieurs et leurs relations avec les algèbres de factorisation et l'homologie chirale topologique. En particulier, on montre que les complexes de Hochschild supérieurs se relèvent en un foncteur d'∞-catégories CH: sSet∞ x CDGA∞ → CDGA∞. On donne une caractérisation axiomatique de CH. On en déduit que les complexes de Hochschild supérieurs sont équivalents aux algèbres de factorisation commutative constantes. On montre aussi que CH est équivalent à l'homologie chirale topologique (sur l'intersection de leurs domaines de définition). Enfin on montre que l'homologie chirale topologique et les algèbres de factorisation localement constantes sont des notions essentiellement équivalentes. On donne aussi une formule de Fubini pour l'homologie chirale topologique et les chaines de Hochschild.
English detailed abstract
Abstract. In this paper, we study the higher Hochschild functor and its relationship with factorization algebras and topological chiral homology, which are concept motivated by Topological Field Theories integrating algebraic structures and higher categories of manifolds. To this end, we emphasize that the higher Hochschild complex is an (∞,1)-functor from the category sSet x CDGA to the category CDGA (where sSet and CDGA are the (∞,1)-categories of simplicial sets and commutative differential graded algebras) and give an axiomatic characterization of this functor. The above (∞, 1)-functor is a (derived/homotopical) lift of the higher Hochschild chain complex defined by Pirashvili that we studied in the Chen Model for mapping spaces and higher Hochschild cohomology papers. The (∞,1)-framework (or said otherwise derived/homotopical framework) is a very convenient setting to state the axioms and in particular a locality axiom. From the axioms we deduce several properties and computational tools for the higher Hochschild functor. We then study the relationship of the higher Hochschild functor with factorization algebras (as studied by Costello and Gwilliam) by showing that in reasonable cases, the Hochschild functor determines a constant commutative factorization algebra. Conversely, we show that every constant commutative factorization algebra is naturally equivalent to the Hochschild chain factorization algebra. Similarly, we also study the relationship with topological chiral homology (TCH), an (derived) invariant for (stably-)framed manifolds with coefficient in En-algebras due to Lurie. In particular, we show that CH_M(A) = TCH(M, A), i.e. that the higher Hochschild functor is naturally equivalent to topological chiral homology, whenever both are defined. As a corollary, we also get a similar statement of the relationship with blob homology. Finally, we prove that topological chiral homology determines a locally constant factorization algebra and that any locally constant factorization algebra on a manifold essentially arises in this way. We also give a Fubini type formula for computing topological chiral homology.
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Résumé: On établit, pour tout groupe de Lie G, une bijection entre les [G → Aut(G)]-fibrés principaux et les G-gerbes au dessus d'un champ. On donne des classes caractéristiques universelles pour les fibrés principaux sur des 2-groupes de Lie et explique comment les calculer à partir de connexions. English abstractAbstract. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group [G → Aut(G)]-bundles over Lie groupoids and, on the other hand, G-extensions of Lie groupoids (i.e. between [G → Aut(G)]-bundles over differentiable stacks and G-gerbes over differentiable stacks). We also introduce universal characteristic classes for 2-group bundles. For groupoid central G-extensions, we prove that the universal characteristic classes coincide with the Dixmier-Douady classes that can be computed from connection-type data.
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Résumé: Le nombre de Helly d'une famille d'ensembles est le cardinal de la plus grande sous-famille minimale d'intersection vide. On donne des bornes des nombres de Helly pour des familles de sous-espaces ouverts d'un espace topologique X dont toutes les sous-familles G vérifient que les intersections (à partir d'une certaine taille) des éléments de G ont au plus r composantes connexes qui n'ont pas d'homologie (excepté en basse dimension). Notre résultat généralise des théorèmes de helly "topologiques" et, comme application, on obtient des bornes pour plusieurs familles de transversales géométriques. English detailed abstractAbstract.
The Helly number of a family of sets is the size of its largest inclusion-wise minimal subfamily with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arcwise connected topological space Γ. Assume that for every sub-family G in 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, for Γ=R^d, d_Γ=d). This bound is best possible. We prove, in fact, a stronger theorem where small subfamilies 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. We use the fact that the Helly number can be bounded by looking at the vanishing of homology groups of subcomplexes of the nerve. Our approach is based on replacing the usual nerve by the multi-nerve which is a simplicial poset (and no longer a simplicial complex as for the nerve), which encodes the various connected components of intersections. Using that the Helly number can be bounded Then we apply some homological machinery to relate the higher homology groups of the multinerve with those of the space given by the union of the elements in F (which is an open subset of Γ). We also explain how to adapt our result for finite families of compact subsets. Hide abstract