Institut de Géométrie, Algèbre et Topologie

Séminaire de topologie 2007/08

le mardi ou le jeudi de 14h15 à 15h30

BCH 5112

## Programme

(Voir aussi le programme du groupe de travail en topologie, le programme du séminaire 2006/07, et le programme du séminaire 2005/06.)

Date Titre Orateur

mardi
02.10.07
à 14h15

K-théorie algébrique rationnelle de ku et 2-fibrés vectoriels Christian Ausoni
Bonn
25.10.07

Tore August Kro
Trondheim (NTNU)

01.11.07
à 16h

Craig Westerland

15.11.07 Produits sur les K-théories de Morava Samuel Wüthrich
EPFL
22.11.07 Catégorie des fonctions de transition pour un groupe G (I) Nicolas Michel
EPFL
29.11.07 On the 1-type of a Waldhausen K-Theory spectrum Andrew Tonks
London Metropolitan University
13.12.07 Catégorie des fonctions de transition pour un groupe G (II) Nicolas Michel
EPFL
29.12.08 Minimal Koszul models of topological space Alexander Berglund
Stockholm

mardi
05.02.08
à 14h15

Homotopie rationnelle et le Lemme de Hirsch (soutenance de PDM) Julian Kellerhals
EPFL
19.02.08 An algebraic model for the chains of a mapping space out of a closed surface I Méadhbh Boyle
Aberdeen
21.02.08 Le lien entre le mapping class groupe du disque et le groupe des tresses
infinies
Jan Brunner
EPFL
26.02.08 An algebraic model for the chains of a mapping space out of a closed surface II Méadhbh Boyle
Aberdeen
20.03.08 Tannaka duality for comodules over a Hopf algebra Steve Bennoun
EPFL
03.04.08 Higher Leibniz formulas Robert Bruner
Wayne State/Oslo
15.04.08 Introduction à l'homotopie rationnelle Yves Félix
Université Catholique de Louvain
22.04.08 Modules différentiels cofibrants: un outil en topologie algébrique Yves Félix
Université Catholique de Louvain
26-31.05.08 Young Topologists' Meeting 2008
05.06.08 Coloured posets and their homology with applications to knot theory

Paul Turner
Heriot-Watt/Fribourg

01.07.08 Bar constructions and Quillen homology of modules over operads John Harper
Notre Dame/EPFL
03.07.08 Introduction à la théorie des groupes p-locaux finis Fabien Junod
Aberdeen
08.07.08 Sur quelques auto-équivalences de groupes p-locaux compacts Fabien Junod
Aberdeen

Abstracts

Tore Kro: We review what elliptic cohomology is. Furthermore, we will mention the various attempts to define it geometrically. In the program initiated by Baas, the idea is to consider 2-vector bundles. We will look at their definition, and the related notion of principal 2-bundles, and give examples. Moreover, we outline the proof showing that the nerve of a topological 2-category classifies principal 2-bundles structured by this 2-category. As a corollary, we will see that the K-theory associated to Baez and Crans 2-vector bundles splits as two copies of ordinary K-theory.

Craig Westerland: We describe an action of Getzler's gravity operad on the S^1-equivariant homology of the free loop space LM of a closed manifold M, using string topology constructions. We will explain each of the terms of the previous sentence. Time permitting, we will explore a (very) conjectural extension of these ideas to topological cyclic cohomology.

Samuel Wüthrich: Les célèbres théorèmes de nilpotence et de périodicité distinguent les K-théories de Morava comme des objets d'un intérêt central pour l'étude de la catégorie d'homotopie stable. Malgré leur rôle prééminent, des problèmes fondamentaux concernant leur nature restent ouverts, tel qu'une bonne compréhension des structures multiplicatives sur les spectres représentants K(n) et K_n. Je vais présenter un résultat de classification pour des structures de produit homotopique, obtenu en collaboration avec Alain Jeanneret. Nous construisons une action libre et transitive d'un certain K_*-module libre de rang n sur l'ensemble des produits sur K, où K désigne K(n) ou K_n. Comme une conséquence assez surprenante, nous trouvons que l'algèbre des coopérations K_*(K) est une invariante complète pour les produits aux nombres premiers impairs.

Nicolas Michel: Je vais commencer par rappeler la classification des G-fibrés principaux par les fonctions de transition. J'expliquerai ensuite comment voir cette classification comme phénomène de descente. Si le temps le permet, je donnerai également une description des fonctions de transition en termes de foncteurs topologiques sur le groupoïde de Cech et en termes de cocycles d'un certain complexe.

Andrew Tonks: (joint work with F Muro) Waldhausen's K-theory of a category C with cofibrations and weak
equivalences extends the classical notions of the K-theory of rings, additive categories and exact categories. The group K_0(C) has a straightforward presentation in terms of generators and relations. Our aim in this talk is to present such an "easy" presentation of the group K_1(C). In fact we are able to give a new small (and functorial) algebraic model D(C) for the 1-type of the Waldhausen K-theory spectrum KC. This model consists of a very special kind of symmetric monoidal category called stable quadratic module. The model is defined in terms of generators and relations, and can be regarded as a presentation simultaneously of K_0, K_1, and the action of the stable Hopf map.

