Introduction aux ensembles simpliciaux

• Sacha Ikonicoff

Room: Amphi Lavoisier

Introduction à l'algèbre homotopique

• Rigel Apolonio JUAREZ OJEDA

Room: Amphi Lavoisier

Autour du théorème B de Quillen (Topologie algébrique et applications)

• Ieke Moerdijk

Room: Amphi Lavoisier Pour un foncteur entre petites catégories, le théorème B de Quillen donne un critère pratique pour déterminer la fibre homotopique de l'application entre espaces classifiants de ces catégories. Dans cette série de trois conférences, je vais donner une démonstration d'une version plus générale, pour un foncteur entre catégories simpliciales. Cette version plus forte a comme applications immédiates plusieurs théorèmes importants, entre autres un théorème de descente pour des espaces simpliciaux, une construction d'univers univalents en théorie homotopique des types, et, last but not least, une démonstration efficace du "group completion theorem" et du théorème de périodicité de Bott. Si le temps nous le permet, j'expliquerai comment les méthodes s'appliquent dans le contexte de la théorie d'homotopie des préfaisceaux simpliciaux, utilisés par exemple en théorie d'homotopie $A^1$.

Espaces d'intersection, homotopie rationnelle et structures de Hodge mixte (Topologie algébrique et applications)

• Mathieu Klimczak (Université de Nantes)

Room: Amphi Lavoisier La théorie des espaces d'intersection permet de restaurer la dualité de Poincaré pour des espaces à singularités isolées, par exemple les variétés algébriques projectives complexes à singularités isolées. Etant donnée un tel espace à singularités isolées $X$, on peut lui associer une famille d'espaces topologiques $I^{\overline{p}}X$, ses espaces d'intersection, vérifiant une "dualité de Poincaré généralisée". Si $X$ est une variété algébrique projective complexe à singularités isolées, alors la cohomologie rationnelle de ses espaces d'intersection peut être munie d'une structure de Hodge mixte canonique, alors même que ces espaces ne sont pas des variétés algébriques projectives complexes à singularités isolées. Après avoir expliqué la construction des espaces d'intersection, on montrera via des techniques d'homotopie rationnelle comment définir ses structures de Hodge mixtes. On utilisera ces dernières pour obtenir des résultats de formalité.

A small projective resolution of complex K-theory (Topologie algébrique et applications)

• Greg Arone

Room: Amphi Lavoisier Around 1982 Nick Kuhn proved that the symmetric powers of the sphere spectrum give rise to a minimal projective resolution of $HZ$. He then asked if there were other interesting examples of small projective resolutions of spectra, in particular of spectra like $bo$ or $bu$. In this talk I will show how to construct a small projective resolution of the connective K-theory spectrum $bu$. Our resolution has many similarities to the classic one that arises from the symmetric powers filtration. We give a unified proof of exactness of both resolutions, that is different from Kuhn’s proof. A key ingredient in our proof is a vanishing result for the Bredon homology of the complex of partitions and the complex of direct-sum decompositions. Joint work with Kathryn Lesh.

Équation de Yang-Baxter, tableaux de Young, factorisations de groupes (Topologie algébrique et applications)

• Victoria Lebed (Trinity College Dublin)

Room: Amphi Lavoisier On commencera par des généralités sur la cohomologie des ensembles pré-cubiques et, plus particulièrement, des solutions idempotentes de l'équation de Yang-Baxter. Cette théorie générale nous permettra de fabriquer des complexes relativement petits calculant la cohomologie de Hochschild de certaines algèbres associatives. On s'intéressera à deux applications : 1) Pour les tableaux de Young munis du produit de Knuth, on simplifiera et précisera les calculs cohomologiques apparus au cours du récent renouveau de l'intérêt envers ces structures. 2) On établira une forme faible de la formule de Künneth pour les groupes factorisés.

A generalized Blakers-Massey Theorem (Topologie algébrique et applications)

• Georg Biedermann (LAGA, Paris 13)

Room: Amphi Lavoisier (joint with M. Anel, E. Finster, and A. Joyal) We present a generalized version of the Blakers-Massey Theorem in the context of $\infty$-topoi. The proof refines a proof of the classical theorem by Finster and Lumsdaine given in the language of Homotopy Type Theory and its "re-engineered" version by Rezk. The main tools are certain factorization systems (modality) and homotopical descent. The classical theorem and a recent generalization due to Chacholski-Scherer-Werndli are easy consequences. As an application we prove a conjecture by Goodwillie: a Blakers-Massey Theorem for the calculus of homotopy functors. From it we obtain an independent proof of the fact that homogeneous functors deloop.

Minimal models for operadic algebras over arbitrary rings (Topologie algébrique et applications)

• Fernando Muro

Room: Amphi Lavoisier The classical theory of minimal models for operadic algebras works when they have projective homology, e.g. if they are defined over a field. In the associative case, Sagave extended the theory to arbitrary algebras over any ring by means of a new kind of structure which merges A-infinity algebras and projective resolutions, called derived A-infinity algebras. We will endow the category of derived A-infinity algebras with a homotopical structure equivalent to that of differential graded algebras. We will show that, in this new homotopy category, any differential graded algebra is equivalent to its minimal model. Moreover, we will extend all this beyond the associative setting by using Koszul duality results from Maes's thesis, which extend work of Livernet-Roitzheim-Whitehouse.

Autour du théorème B de Quillen, II

• Ieke Moerdijk

Room: Amphi Figlarz Voir le résumé principal.

Le modèle de Lambrechts–Stanley des espaces de configuration (Topologie algébrique et applications)

• Najib Idrissi (Université Lille 1)

Room: Amphi Figlarz Nous prouvons la validité sur R d'un modèle en CDGA pour les espaces de configurations des variétés simplement connexes dont la caractéristique d'Euler est nulle, répondant ainsi à une conjecture de Lambrechts et Stanley. Cela entraîne que le type d'homotopie réel de ces espaces de configuration ne dépend que d'un modèle à dualité de Poincaré de la variété. En nous fondant sur la preuve de Kontsevich de la formalité des opérades des petits disques, nous prouvons également que le modèle est compatible avec l'action de l'opérade de Fulton--MacPherson quand la variété est parallélisée en utilisant un complexe de graphes étiquetés. Nous utilisons ce résultat plus précis pour obtenir un complexe calculant l'homologie de factorisation. Référence : http://arxiv.org/abs/1608.08054

The loop space of a p-local group (Topologie algébrique et applications)

• Ran Levi

Room: Amphi Figlarz The homotopy type of the loop space on the p-complete classifying space of a finite group was studied by myself and a few other researchers since the early 90s. The homology of such loop spaces is of particular interest from the homotopy theoretic point of view, as it exhibits a rather rigid behaviour, yet not very well understood. From the algebraic point of view, works of Benson-Greenlees-Iyengar suggest the loop space homology of p-completed classifying spaces provides interesting examples of much more general phenomena. In his 2009 paper "An algebraic model for chains on $\Omega BG^\wedge_p$", Benson showed that the homology can be defined purely algebraically through what he named a "squeezed resolution" for the group in question. The theory of p-local finite and compact groups allows one to study homotopy theoretic and algebraic properties of p-completed classifying spaces in a very general context, and where a genuine group is not necessarily available. Thus the question that arises naturally is whether one can construct an analog of a squeezed resolution for p-local groups. The answer to this question turns out to be positive in a more general sense. In an ongoing project with Broto and Oliver we show that for any small category $\mathcal{C}$ satisfying certain conditions, the homology of the loop space of its p-completed nerve can be understood algebraically by means of a squeezed resolution. In this talk I will present the construction of squeezed resolutions in this context and discuss some of their properties. I will also relate this to a number of interesting homotopy theoretic questions.

Intersections complètes (Topologie algébrique et applications)

• Jean Fasel (Université Grenoble-Alpes)

Room: Amphi Figlarz Soit X=Spec(R) une variété affine lisse sur un corps k, et soit Y une sous-variété fermée correspondant à un idéal I. Il est en général difficle de donner un ensemble de générateurs de I, même dans le cas où X est un espace affine. Néanmoins, le lemme de Nakayama montre que le nombre de générateurs de I est au moins égal au nombre de générateurs de son fibré conormal et au plus égal à ce nombre plus 1. Dans cet exposé, nous utiliserons des idées “topologiques” au sens large pour déterminer le nombre de générateurs de I, donnant au passage une réponse positive à une vieille conjecture de Murthy.

Sur les K-théories de Morava des 2-groupes abéliens élémentaires (Topologie algébrique et applications)

• Le Chi Quyet NGUYEN (LAREMA - Angers)

Pour chaque nombre premier p et chaque entier naturel n, il existe une théorie cohomologique complexe orientée K(n) que l'on appelle la n-ième K-théorie de Morava modulo p. Dans cet exposé, on étudie le cas p=2. On utilise des techniques d'Atiyah-Segal et la loi de groupe formel associée à K(n) pour obtenir une description du foncteur $V \mapsto K(n)^*(BV^{\sharp})$ où V est un espace vectoriel de dimension finie quelconque, et $\sharp$ désigne le dual linéaire. Pour n=2, on en déduit que ce foncteur est analytique. Il correspond alors à un module instable à gauche d'après le dictionnaire donné par Henn-Lannes-Schwartz. Le dual linéaire de ce module est détecté dans la structure d'anneau de Hopf de l'homologie de l'Omega-spectre qui représente la théorie de Morava.

Topological coHochschild homology (Topologie algébrique et applications)

• Kathryn Hess Bellwald (EPFL)

Room: Amphi Figlarz (Joint work with Brooke Shipley) Topological Hochschild homology (THH) is a version for ring spectra of classical Hochschild homology of rings, the importance of which is due primarily to its close connection to algebraic K-theory, mediated by the Dennis trace map. In this talk, I will introduce a dual version of THH for coalgebra spectra, called topological co-Hochschild homology (coTHH), which is a spectral version of coHochschild homology (coHH) of dg coalgebras, and explain its relationship to both THH and algebraic K-theory, also expressed in terms of a trace map. As a warm up, I will begin by reviewing coHH of dg coalgebras, with an extension to dg cocategories, and will then present new results in the dg case. To conclude, I will present recent computational results fo coTHH, due to Bohmann, Gerhardt, Høgenhaven, Shipley, and Ziegenhagen.