We will discuss the multiplicative properties of this model with respect to biexact functors, showing that for monoidal Waldhausen categories our model determines the 1-type of K-theory not only as a spectrum but as a ring spectrum.

Alexander Berglund: Sullivan's rational homotopy theory assigns to each simply connected space X a Sullivan model --- a differential graded algebra
over Q whose underlying algebra is the symmetric algebra on some graded vector space and whose differential raises polynomial degree. Apart from the important fact that isomorphism classes of Sullivan models serve as representatives of rational homotopy types,algebraic invariants such as the rational homotopy groups may be computed from the Sullivan model of the space X.

In this talk I will propose a definition of a `Koszul model' of a space. Koszul models are more general than Sullivan models, but can also be used to compute rational homotopy groups. This fact depends on an extension of Koszul duality between cocommutative coalgebras and Lie algebras to a duality between cocommutative dg-coalgebras and L_\infty-algebras.

One can not in general expect the Sullivan model to be finitely generated, not even if the space is, say, a finite CW-complex. However, it seems reasonable to ask whether one can find finitely generated Koszul models. Examples to illustrate this are provided by moment-angle complexes associated to simplicial complexes, studied in toric topology. These admit finite dimensional Koszul models, whereas their Sullivan models are not finitely generated, let alone finite dimensional, except in trivial cases.

Méadhbh Boyle: Let X be a 2-connected space, which is formal in an appropriate sense. Let S_g denote a closed surface of genus g. I will present a construction of a model for the chains on the pointed mapping space Map_*(S_g, X). In the first talk I will introduce the usual cobar construction as a model for the based loops on X, and the Hess-Levi model for the double loop space on X. In the second talk I will construct the model for Map_*(S_g, X) as a twisted tensor product of $2g$ copies of the cobar construction, and of the Hess-Levi model.

Jan Brunner: Pour chaque nombre naturel n, il existe un isomorphisme entre le mapping class groupe du disque avec n points enlevés, MH(n), et le groupe de
tresses B(n). Il s'avère que, dans le cas infini, ces deux groupes ne sont pas isomorphes. Toutefois, en introduisant une description algébrique adéquate de B(\infty), le groupe de tresses sur un nombre infini de brins, il est possible d'identifier MH(\infty) comme un sous-groupe de B(\infty).

Steve Bennoun: In this talk, I shall first recall the notions of algebra, coalgebra and bialgebra in a monoidal category. I shall then define the category of comodules over a bialgebra A and prove that it is autonomous when A is a Hopf algebra. Next, starting with this category, I shall use Tannaka duality to reconstruct the bialgebra A. This will naturally lead to a construction for freely adjoining an invertible antipode to a bialgebra.

Robert Bruner: The Leibniz formula tells us how differentials behave on products. When considering an S-algebra, there are higher order operations (Dyer-Lashof operations and their generalizations) and it is possible to work out formulas for differentials on these. They have been worked out in detail in two important cases, the Adams spectral sequence and the spectral sequence(s) for the homology of the homotopy fixed points, orbits or Tate construction of an S1-equivariant S-algebra. In both cases, they provide a great deal of information about the differentials and extensions in the spectral sequence.

Yves Félix: Premier exposé: Construction du modèle minimal en homotopie rationnelle, son utilisation et des applications.
Deuxième exposé: Définitions et propriétés de modules différentiels cofibrants; Les constructions bar et cobar, et fibrations; Application aux immersions et à la string topology; Application à la LS catégorie.

Paul Turner: Motivated by the definition of Khovanov homology I will present the notion of a coloured poset and describe a homology functor for these objects. I will explain how this is related to, and indeed generalises, Khovanov's construction. I will then outline a theory of bundles for coloured posets, producing in certain cases a Leray-Serre type spectral sequence. The latter will then be used to produce a new spectral sequence for computing the Khovanov homology of a knot.

John Harper: Even in the case of a simple algebraic structure such as commutative algebras, homology in the derived sense of Quillen provides interesting invariants; in Haynes Miller's proof of the Sullivan conjecture on maps from classifying spaces, derived homology of commutative algebras is a critical ingredient. This suggests that homology for the larger class of algebraic structures parametrized by an operad will also provide interesting and useful invariants. In recent work, for the two contexts of symmetric spectra and unbounded chain complexes, I have established a homotopy theory for studying Quillen homology of modules and algebras over operads, and have shown that homology can be calculated using simplicial bar constructions. A larger goal is to determine the extra structure that appears on the derived homology and the extent to which the original object can be recovered from its homology when this extra structure is taken into account. This talk is an introduction to these results with an emphasis on several of the motivating ideas.

Fabien Junod: Dans le premier exposé je vais présenter les notions de base de la théorie des groupes p-locaux finis. Cette théorie a été développée par C. Broto, R. Levi et B. Oliver dans le but de donner un contexte général dans lequel on peut étudier l'homotopie de la p-complétion des espaces classifiants de groupes finis.

Dernière mise à jour: 30.06.08