Autour du théorème B de Quillen, III

• Ieke Moerdijk

Room: Amphi Haüy Voir le résumé principal.

Eilenberg-MacLane mapping algebras and higher distributivity (Topologie algébrique et applications)

• Martin Frankland (Universität Osnabrück)

Room: Amphi Haüy Primary cohomology operations are given by homotopy classes of maps between Eilenberg-MacLane spectra. Composition of such maps is bilinear up to homotopy, but not strictly: it is strictly linear in one variable and linear up to coherent homotopy in the other variable. In joint work with Hans-Joachim Baues, we introduce the notion of weakly bilinear mapping theory to encode this structure. I will describe the higher distributivity laws satisfied by this structure, along with some examples in mod 2 cohomology.

Topological complexity of configuration spaces (Topologie algébrique et applications)

• Mark Grant

Room: Amphi Haüy Topological complexity is a numerical homotopy invariant whose definition (due to M. Farber) was inspired by the motion planning problem in Robotics. It has enjoyed much recent attention from homotopy theorists, partly due to its potential applicability, and partly due to its close resemblance to another more classical invariant, the Lusternik-Schnirelmann category. Classical configuration spaces (whose points are tuples of pairwise distinct points in some manifold, either ordered or unordered) are a natural class of spaces to consider, from either the homotopy theory or the Robotics point of view. We will survey some known results about their topological complexity, some of which were obtained in recent joint work with D. Recio-Mitter. The bounds we employ depend only on the fundamental group, allowing appealing geometric arguments using braids.

Bifibrations of model categories (Topologie algébrique et applications)

• Pierre Cagne (Univeristé Paris 7)

Room: Amphi Haüy In this talk, I will explain how to endow the total category $\mathcal E$ of a well- behaved Grothendieck bifibration $\mathcal E \to \mathcal B$ with a structure of a model category when both the basis $\mathcal B$ and all fibers $\mathcal E_b$ of the bifibration are model categories. The motivating example is the well-known Reedy model structure on a diagram category $[\mathcal R,\mathcal M]$. The crucial step in its construction by transfinite in-duction lies in the successor case, which is usually handled by reasoning on latching and matching functors. A first observation is that those functors define a Grothendieck bifibration on the restriction functor $[\mathcal R_{\lambda+1},\mathcal M] \to [\mathcal R_\lambda,\mathcal M]$ where $\mathcal R_\lambda$ denotes the full subcategory of $\mathcal R$ whose objects have degree less than $\lambda$. Unfortunately, this bifibration fails to fulfil the conditions of application of existing theorems in the litterature ([1], [2]), which would have allowed to lift the model structure from the base category $\mathcal B=[\mathcal R_\lambda,\mathcal M]$ to the total category $\mathcal E=[\mathcal R_{\lambda+1},\mathcal M]$. I will explain how to relax the hypotheses appearing in [1] and [2] by focusing on (co)cartesian lifts over acyclic (co)fibrations rather than over weak equivalences. This idea leads us to a simple and elegant condition for our new construction: some commutative squares in the base category are required to satisfy a homotopical version of the Beck-Chevalley condition. To conclude, I will apply the result to the Reedy construction and its generalizations ([3], [4]). --- [1] Stanculescu, A.E., Bifibrations and weak factorization systems, Applied Categorical Structures, 20(1):19-30, 2012 [2] Harpaz, Y, and Prasma, M., The Grothendieck construction for model categories, Advances in Mathematics, 218:1306-1363 (August 2015) [3] Berger, C., and Moerdijk, I., On an extension of the notion of Reedy category, Mathematische Zeitschrift, 269(3):977-1004, December 2011 [4] Shulman, M., Reedy categories and their generalizations, arXiv preprint, arXiv:1507.01065 (2015)

Local structure of finite groups and their p-completed classifying spaces (Topologie algébrique et applications)

• Bob Oliver

Room: Amphi Haüy I plan to describe the close connection between the homotopy theoretic properties of the $p$-completed classifying space of a finite group $G$ and the $p$-local group theoretic properties of $G$. One way in which this arises is in the following theorem originally conjectured by Martino and Priddy: for finite groups $G$ and $H$, $BG{}^\wedge_p\simeq BH{}^\wedge_p$ if and only if $G$ and $H$ have the same $p$-local structure (the same conjugacy relations among $p$-subgroups). Another involves a description, in terms of the $p$-local properties of $G$, of the group $\mathrm{Out}(BG{}^\wedge_p)$ of homotopy classes of self equivalences of $BG{}^\wedge_p$. After describing the general results, I'll give some examples and applications of both of these, especially in the case where $G$ and $H$ are simple Lie groups over finite fields.